a body in which flexoelectrical phenomena are observed always has a finite size . in some casesthe finite sample size is not significant , and the sample can be described as an infinite medium .examples are the influence of the flexoelectric effect on the dispersion of acoustic phonons or its influence on structure of ferroelectric domain wall . in other cases , such as flexoelectricbending of a plate , finite size of a sample plays a fundamental role .obviously , the effects associated with the finite size of the sample caused by the presence of the sample boundary . in particular , this raises the surface piezoelectricity which , as it turns out , leads to the effects of the same order as the flexoelectric effect .less trivial fact is that even without surface piezoelectricity , in the presence of flexoelectricity classical theory of elasticity should be modified . in this caseit turns out that , generally speaking , one should use elastic boundary conditions of non - classical form .these boundary conditions are consistent only if one considers the spatial dispersion of the elastic moduli ( contribution to the thermodynamic potential which is a bilinear in elastic strain gradients ) .the latter leads to that not only the boundary conditions , but also differential equations of elastic equilibrium should be non - classical .equations of elastic equilibrium are needed to describe the converse flexoelectric effect ( mechanical response to the polarization ) . as regards the direct flexoelectric effect ( polarization response to an inhomogeneous elastic strain ) , to describe itthe equations of polarization equilibrium are needed .it is essential that this equations and boundary conditions for them are modified by flexoelectricity not so radically as the equations and boundary conditions describing the elastic response ( converse flexoelectric effect ) .this is related to the fact that the flexoelectric contributions to the thermodynamic potential contains only the first spatial derivatives of the polarization , while it contains the second spatial derivatives of the elastic displacements .equations of polarization equilibrium and boundary conditions for them in the presence of flexoelectricity were derived in paper .thus , non - trivial features of flexoelectricity in finite samples reveal themselves mainly in the converse flexoelectric effect .this is why we do not discuss much equations of polarization equilibrium , focusing mainly on the converse flexoelectric effect at a given polarization .moreover , in the particular examples ( but not in the general theory ) further we assume a homogeneous polarization at which non - trivial features of flexoelectric effect in finite samples reveal itself most dramatically .the case of homogeneous polarization is interesting in the sense that a naive analysis based on constitutive equations leads to the ( erroneous ) conclusion that the homogeneous polarization does not cause a flexoelectric deformation of a body .the consequence of this conclusion is the ability to create a sensor which not behave as an actuator . such an ability contradicts the general principles of thermodynamics .but in reality violations of the principles of thermodynamics is not happening , homogeneous polarization does lead to a flexoelectric deformation of the body .  in the paper the conclusion that homogeneous polarization leads to a flexoelectric deformation of a thin platewas made on the basis of the particular ansatz for elastic displacement distribution over the sample and direct minimization of the thermodynamic potential .a more rigorous analysis of this problem requires the solution of differential equations of elastic equilibrium with appropriate boundary conditions .such a solution for the case of homogeneously polarized ball was presented in the paper . as was expectedthe solution confirmed the presence of flexoelectric strains caused by homogeneous polarization .it is worth mentioning that in the presence of flexoelectricity the exact ( within the framework of a continuum media theory ) equations of elastic equilibrium are too complex to be used for a case differs from a simple case of a ball . even for a ballthe solution is very cumbersome and requires an introduction of non - standard functions presented by a series in powers of radial coordinate . so that the development of approximate method is worth .such a method was developed in the paper .papers on the subject are too brief .more detailed description is presented here .general aspects of derivation of the equations of elastic equilibrium and the boundary conditions for them in the case when the thermodynamic potential depends on higher derivatives discussed in another paper . herewe apply these results to the case of flexoelectricity and develop the theory further . besides, it is presented a comparison of exact solution and the solution of the same problem within the approximate theory .some additional examples of approximate theory application are presented also .it is well known that within the theory of continuum flexoelectricity is described by a contribution to thermodynamic potential density : here is polarization , is elastic displacement , and are material tensors , , and einstein summation rule is adopted . note that usually strain tensor used instead of displacement gradients ( distortion tensor ) .but since is convolved with material tensors which are symmetrical in corresponding indices , it does not matter to write down or .note also that here it is used definition of slightly different from that adopted in , pairs of indices and are swapped .this definition corresponds to and more widely accepted .certainly the opposite definition is also applicable .thermodynamic potential of a body is integral over body volume from thermodynamic potential density . using integration by partsits flexoelectric contribution can be represented in the form : where is so - called flexocoupling tensor , is surface piezoelectric tensor of special form : hereinafter is unit vector normal to body surface .note that is different in different points of the surface and it is symmetric in indices .thus flexoelectric part of thermodynamic potential density ( [ eq : hflx1 ] ) can be reduced to the lifshitz - type form together with additional surface piezoelectricity described by piezoelectric tensor ( [ eq : dijk ] ) .further effects of surface piezoelectricity are not described so that lifshitz - type form ( [ eq : hflxlt ] ) is used .certainly besides flexoelectric potential ( [ eq : hflxlt ] ) one should take in account elastic energy .it will be clear from what follows that besides conventional elastic energy higher elasticity term should be added .so that total volume density of the thermodynamic potential is .\ ] ] possible surface contribution to the thermodynamic potential is not considered here .so that volume integral from ( [ eq : hb ] ) is total thermodynamic potential of a body .starting from the equation ( [ eq : hb ] ) it is possible to write down the equations of elastic equilibrium and the boundary conditions for them .in fact , one just need to calculate the tensors  and use the results of . in this way it turns out that `` physical stress '' which obey the equations of equilibrium are determined by equation : conditions for ( [ eq : equilib ] ) can be written in different equivalent forms , one should only substitute ( [ eq : tij ] ) and ( [ eq : thetaijk ] ) to equations from .particularly they can be written in the form which coincides with that presented in :  naturally terms contained and do not appear because surface contributions to thermodynamic potential is not described .it is obvious from ( [ eq : thetnn ] ) and ( [ eq : thetaijk ] ) that if and then generally . so that the limit is singularthis fact explains the above statement that generally in the presence of flexoectricity one should take into account the higher elasticity ( spatial dispersion of elastic moduli ) .higher elasticity term can omitted only in very special ( and physically doubtful ) case when on the surface of a body . in this special caseone can use classical boundary conditions , additional terms in them disappear .  regarding the description of the direct flexoelectric effect , to obtain the corresponding differential equations and boundary conditions , one should add to ( [ eq : hflxlt ] ) the following expression : where is electric field .if surface contributions are not considered then it remains only to vary such thermodynamic potential in and to obtain by standard way the following differential equations and boundary conditions for them it does not constitute a serious problem to take into account the surface contributions here . usually such contributions contain only the polarization but not polarization gradients .this is why to find the change of the equation written above is a simple exercise in this case , in which we are not resting here .when solving a particular problem it is convenient to use covariant formalism ( see ) . to transform equations from cartesian form to covariant form one should only change partial derivatives to covariant derivatives and provide an index balance ( only superscript can be convolved with subscript ) . so that equations of mechanical equilibrium and boundary conditions for them take the form :  here is covariant representation of unit vector normal to body surface , tensors and are determined as follows :  it is also convenient to use curvilinear coordinate system of special class in which equation of body surface has the form . in this coordinate system tensor can be presented in terms of cristoffel symbols : besides has the only component :   for a case of a ball it is naturally to use spherical coordinate system : here , and are cartesian coordinates , , , and are curvilinear ones .equation of the ball surface has the form , so that these curvilinear coordinates belong to the special class mentioned above .  instead of curvilinear indices , , and it is used also the indices , , and respectively .so the letters , and are excluded from notation of `` running '' indexes .particular cartesian indices are further denoted as , and .metric tensor components , its determinant , and cristoffel symbols can be derived directly from ( [ eq : xyz ] ) : other components are zero . since tensor is diagonal , inverse tensor is diagonal also and corresponding components are simply equal to the inverse diagonal components of .unit vector normal to the surface has the only component .further one should specify the form of the material tensors for isotropic media .it is well - known that elastic tensor of such a media is defined by two parameters , say its components and ( voigt notation is used ) , and can be expressed as follows : it is possible to show that flexocoupling tensor of isotropic media can be expressed analogously : for six - rank tensor of higher elastic moduli of isotropic media further it is used a simplified expression where and are constant parameters .  above the expressions of material tensors are presented in cartesian components .to convert them to curvilinear components one should only change kronecker delta to metric tensor with corresponding location ( up or bottom ) of indices .if the ball is polarized along -axis with polarization then polarization vector has two covariant components :   the equations above mathematically completely determine the problem under study . to solve this problem one must first separate the variables .generally this is done by decomposition of the vector field in a series of spherical vectors . notethat since the covariant formalism is used here , spherical vectors do not coincide with their usual form .however , the required form of these vectors can easily be obtained from conventional .covariant spherical vectors can be defined as follows where is scalar spherical functions expressed in the usual manner in terms of associated legendre polynomials .  in the caseconsidered here , i.e. when the polarization is homogeneous , expansion in spherical vectors is simplified radically .indeed , it is easy to see that in such expansion of the polarization field there are only the terms with , and top indices 1,3 .since the theory is linear , it follows that the expansion of the field contains only the similar terms . as a result, this leads to the fact that , , .this is why one can use a substitution :  factor added into ( [ eq : utheta ] ) to make radial function analytical at .the consequence of using the covariant formalism is that is not a physical displacement in the meridional direction , such physical displacement is , while .certainly the physical component of the displacement should not have a singularity at the ball center .note that covariant form of contains the same factor .thus , the problem is reduced to finding the radial functions and . to find the corresponding differential equations and boundary conditions one should substitute ( [ eq : upsi])([eq : ur ] ) in the above equations and make a very cumbersome but absolutely straightforward algebraic transformations . as the resultit yields the system of two differential equations  and four boundary condition for it   - \\ \\\displaystyle v_1(9f'''_1 + 18 f''_1 - 46  f'_1 + 36 f_1  -  14  f''_2 + 32  f'_2 - 36 f_2 ) - \\ \\\displaystyle \left.2 v_2(3f'''_1 + 6 f''_1 - 22  f'_1 + 28 f_1 -  2  f''_2  + 16  f'_2 - 28f_2)\right|_{\xi=1 } = r^2 f_{12}p  \ , , \end{array}\ ] ]  here is dimensionless radial coordinate , prime denotes the derivative in , , .  in general , the system of two fourth - order equations ( [ eq : feq1 ] ) ( [ eq : feq2 ] ) requires eight boundary conditions , but here we have only four boundary conditions ( [ eq : fbc1 ] ) ( [ eq : fbc4 ] ) .however , as is usually in the case of such problems , the missing boundary conditions are replaced by the conditions that a solution should be analytic at zero . therefore , the solution should be expressed in the form of series in non - negative powers of .practically it is more convenient to find the complete basis set of solutions expressed as such series , and to present a general solution as a linear combination of such basis functions .if the number of basis functions is equal to the number of boundary conditions then the coefficients in the linear combination can be easily founded by solving a system of linear algebraic equations .in accordance with the above , we express a solution in the form :  the constant coefficients are found from the condition that obeys the system of differential equations ( [ eq : feq1 ] ) - ( [ eq : feq2 ] ) .this condition leads to the following linear algebraic equations for the coefficients ( index which is numbering the solutions is omitted ) : {1\,n }  \\\displaystyle + [ v_3(8n^2 - 8n-16 )  -v_4(2n^3 - 8n^2 + 2n+12 ) ] a_{2\ , n }  = \\ \displaystyle [ c_{44}(2n^2 - 6n-2 ) + c_{12}(n^2 - 3n ) ] a_{1 \ , n-2 }  - \\\displaystyle  [ c_{44}(2n-10)+c_{12}(2n-6)]a_{2 \ , n-2 } \ , , \\\\ \displaystyle [ v_3(4n^2 - 4n-8)+ v_4(n^3+n^2 - 4n  -  4)]a_{1\,n } + \\ \displaystyle [ v_3(n^4 - 2n^3 - 5n^2 + 6n+8 ) - v_4(2n^2 - 2n-4)]a_{2\,n } = \\ \displaystyle [ c_{44}(n+2 )  + c_{12}n]a_{1\,n-2}+[c_{44}(n^2 - 3n-2 ) - c_{12}2]a_{2\,n-2 } \ , .\end{array } \right.\ ] ] from ( [ eq : aeq ] ) one can see that the system of linear algebraic equations , even though it is infinite , has the characteristic structure : the coefficients with greater are expressed in terms of the coefficients with smaller .therefore , one can find for any finite step by step , one needs only to analyse the case of small and define some of the for these small .naturally for small system of equations ( [ eq : aeq ] ) is degenerate , some with small can be defined arbitrarily .this arbitrariness leads to that it may be several functions and their normalization may be arbitrary .detailed analysis shows that the system of equations ( [ eq : aeq ] ) is degenerate for only .it turns out five arbitrary constants , and so it is the same number of basis functions .however , one of this functions is physically meaningless and should be omitted ( more on this see below ) .  from an abstract point of view, arbitrary constants can be defined as anything , it is just needed that obtained basis functions be linearly independent .however , there is an important question : is it possible to choose these constants in such manner that the series representing at least some of the basic functions cut short , would constitute a finite sums ?it turns out that this choice of constants is really possible .  from the system of equations ( [ eq : aeq ] ) it is clear that in order to series cut short , it should be .for this to be possible , in any case , right - hand sides of ( [ eq : aeq ] ) should be zero for .together with ( [ eq : aeq ] ) for these conditions lead to a system of linear algebraic equations which is quite amenable to analysis .the result is that there may be two basic functions which are finite sums .one of them is determined by the conditions , .this function corresponds to the displacement of the ball as a whole .such a displacement is physically meaningless and this basis function should be omitted .certainly such a displacement does not change the thermodynamic potential , so that this function indeed is a solution of the equations derived from a stationary condition of this potential .but it is meaningless solution .another basis function , which is finite sum , is determined by condition , other coefficients are zero .so that first basis function is determined as follows : other are zero .it should be stressed that the function obeys not only the equations ( [ eq : feq1 ] ) ( [ eq : feq2 ] ) but also the equations of the classical theory of elasticity , which can be obtained from ( [ eq : feq1 ] ) ( [ eq : feq2 ] ) by setting .it can be verified by direct substitution .further analysis shows that there are three additional basic functions which are expressed by infinite series and can be chosen , for example , as follows .  coefficients with are calculated by means of ( [ eq : aeq ] ) .  coefficients with and are calculated by means of ( [ eq : aeq ] ) .  coefficients with are calculated by means of ( [ eq : aeq ] ) .graphs of the basis functions defined above for some set of parameters are shown in fig.[fig : b1 - 4 m ] .turn our attention that in contrast with the functions  are concentrated near the surface and decay rapidly in the ball volume .  , , , , , , ., width=491 ]  , , , , .solid line is for , , dashed line is for , ., width=377 ]  above there is explicit representation of four basic functions  .it only remains to substitute ( [ eq : series ] ) in the boundary conditions ( [ eq : fbc1 ] ) ( [ eq : fbc4 ] ) , and from the resulting system of four linear algebraic equations to find numerically .thus , the problem is solved .result of numerical calculations for some parameters is presented in the fig.[fig : exaball ] .note that the graphs shows that , except for a thin layer near , the curves are parabolic , the more so the less and are . in the bulk of the ballthe only function remains with a good approximation .thus , for a small , except for a thin surface layer , solution approximately obeys the equations of classical elasticity theory ( see discussion of function properties above ) .this observation is used in the next section to construct an approximate method of flexoelectric deformations calculations .this method is needed to describe more complicated geometry then spherical geometry of a ball .exact ( within the framework of a continuum media theory ) equations of elastic equilibrium used in previous section are partial differential equations of the fourth order .they are too complex to be used in the cases interesting for applications . even for a homogeneously polarized ball of isotropic media solution of themis cumbersome and requires a functions expressed by a series in powers of radial coordinate . so that to find flexoelectric deformation in more complicated cases one needs an approximate method .such a method was proposed in and it is described in this section .  when solving the problem for a ball ( see previous section ) we have seen that the solution is a sum of two parts .the first part corresponded basis function obeys the equations of the classical theory of elasticity , the second ( contribution of remaining basis functions ) is a non - classical part which , for small , is concentrated near the surface and rapidly decays inside the body .hereafter the first part is called a volume or a classical part , and the second one a non - classical or a surface part .it is naturally to assume that this property should be preserved in more general cases .the aim is to use this observation for construction of approximation that is useful for solution of more general problems than the case of a ball .  by means of ( [ eq : sigma - def - covar ] ) equations ( [ eq : el - eq - covar ] ) can be represent in following form boundary conditions for this differential equations here are used in the form different from used in previous section .particularly they are  here the indices enclosed in parentheses runs only the values 1 and 2 , if there are multiple indexes in one pair of parentheses then these indices are not equal to 3 simultaneously .equivalence of this form of boundary conditions to ( [ elbond3 ] ) is proved in .certainly it is assumed coordinate system where equation of the body surface has a form , is some constant .  according to the observation mentioned above the solution of the equation ( [ eq : eqvil0 ] ) is presented in the form , where obey the classical equations : from ( [ eq : tildeeq ] ) and ( [ eq : eqvil0 ] ) it follows that obey differential equations : substitution and some transformations also yield that the boundary conditions ( [ eq : bond1 - 0 ] ) and ( [ eq : bond2 - 0 ] ) take the form :   equations above are exact . in order to turn into approximationone should note that real values of are small .it is convenient to assume that all of them are proportional to a scalar .it is assumed also that all is proportional to a scalar . if then the right - hand side of ( [ eq : hateq ] ) and the second term on the right hand side of ( [ eq : bond1 m ] ) can be neglected .indeed , since obey the classical equation ( [ eq : tildeeq ] ) , in the case of sufficiently smooth surface and in the absence of polarization gradients tend to infinity in the limit , the classical part can not have large gradients which can compensate smallness of . as for the non - classical part , the situation is different : here such compensation is possible , but only if the derivatives are in .moreover , in a thin layer near the surface covariant derivatives in can be replaced by the usual derivatives , -terms give only small corrections here .thus , ( [ eq : hateq ] ) and ( [ eq : bond1 m ] ) can be approximately replaced by  naturally decomposition is not unique . by requiring that obey ( [ eq : tildeeq ] ) , this non - uniqueness is restricted , but is not eliminated completely .it is clear that does not contain a non - classical part , but it does not mean that does not contain a classical part . to eliminate remaining non - uniquenessone should require that exponentially decay inside the body . under such condition obey not only ( [ eq : hateqm ] ) but also equations ( [ eq : hateqmint ] ) are a system of three ordinal differential equations , the dependence on and is parametric here . moreover , in a thin layer near the body surface one can assume that the coefficients do not depend on , they are approximately equal to the surface values .solution of such a system is easy . in a standard way one should find a fundamental basis set of solutions in the form .obviously equations for amplitudes are equation ( [ eq : eigen ] ) is a standard generalized eigenvalue problem for symmetric positive definite matrices .therefore and are calculable by standard way .if the body bulk corresponds to for definiteness , then equals the positive square root of the -th eigenvalue , and general representation of non - classical part is where only the coefficients are unknown .the latter can be found easily by ( [ eq : bond1m2 ] ) which yields a simple system of linear algebraic equations : note that the smallness of in ( [ eq : anfind ] ) is compensated by , and using ( [ eq : eigen ] ) this system of equations can be rewritten as follows : thus , the non - classical part of elastic displacements is found completely in explicit form .  as it follows from above , to find completely the non - classical part of elastic displacements one needs only the boundary conditions ( [ eq : bond1 m ] ) .the remaining boundary conditions ( [ eq : bond2 m ] ) yield the boundary conditions for the classical equations ( [ eq : tildeeq ] ) in this way , one should only substitute known to ( [ eq : bond2 m ] ) and select the terms of the appropriate order of accuracy .however , within this derivation of the boundary conditions for the equations ( [ eq : tildeeq ] ) a singularity arises requiring additional analysis .this singularity is related to the fact that it appears a subtraction of two close but large terms in which it is impossible to use the approximate expression for derived above . in these ( and only these ) termsit is necessary to consider a corrections to such .the appropriate analysis is made below and while means the exact non - classical part of elastic displacements rather then approximate one defined by ( [ eq : hatuapprox ] ) .  according to the above , all the terms in ( [ eq : bond2 m ] ) , where gradients of are convolved with ,should be omitted for the same reasons as in the case of the equation ( [ eq : hateq ] ) .further it is convenient to introduce the notation : with this notation the boundary conditions to the classical equations can be written as follows : now all the terms to be further simplified are contained in . carrying out simplification , first of all one should leave in the last term only the partial ( not covariant ) derivatives in .the reason for this is the same as above .next , one should express the remaining covariant derivatives in terms of partial derivatives and -terms .the exact expression for the third covariant derivative of the vector is very cumbersome . butkeeping in mind that these third covariant derivatives are convolved with , one can omit all the terms where initial vector is not differentiated at least two times .so it turns out : with this equation and simplifications mentioned above we get : again , using the fact that to compensate smallness of one needs at least two derivatives in , this equation is further simplified : it is convenient to introduce yet another notation : using the symmetry properties of it can be rewritten as follows : in this notation the expression for can be written as follows : now one can proceed to the analysis of the singularity mentioned above .it appears in the first and third term of ( [ eq : sdefm4 ] ) , when all the derivatives are in . for all other termsthere is no singularity and one can use the expression ( [ eq : hatuapprox ] ) for .it is also clear that singular terms exactly cancel each other out because if approximate expression ( [ eq : hatuapprox ] ) is used .this is due to the explicit form of this expression. one might think that these terms should simply be excluded , but in reality the situation is more complicated .the fact is that is equal to zero only for the main approximation for .but the corrections to this approximation may result in the fact that this expression becomes finite .henceforth we denote such corrections as keeping notation only for the main approximation . in this notationit turns out : to compensate the smallness of in the fifth term of this equation there should be at least two differentiations in .thus , after some reductions one can rewrite ( [ eq : sdefm5 ] ) as follows : it is clear that in ( [ eq : sdefm8 ] ) one should use only corrections of order .only such corrections yield a contribution of the same order as the other terms . to find the corresponding contributions , we should keep in the exact equation only the terms of the order . indeed if then the main terms and have just such an order . note that although there are terms and in the equation , which are of the order , now they cancel each other completely due to the fact that now denotes the main approximation .right hand side of ( [ eq : hatuweq ] ) is , so it should be omitted .the terms and should be omitted by the same reasons .thus ( [ eq : hatuweq ] ) is reduced to equation ( [ eq : hatuwsimpl ] ) requires further simplification : in the right - hand side we should keep only the terms . for this in the first term there should be one partial derivative in , and in the second term there should be three derivatives in .this is why we simplify second and fourth covariant derivatives as follows :     eventually the simplified equation ( [ eq : hatuwsimpl ] ) takes the form :   taking into account that near the body surface the christoffel symbols and material tensors can be considered as constants , the equation ( [ eq : endweq ] ) has a very specific form : all of its terms are differentiated with respect to at least once .therefore , there exists a particular solution of this equation which obeys an equation with one less derivation :   note that in the left - hand side of ( [ eq : endweqred ] ) there is exactly the expression that is needed in ( [ eq : sdefm8 ] ) .the general solution of the inhomogeneous differential equation is the sum of a particular solution of the inhomogeneous equation and the general solution of the homogeneous equation .but the remarkable fact is that any solution of the homogeneous equation decaying away from the surface when substituted into yields zero .so it is quite enough to consider only a particular solution , and one can simply replace in ( [ eq : sdefm8 ] ) by right - hand side of ( [ eq : endweqred ] ) .having made the replacement , we obtain in the form : elementary transformations lead ( [ eq : snotready ] ) to a rather simple form : where is redefined as this equations together with ( [ eq : bond3m2 ] ) completely solves the problem considered in this section .remember that in used coordinate system unit vector normal to the body surface has the only component .so that with respect to the classical part of strain , actually defines the external force density on the body surface , one should only multiply it by .this is why differential equations ( [ eq : tildeeq ] ) appended by ( [ eq : bond3m2 ] ) correspond to a standard problem of classical theory of elasticity for a body under external forces on its surface .methods of solution of such problems are well - known and do not require special consideration .it should be emphasized that physically there are no external forces on the surface .forces mentioned above are purely formal in nature and describe the interaction of non - classical and classical parts of deformation .it worth to find flexoelectric deformations of homogeneous polarized ball by approximate method described in previous section and compare this deformations with ones known from exact solution .this is a test of approximate method at least .  for isotropic material the matrices and are diagonal .so that in order to find one does not even need to solve the generalized eigenvalue problem ( [ eq : eigen ] ) , the result is obtained at once : where  for comparison with the exact solution it is useful to present in terms of the functions and :  next step is to find on the ball surface . to do this, one should just use the equations of the previous section .it turns out \ , , \ ] ] \ , .\ ] ] certainly .this is obviously from symmetry but can be also obtained by direct calculations .  to solve the differential equations ( [ eq : tildeeq ] )actually is not necessary here .it is clear that in the terms of functions and the solution is proportional to the previously defined function .eventually the validity of this statement can be verified by direct substitution .so that one just needs to find the coefficient of proportionality , it is easily done by using ( [ eq : tildes33ball ] ) or ( [ eq : tildes23ball ] ) .one can use any of these conditions , they both give the same result :  it is also useful to derive the expression for the classical part of the elastic displacement in cartesian components as a function of the cartesian coordinates .direct conversions yield where  we can find and compare it with the results of exact calculations .this comparison is shown in fig.[fig : ballcompare ] .this figure shows a good agreement between the approximate and the exact solution , the smaller higher elastic moduli match those better .it should also be emphasized that the latest version , when the difference is badly distinguishable , corresponds to most physically reasonable values of higher elastic moduli .thus , the approximate method works very well at least for a homogeneously polarized ball with the given parameters .  , , , , . for , ;  for , ;  for , .solid line is approximate solution , dashed line is exact solution . , width=377 ]in this section we apply the above - described method to find the bending of a homogeneously polarized circular rod of an isotropic material .such a rod may be polarized in different directions . however , due to the fact that the flexoelectric effect is linear , we can restrict ourselves to the case of longitudinal and transverse polarization only .the response of the rod to the polarization of an arbitrary direction obviously is a superposition of responses to the longitudinal and transverse polarization .besides , axial symmetry allows us not distinguish different directions of the transverse polarization .as stated above , the classical part of strain is determined by the formal forces on the body surface .the surface of the rod consists of a cylindrical surface and end surfaces .first we consider the cylindrical surface of the rod . hereit is natural to use a cylindrical coordinate system .let cylinder be located along the longitudinal axis corresponded to coordinate .this coordinate is also denoted as .the other two curvilinear coordinates are the angular and radial .angle is measured in respect to the direction of the cartesian -axis .equation of the surface is where is rod radius . so that this coordinate system belongs to the class that is needed in accordance with the method of calculations .  for such coordinates simple geometrical reasoning lead to the following equations defining the relationship between cartesian and curvilinear coordinates : by means of simple differentiation it is easy to find the metric tensor , its determinants and the christoffel symbols : the remaining components are zero . since the metric tensor is diagonal to find the inverse metric tensor is easy : one just need to take the reciprocals of the diagonal components .note also that the vector normal to surface has the only component .  by making perfectly straightforward calculations similar to the case of a ball described above, the following expression can be obtained for the non - classical part of the elastic displacement near the cylindrical surface .if the polarization is longitudinal along the -axis then the expression is as follows : for transverse polarization along -axis is determined by the following equations : here is dimensionless radial coordinate ,  further , direct application of the equations of section [ sec : aprox ] gives the following boundary conditions for the classical part of the displacement . for transverse polarizationthey are \ , , \\ \\\displaystyle \left.\tilde{\sigma}^{33}n_3\right|_s = \frac{p\cos\phi}{r}\left [ \frac{c_{44}(f_{12}+2f_{44})}{c_{12}+2c_{44 } } - f_{44 } \right ] \ , .\end{array } \right.\ ] ] if polarization is longitudinal then all are zero .boundary conditions ( [ eq : rodbc1 ] ) can be also expressed in terms of the cartesian tensor component .elementary transformations yield : in this form of boundary conditions it is immediately clear that the total force acting on the cylindrical surface is equal to zero , the integration over the angle turns these expressions to zero .it also can be easily calculated that the total bending moment of these forces vanishes while integrating over the angle .  thus , not only the sum of formal surface forces , but also the sum of the bending moments of these forces , acting on a small part of the cylindrical surface length , are equal to zero .this means that the bending moment does not change along the homogeneously polarized rod .we emphasize that even so , the bending moment appears , but entirely by the boundary effects at the ends of the rod .so that in the case of homogeneously polarized rod the problem is reduced to the standard problem of the classical theory of elasticity , i.e. to the determination of rod bending under the bending moments applied to its ends .certainly the surface forces ( [ eq : rodbc1 ] ) slightly deform cross - section of the rod .since the main effect is the bending of the rod , we do not discuss this deformation .note only that qualitatively it is similar to the deformation of the ball in the meridian cross - section .  to calculate the bending moment appeared at the ends of the rod ,we need to specify the shape of these ends .we emphasize that we can not limit the rod by planes because this gives sharp edges while the theory requires smooth surface .we should smooth out these sharp edges , say by a quarter of toroidal surface , or assume that the rod terminates by halves of the ball .calculations were made for both these variants , and it was obtained the same result . omitting the details of these rather simple calculations , we note only that in the case of spherical ends one can use the results of the previous section . for edges slightly smoothed by the toroidal surface one can approximately describe a small part of this toroidal surface as cylindrical one and use the equations given above in this section . eventually it turns out that the bending moment is the other components are equal to zero .it is implied here that the rod is transversely polarized along -axis and the equation is written for the end in the positive direction of -axis . for the other end of the rod has the opposite sign .as is already clear from symmetry , for the longitudinal polarized rod there is no bending moment at all , this can be also proved by direct calculations .such calculations also show that the total force acting on each end of the rod is zero as for longitudinal and for transverse polarization .equation ( [ eq : myrod ] ) actually solves the problem of flexoelectric bending of the homogeneously polarized rod .one should only substitute this bending moment into standard equations from textbooks ( see for instance ) .in this section we apply the above - described method of approximate calculation to find the flexoelectric bending of a thin circular plate with a radius and thickness , uniformly polarized normal to its plane .material of the plate is assumed isotropic . for definiteness, we assume that the average surface of the plate lies in the coordinate plane .calculations for the plate are generally similar to those made in the previous section for the rod .however , most part of the plate surface is flat , so that one can immediately conclude , without any calculations , that the formal forces on this part of the surface are zero .as in the case of the rod , bending of the plate is determined by edge effects which are discussed below .thus , for most part of the plate , except for the edges , the classical part of the plate deformation is determined by standard equations without surface loading .it is well known that under such conditions the displacements of the average surface of the plate obey two - dimensional differential equation where is two - dimensional laplace operator .the components of the strain tensor can be expressed in terms of as follows : using polar coordinates it is easy to find an axially symmetric , regular at origin , solution of the equation ( [ eq : bilaplace ] ) : this solution contains a single integration constant which is nothing but the curvature of the plate .  to find integration constant needs first calculate the components of the strain tensor by means of ( [ eq : uab ] ) and second calculate the components of the strain tensor by means of standard equations .thereafter it becomes obvious that there is the linear density of the bending moment on the boundary of the plate .for instance , at the point of intersection of the plate boundary with the coordinate axis this bending moment density is due to axial symmetry is the same at all other points of boundary .  on the other hand be calculated in terms of formal forces arising due to flexoelectricity on curved surfaces on the plate boundary . as in the case of the rod ,it is necessary to smooth out sharp edges using one - quarter of toroidal surface ( on each sharp edge ) or one - half of the toroidal surface for the whole boundary .a small part of a toroidal surface can be considered as a cylindrical one , so that we can apply the equations ( [ eq : foncylcart ] ) ( coordinate system should be rotated ) . for bothvariants of sharp edges smoothing it turns out it only remains to equate the two expressions for and express the curvature of the plate in terms of polarization :   it is interesting to compare the equation ( [ eq : flexg ] ) with the result obtained in by direct minimization of the plate energy .elementary transformations of the equations from yield the equation different from ( [ eq : flexg ] ) only in that there is 12 instead 6 in the numerator .however , it should be kept in mind that in it is considered the case when in a thin layer near the plate surface polarization falls to zero .if , in accordance with calculations presented here , we modify the calculations to the case , when the polarization is strictly homogeneous , then it appears exactly the equation ( [ eq : flexg ] ) .a similar dependence on the boundary conditions imposed on the polarization also holds for direct flexoelectric effect , it is discussed in detail in another paper .above we have considered the continuum theory of the flexoelectric effect in the finite - size bodies . in the part of description of the converse flexoelectric effect , this theory turns out to be quite complicated .this is due to the fact that the independent variables are the elastic displacement .flexoelectric part of the thermodynamic potential depends on the second spatial derivatives of the elastic displacement .moreover in the case of flexoelectricity the elastic energy should be considered taking into account its dependence on the second spatial derivatives of elastic displacement ( elastic spatial dispersion ) , or the theory is self - contradictory in the general case .all of this leads to several theoretical problems .the first theoretical problem is that derivation of the boundary conditions for the differential equations of elastic equilibrium becomes non - trivial in such a case . while the volume integration by parts , which is necessary to express the variation of the thermodynamic potential in terms of independent variates , its appear the surface integrals containing the gradients of these independent variates . needed transformation of these surface integralsrequire special mathematical tools .for this purpose it was offered to use additional surface curvilinear coordinate system or to solve the problem in curvilinear coordinates of the general form from the beginning . from a practical point of viewthe second approach is more convenient although it has been proved that this two approaches are mathematically equivalent .it should also be noted that its appear that the curvature of the boundary surface plays an important role in the boundary conditions obtained in both approaches .  solving the problem of elastic boundary conditions derivation, we get the opportunity to solve , in principle , any boundary problems needed to describe flexoelectricity in finite - size bodies .however , the corresponding boundary problems are extremely complex and difficult to solve .such a boundary problem has been solved for a homogeneously polarized ball but for more complex geometries to make a similar is unrealistic . even for the ballthe solution is extremely cumbersome .therefore the development of approximate methods for solving such boundary problems is desired .this is the second theoretical problem .the problem of developing a method for the approximate solution of the corresponding boundary problems were also solved and the solution is described above .it turns out that in the framework of such problems elastic displacements can be approximated as the sum of two parts .the first part was called non - classical , it is concentrated near the surface of the body and decays exponentially inside the body . to find this partit is necessary to solve fairly simple one - dimensional equations . the second , classical partis determined by the equations of the classical theory of elasticity .it turns out that the boundary conditions for this equations have a standard classical form of the boundary conditions for the body under the external forces on the surface .thus , the finding of the classical part is reduced to the standard classical problem which does not require a separate discussion .it should be noted that these forces on the surface are formal .physically there is no forces on surface , formal forces describe the interaction between classical and non - classical part of the elastic displacement . so that these formal forces are expressed in terms of non - classical part of the displacement .the corresponding equations are given .it is important to note that for homogeneous polarization the formal forces , describing the influence of the non - classical part of displacement to the classical one , appear only on the curved parts of the surface of the body .thus , the above statement that curvature of the surface plays an important role gets a clear physical meaning .note that the body can be , say , a polyhedron . in this casethe formal forces appear only on the edges which can be treated as a limiting case of a curved surface .while calculating one should slightly smooth these edges and tend to zero the radius of smoothing at the end of the calculations .the similar approach is applied above for the particular case of the rod and thin plate which also has the sharp edges .thus , in framework of the theory of continuum the description flexoelectricity in finite - size bodies actually requires special theory .this theory is described above in detail .the application of this theory is illustrated in particular problems of calculation of the bending of homogeneously polarized rod and a thin plate .tagantsev is acknowledged for reading the manuscript .

Abstract: in many cases the correct theoretical description of flexoelectricity requires the consideration of the finite size of a body and is reduced to the solution of boundary problems for partial differential equations . generally speaking , in this case one should solve jointly the equations of polarization equilibrium and equations of elastic equilibrium . however , due to the fact that typically flexoelectric moduli are very small , usually one can consider the solution of polarization equilibrium equations at a given elastic strain ( direct flexoelectric effect ) or the solution of elastic equilibrium equations at given polarization ( converse flexoelectric effect ) . derivation of the polarization equilibrium equations and boundary conditions for them can be made in the quite usual way . solution of these equations usually is not too difficult problem . on the contrary description of converse flexoelectric effect is more complicated problem . inter alia effective solution of corresponded boundary problems requires the development of special mathematical methods . the subject of this paper is a detailed discussion of the relevant theory focuses on the converse flexoelectric effect . it is also considered some particular examples illustrating the application of the general theory .


massive multiple - input multiple - output ( mimo ) systems promise tremendous performance gains in terms of network throughput and energy efficiency by employing simple coherent processing across arrays of hundreds or even thousands of base station ( bs ) antennas , serving tens or hundreds of mobile terminals . as an additional benefit ,massive mimo is inherently more secure than conventional mimo systems , as the large - scale antenna array equipped at the bs ( alice ) can accurately focus a narrow and directional information beam on the intended terminals ( bob ) , such that the received signal power at bob is several orders of magnitude higher than that at any incoherent passive eavesdropper ( eve ) .unfortunately , this benefit may vanish if eve also employs a massive antenna array for eavesdropping . in this case , unless additional measures to secure the communication are taken by alice , even a single passive eve is able to intercept the signal intended for bob .  since security is a critical concern for future communication systems , facilitating secrecy at the physical layer of massive mimo systemshas received significant attention recently . artificial noise (an ) generation was employed to provide physical layer security in a multi - cell massive mimo system with pilot contamination in .thereby , it was shown that secure communication can be achieved even with simple matched filter ( mf ) precoding of the data and null - space ( ns ) precoding of the an .nevertheless , it was revealed in that significant additional performance gains are possible with more sophisticated data and an precoders , including polynomial precoders .furthermore , an - aided jamming of rician fading massive mimo channels was investigated in .all aforementioned works on secure massive mimo are based on the assumption that the transceivers of the legitimate users are equipped with perfect hardware components , i.e. , the effects of hardware impairments were not taken into account .nevertheless , all practical implementations do suffer from impairments arising from non - ideal hardware components .these impairments are expected to be particularly pronounced in massive mimo systems as the excessive number of bs antennas makes the use of low - cost components desirable to keep the overall capital expenditures for operators manageable .several works have investigated the impact of hardware impairments on massive mimo systems - .these works demonstrated that hardware impairments can severely limit the performance of massive mimo systems , and that phase noise originating from free - running oscillators is the main contributor in degrading the quality of the channel state information ( csi ) estimates needed for precoder design . on the one hand, phase noise causes the csi estimates to become outdated more quickly , and on the other hand , it may cause a loss of orthogonality of the pilot sequences employed by the different users in a cell for uplink training .if communication secrecy is considered , an additional challenge arises : whereas the legitimate users of the system will likely employ low - cost equipment giving rise to hardware impairments , the eavesdropper is expected to employ high - quality equipment and avoid hardware impairments .this disparity in equipment quality was not considered in the related work on physical layer security - nor in the related work on hardware impairments - and necessitates the development of a new analysis and design framework .motivated by the above considerations , in this paper , we present the first study of physical layer security for massive mimo systems in the presence of phase noise .thereby , we focus on the downlink and investigate the effects of multiplicative phase noise at the bs and the users . as a worst - case scenario ,the eavesdropper is assumed to employ ideal ( phase noise - free ) hardware .we derive a tight lower bound for the ergodic secrecy rate achieved by a downlink user in the presence of phase noise when mf data precoding and ns an precoding are employed at the massive mimo bs .the derived bound provides insight into the impact of various system and channel parameters , such as the phase noise variance , the amount of power allocated to the an , the number of users , and the number of deployed local oscillators ( los ) on the ergodic secrecy rate ._ notation : _superscripts and stand for the transpose and conjugate transpose , respectively . is the -dimensional identity matrix .the expectation operation of a random variable is denoted by  ] denotes the element in the row and column of matrix .finally , ^+=\max\{x,0\} ] of length ( ) .we have for , and otherwise .the received vector , at the bs is given by here , the channel vector of the mt , , is modelled as block rayleigh fading , where denotes the path - loss .thereby , is assumed to be constant during coherence time and change independently afterwards . denotes the thermal noise at the bs . in ( [ yult ] ) ,the term characterizes the phase noise affecting the uplink training , and is given by , where denotes the number of free - running los equipped at the bs .thereby , we assume that at the bs each group of antennas is connected to one free - running lo . is the phase noise that affects the link between the lo at the bs and the mt . adopting the discrete - time wiener phase noise model , in time interval , the phase noises at the lo of the bs and the mt are modelled as , , and , , where and are the phase noise ( increment ) variances at the bs and the mts , respectively .+ for channel estimation , we collect the signal vectors received during the training phase in vector ^t \in \mathbb{c}^{bn \times 1} ] , .considering the properties of lmmse estimation , the channel can be decomposed as , , where denotes the lmmse channel estimate given in ( [ hatgkt1 ] ) and represents the estimation error . and are mutually uncorrelated and have zero mean ( * ? ? ? * theorem 1 ) .the error covariance matrix is given by =\beta_k \left(1-\beta_k \boldsymbol{\omega}_k^h \boldsymbol{\theta}_{\sigma(t ) } \boldsymbol{\sigma}^{-1 } \boldsymbol{\theta}_{\sigma(t ) } \boldsymbol{\omega}_k\right ) { \bf i}_n.\ ] ] eqs .( [ yult])-([error ] ) reveal that , in general , for , the channel estimate of the mt contains contributions from channels of other mts , i.e. , pilot contamination occurs although the emitted pilots are orthogonal .furthermore , ( [ hatgkt1 ] ) reveals that the channel estimate depends on time . as a consequence ,ideally , the channel - dependent data and an precoders employed for downlink transmission should be recomputed for every symbol interval of the data transmission phase , which entails a high computational complexity . therefore ,in the following , we assume that the data and an precoders are computed based on the channel estimate for one symbol interval ( e.g.  ) and are then employed for precoding during the entire data transmission phase , i.e. , for . for notational conciseness, we denote the corresponding channel estimate by , .   assuming channel reciprocity , during the downlink data transmission phase , the received signal at the mt in time interval is given by where represents the thermal noise at the mt . the downlink transmit signal in ( [ ykt ] ) is modeled as where the data symbol vector and the an vector , , are multiplied by data precoder and an precoder , respectively .as we assume that the eavesdropper s csi is not available at the bs , an is injected to degrade the eavesdropper s ability to decode the data intended for the mts .thereby , it is assumed that the components of and are independent and identically distributed ( i.i.d . )circularly symmetric complex gaussian ( cscg ) random variables , i.e. , and . in ( [ x ] ) , and denote the power assigned to each mt and each column of the an , where is the total power budget and  ] for signal detection , while treating the deviation from the average effective channel gain as gaussian noise having variance -|\mathbb{e}\left[{\bf g}^h_k(t ) { \bf f}_k\right]|^2 ] with and given by ( [ gammak ] ) and ( [ ce ] ) , respectively , by monte carlo simulation based on independent channel realizations . in this paper , we adopt spatially orthogonal pilot sequences of length  .an example for this type of pilot sequences is provided in ( * ? ? ?* eq . ( 7 ) ) . for simplicity , in this section, we assume the path - losses for all mts are identical , i.e. , , , and the coherence block length is equal to time slots .typical values for the phase noise increment standard deviations , , , used include , which was adopted in the long - term evolution ( lte ) specifications , and , which corresponds to strong phase noise according to .the specific values of other adopted system parameters are provided in the captions of the figures .[ fig1 ] shows the achievable ergodic secrecy rate as a function of the power allocation parameter for different numbers of deployed los ( ) and different numbers of mts ( ) for the case of strong phase noise ( ) .we observe that distributed los ( ) outperform a common lo ( ) for most values of , because the desired signal power is independent of , while the multiuser interference power is reduced when distributed los are deployed , cf .lemma 2 and remark 1 .this is in accordance with the observation made in for the mts achievable rate . on the other hand , for small values of ,a common lo is able to achieve a similar ( or even slightly higher ) secrecy rate as distributed los .this is because the an leakage , represented by in ( [ rkunder ] ) , increases in , and becomes dominant when a large amount of power is assigned for an emission , cf .remark 2 .   for a system with , , , , and db.,title="fig:",width=268 ] +  fig .[ fig2 ] depicts the achievable ergodic secrecy rate as a function of the phase noise standard deviation for different numbers of eavesdropper antennas .the cases of a common lo and distributed los are considered .as expected , the secrecy rate is monotonically decreasing in the phase noise variance as well as in .strong phase noise degrades the mts achievable rate , while a larger number of eavesdropper antennas improve eve s capacity .both effects are detrimental to the achievable secrecy rate .furthermore , this figure also suggests that distributed los achieve a higher secrecy rate compared to a common lo when the optimal maximizing the secrecy rate is employed .   for a system with , , , and db . the optimal maximizing the secrecy rate is adopted.,title="fig:",width=268 ] +in this paper , we have investigated the impact of multiplicative phase noise on the secrecy performance of downlink massive mimo systems employing mf data precoding and ns an precoding at the bs . for the considered system ,a closed - form lower bound on the achievable ergodic secrecy rate of the users was derived .this bound can be used to obtain insights regarding the impact of system parameters , such as the number of los and the phase noise variances , on system performance .our analytical and simulation results revealed that distributively deployed los can achieve a better performance than one common lo as long as the power assigned for data transmission is sufficiently large .20 f. rusek , d. persson , b. k. lau , e. g. larsson , t. l. marzetta , o. edfors , and f. tufvesson , `` scaling up mimo : opportunities and challenges with very large arrays , '' _ ieee sig .proc . mag .40 - 46 , jan . 2013 .d. kapetanovic , g. zheng , and f. rusek , `` physical layer security for massive mimo : an overview on passive eavesdropping and active attacks , '' _ ieee commun . mag .53 , no . 6 , pp .21 - 27 , jun . 2015 .e. bjrnson , j. hoydis , m. kountouris , and m. debbah , `` massive mimo systems with non - ideal hardware : energy efficiency , estimation , and capacity limits , '' _ ieee trans .inform . theory _60 , no . 11 , nov . 2014 .r. krishnan , m. r. khanzadi , n. krishnan , y. wu , a. graell i amat , t. eriksson , and r. schober , `` linear massive mimo precoders in the presence of phase noise- a large - scale analysis , '' _ ieee trans .vehicular tech .pp , no . 99 , pp . 1 - 1 , jun .e. bjrnson , m. matthaiou , and m. debbah , `` massive mimo with non - ideal arbitrary arrays : hardware scaling laws and circuit - aware design , '' _ ieee trans .wireless commun .14 , no . 8 , pp .43534368 , aug .a. pitarokoilis , s. k. mohammed , and e. g. larsson , `` uplink performance of time - reversal mrc in massive mimo systems subject to phase noise , '' _ ieee trans .wireless commun .711 - 723 , feb . 2015 .x. zhou and m. r. mckay , `` secure transmission with artificial noise over fading channels : achievable rate and optimal power allocation , '' _ ieee trans . veh .3831 - 3842 , jul . 2010 .j. zhu , d. w. k. ng , n. wang , r. schober , and v. k. bhargava , `` analysis and design of secure massive mimo systems in the presence of hardware impairments , '' submitted to _ ieee trans .wireless commun ._ , feb . 2016 .available : http://arxiv.org/pdf/1602.08534v1.pdf .

Abstract: in this paper , we investigate the impact of phase noise on the secrecy performance of downlink massive mimo systems in the presence of a passive multiple - antenna eavesdropper . thereby , for the base station ( bs ) and the legitimate users , the effect of multiplicative phase noise is taken into account , whereas the eavesdropper is assumed to employ ideal hardware . we derive a lower bound for the ergodic secrecy rate of a given user when matched filter data precoding and artificial noise transmission are employed at the bs . based on the derived analytical expression , we investigate the impact of the various system parameters on the secrecy rate . our analytical and simulation results reveal that distributively deployed local oscillators ( los ) can achieve a better performance than one common lo for all bs antennas as long as a sufficient amount of power is assigned for data transmission .


for a study of the asymptotic behavior of solutions of a system , the area of dynamical systems has developed a lot of tools , but in many phenomena as financial markets , equations that model them are unknown and the only available information is a temporal set of measures .  a time series is a function , and its image could be considered as observations taken overtime in a particular phenomenon .those pieces of observations have information about the system and we are ready to answer questions like does that serie have enough information to rebuild the dynamics ? if it is so , can we predestine its future behaviour ? .but there is one more general question .will the financial market have a random behaviour or not ? .the chaos theory could give us an answer where systems with an apparently ramdom behaviour , comes from a deterministic system , it means a completely modeling system by equations .chaotic system complexity lies primarily in sensitivity to initial conditions , small variations produce big changes.this sensitivity is quatified by lyapunov exponents .we will do our research in a time series of the stock prices in the mining company minsur.sa to determine concrete evidence of chaos presence in that series .graphic([fig1 ] ) shows evolution of stock prices of the company .to show the use of chaos theory in time series , we consider the lorenz system , which is given by : the dynamics of system solutions is shown in the figure([fig2 ] )    it is well known lorenz system is sensitive to small variations with initial data , it means , solutions with very close initial data is not exact exponentially .figure([fig3 ] ) shows such a behaviour in the first component of solutions for very close data initial .the series in the figure([fig3 ] ) were generated with initial data in points and in time with variation in mili seconds .notice the trajectories differ after 3 seconds , it means , small variations generate big changes quickly .so that , chaos theory could explain some phenomena in the financial market , where variations like speculations generate changes in a short term .one of the goals in this work is to investigate chaotic behavior on time series in the financial market precisely applied in the peruvian financial system    the main task to study the chaotic behavior in a time series is the rebuilding of the dynamics it has .if a time series comes from a deterministic system , so that it has dependence between its components given by the system of equations that modeling and contains geometric information considering that trajectories converge to an attractor which will be inmerse in some euclidian space . using such information and the whitney immersion theorem ideas in which a dimensional compact manifold can be inmersed in and its generalizations for fractal sets given for tim sauer , we will rebuild the dynamics knowing the information of one of the components of lorenz system only .now we can see the probabilistic version of theorem of withney[ preva](prevalent theorem of whitney ) if is a smooth compact manifold of dimension contained on .almost all smooth map , , is an immersion of .the theorem([preva ] ) says it is possible to rebuild the existing dynamics with projections in euclidian spaces .despite we have the theorem of whitney , we still have a practical problem .we only can get a temporal observation of the dynamics in the financial market , the evolution of the stock prices and we will need 1observations of the phenomenon .so , the obtained results are not enough .takens assumed this problem adding the contained dynamics in the time series in the theorem of whitney .it is a projection via a function of observation of the phase space where the dynamical system is developed for .therefore it contains information about the dynamics . for this, it defines the _ delay cordinates _ , which only need a temporary observation .[ delacor ] if is a dynamics system over a manifold , a positive integer ( called of * _ delay _ * ) , and a smoth function .we defines the map of delaying coordinates   taken demonstrated a new version of theorem of whitney for the delaying coordinates  ( takens ) is a dimensional compact manifold , a discrete dynamical system over generated by , and a function of classe  .is a generic characteristic that of map defined by is an immersion .the final generalization used in this article was given by b.hunt , t. sauer and j. yorke , that is a fractal version of theorem of whitney for the delaying maps set with being a fractal set .[ delafraw](fractal delay embedding prevalence theorem ) a dynamics system over an open subset of , and a compact subset of with box dimension . an integer and .asume that contains only a finite number of equilibrio points , it does not contains periodic orbits of of period o , a finit number of periodic orbits of of period , , and these periodic orbits linearizations have different eigenvalues .so for almost every smooth function(in the sense prevalente ) smooth function over , the delay coordinates map is injective over   the theorem([delafraw ] ) does not provide an estimative about the smaller dimension for which almost every delaying coordinates map is injective .however there are numerical algorithms which allow calculate the immersion dimension and the delaying time in the reconstruction.following , we show the reconstruction of the dynamics generated for the system of lorenz using only the coordinate of the systemwe use the lorenz s attractor to show the technique of delaying coordinates .the function of observation , was the projection in the axis . the time series will be formed by coordinates of the trajectories that are numerical solutions of the equation . fixed the dimension , , we change the value of t . according to lorenz s we use delaying times , , , , . in the case points in the space are highly correlated and the graphic is almost a straight line . at the other extreme , ,points are not correlated and the gotten graphic does not represent the reconstruction of the attractor .the optimal delay time was .       +      to determine the dimension where is rebuilt the dynamics , the technique of the neighbours is usedthe idea is about setting a dimension and calculate the distance between close points , if the attractor does not have enough freedom , so when we raise the dimension of the space , those next points get separeted and are pointed as false neighbours .the procedure goes on until the percentage of false neighbours is small . in the series formed with the first components of lorenz s system ,the number of false neighbours is almost zero when the dimension of immersion is , figure([fig10 ] ) .first of all , we will give the definition of chaos according to devaney .[ defcaos ] if is an interval .we say that is chaotic over if :  * has sensibility to initial conditions ; * is transitive ; * the periodic points are dense in .we say that has sensibility to initial conditions when the solutions with initial conditions very close , diverge quickly .a map \rightarrow [ 0,1] ] , containing , there is a and such that   a map \rightarrow [ 0,1] ] there is such that   intuitively , is transitive implies the orbit of an interval is dense on $ ] .  given chaos definition , we will investigate the existence of characteristics in a time series . in this articlewe are interested in searching the characteristic which is characterized by lyapunov exponents .it is important to know lineal systems can not be chaotic .we will identify no lineal dependences on data .the most used techniques are : method of surrogates data and the test bds .+ kantz e schreiber recommend that before doing any non lineal analysis on a data set , it is good practice to check if there is no linearity .because the linear systems are not chaotic and the linear stochastic processes can produce very complex series which can be misunderstood as a product of chaotic system .we presented the technique of surrogate data and the bds test .a very useful technique used by physicist is surrogate generation of _ surrogate data _ , a procedure done by theiler , eubank , longtin , galdrikan y farmer .from the original data , it generates a set of random series so that , these keep the linear properties of the original series ( average , variance , fourier spectrum ) but eliminating the possible nonlinear dependencies.then an indicator is evaluated , which is sensitive to nonlinear dependencies and try to reject the null hypothesis , which states that data are obtained by a stochastic linear process . if the null hypothesis is true , then the substitute series procedure would not affect the indicator  the most used indicators are correlation dimension .it was used by small et .al. and no linear prediction which was used by kaplan .we will illustrate the technique of substitute data applied in lorenz series and use the nonlinear prediction .the election was due to deterministic systems the prediction on a short term is possible , in contrast to stochastic systems in which the prediction in a short term is impossible . for the prediction nonlinear we will use the method developed by hegger , kantz and schreiber , where the value of predicting is the average of `` future '' values .this state , is defined by average of the closed values .it is calculated by the following equation   where is the neighborhood , are values within and are the values at time .held the prediction , this is compared with the quadratic average error of prediction on future value of the original serie with the surrogate series .if the prediction error of the series original is smaller than all the surrogate series , then we reject the null hypothesis .this graphic shows the results .the projection errors of the original series were smaller than its substitutes . like this , we reject the null hypothesis .  the following technique is more about statistics .it was developed by brock , dechert and scheinkman , it s based on the correlation dimension to detect nonlinear structure in a time series .additionally this test can be used to test how good is the fit estimation model  given a time series , using the method of delay coordinates , we have vectors in , , where is the delay time , is the immersion dimensin .the correlation dimension is given by the formula , where is the function point mass , which indicates the probability where the points are close to each other .  for practical and didactic issues we consider .now if we will get then , implies and .  this allows to say that if the points e are closing to each other , then , the points from series and as well .it happens the same with points and .thus , where indicates the probability .if the data are iid(independent and identically distributed ) , then and we have , thus , in dimension we have then , as .this suggests that the data is iid , then ,   where is the integral of correlation in dimension and is the integral of correlation in dimension one .  in , brock ,dechert y scheinkman , statistical bds test is defined by :   where is the estimation of the standard asymptotic error : .they proved that :   introducing to bds test applied in lorenz series and a series consisting of random numbers .cc dimension of immersion(m ) & bds + 1 & 165.270466 + 2 & 447.623951 + 3 & 578.535241 + 4 & 772.617763 + 5 & 1078.467687 + 6 & 2338.679429 + 7 & 3593.653394 + 8 & 5640.618871 +   cc dimension of immersion(m ) & bds + 1 & 0.560594 + 2 & 0.560594 + 3 & 0.425293 + 4 & 0.069765 + 5 & 0.116118 + 6 & 0.336857 + 7 & 0.633835 + 8 & 0.524584 +  to understand the results , let s start talking about the degree of significance of a hypothesis . to ask a null hypothesis , in the case of the bds testthe null hypothesis is that the observations are independent and identically distributed ( iid ) , it can be make a mistake of rejecting the null hypothesis being true .suppose that the probability of committing this error is , namely :  is called the degree of significance of this mistake .now , in the bds test : by the test bds , know that then ,  so ,   if we look at the tables of the normal distribution , we see that you to a degree of signicancia of , . as, we reject the null hypothesis , with degree of signicancia of , if .noting again the tables we can observe that in the case of lorenz series , we reject the null hypothesis , the data are iid . otherwise case series formed by random numbers .detected non - linealidade in the time series , the next step is to search sensitive dependence on initial conditions .this section we obtain results that characterize the sensitive dependence on initial conditions .the study of the attractor and the dynamics is performed in the space reconstructed using delay coordinates .this section we obtain results that characterize the sensitive dependence on initial conditions .the study of the attractor and the dynamics is performed in the space reconstructed using delay coordinates .chaotic systems are , generally , strange attractors which are generated by stretching and contractions .stretches are responsible for sensitivity on initial conditions and are characterized by the exponential distance in finite time , paths with initial conditions very next .contractions allow the pathlines may not move away indefinitely and that they remain confined to a bounded phase space region .distance speed is measured by the lyapunov exponents .  in the case of a discrete dynamic defined by a map system , is the jacobian matrix of .then writing the largest lyapunov exponent is given by :   for almost all vector , for more details see .as we can see the first problem for the calculation of lyapunov exponents is to obtain the matrizes . in the case of having only a time seriesthe calculation of lyapunov exponent is even more complicated .ruelle, proposed the first algorithms based on the following ideas .consider a dynamic generated by the differential equation : in . if we do in the previous equality we get ,   the retrieved system is linear in .then , if the system([sisnolinear ] ) solutions are , then the solutions of the system([sislinear ] ) are .so the matrix , , can be calculated integrating the equations ( [ sisnolinear ] ) .this idea for a linear system to calculate the matrix , , was used by eckman and ruelle for estimating of lyapunov exponents from a time series .  consider a time series , . using delay coordinates ,obtain points , in the space -dimensional of reconstruction .if we develop around point , we would have that for next to it , , is estimated by the best linear approximation of the map that for points next to point , takes in points close .this process can be thought as the linear approximation of the derivatives at each point in space lathe overhauled do . with the same ideas , authors such as wolf , swift , swinney vastano , m. sano , y. sawada proposed an algorithm for the calculation of lyapunov exponents .another factor to consider is that the lyapunov exponents are defined in terms of limits , which guarantees its invariance by smooth transformations .if we think of a finite series , there is no guarantee that interference ( noise ) data , lead to the same exponents .rosenstein , collins and de luca and holger kantz directly presented new techniques based on the idea that the greatest exponent of lyapunov measure the average speed of estrangement of the paths with conditions next .the estimation of the maximum exponent of lyapunov using the algorithm of rosenstein was made in the reconstructed attractor of lorenz .the value of was obtained .the series studied was the series of prices of the shares of the mining company minsur , counting on prices from up to with a total of years of daily observations .   for the use of the surrogatesare generated series substitute which preserve linear properties of the original series , for example , the graph([surrohistpet ] ) shows histograms of the original series as well as his surrogate series .[ surrohistpet ]   the next step is the calculation of the predictions of each of the series , original and substitute , and comparing related errors .the figure([surropredictpet ] ) shows that errors of prediction of the original series presents a relatively small short - term prediction error and substitute series shows a high prediction error from the first moment .therefore the null( : error of the original series prediction = surrogate series prediction error)hypothesis is rejected since the original series prediction error is less than all their substitute series prediction error .therefore , we reject the fact that the original series is generated by a linear stochastic process .[ surropredictpet ]   on the other hand , the bds test is an alternative more statistics to detect non - linear units , perform the test for several dimensions of immersion . in the chartare the values of the bds statistics to dimensions of to .cc dimension of immersion(m ) & bds + 1 & 165.270466 + 2 & 195.653634 + 3 & 236.554950 + 4 & 296.226274 + 5 & 382.825922 + 6 & 508.150676 + 7 & 690.060422 + 8 & 955.415301 + 9 & 1344.764026 + 10 & 1919.802065 + 11 & 2774.198843 + 12 & 4050.599850 + 13 & 5967.524832 + 14 & 8860.578053 + 15 & 13419.054424 +  we see the table that .therefore , we reject the null hypothesis ( that the data are iid ) , with a degree of significance of .  identified the non - linearity in the series ,the next step is the search for evidence of chaotic dynamics .finally , we calculate the greatest exponent of lyapunov as an indicator of chaos . the theorem ergodic justice oseledec method rosenstein and kantz to estimate the maximum exponent of lyaunov by the formula([separaexpo]).in practice it is convenient to only calculate the maximum exponent of lyapunov , since the whole spectrum depends on the dimension of immersion and colors more exponents than actually exist , or less .using the analysis nonlinear and chaos theory applied in the series of prices of the shares of the mining company minsur obtained the following results :  1 .the data contains non - linear dependence .2 .  we can not conclude the existence of chaos in the series .  even considering non - linear dependence in the data results to detect chaos are not conables .some authors estimated the largest lyapunov exponent using only the first iteration of estrangement .a job to follow is the search for presence of orbits periodically unstable , using for this purpose , the recurrence maps or maps of poincar , as well as the presence of a fractal attractor .the authors are so grateful to the department of mathematics applied of the universidad federal de rio de janeiro , ufrj , and the department of sciences of the university private of the north , upn , trujillo - per , where this study was developed .  a holger kantz , thomas schreiber , _ nonlinear time series analysis . _ cambridge university press 2004 .eckmann , d.ruelle , _ _ ergodic theory of chaos and strange attractor , reviews of modern physics , vol 57 no 3 , part i , 1985 .michael small , dejin yu , jennifer simonotto , robert g harrison , neil grubb , k.a.afox.(2002 ) . uncovering non - linear structure in human ecg recordings .chaos , solitons , fractals volume 13 , issue 8 , pages 1755 - 1762 .michael t. rosenstein , james j. collins , carlo j. de luca(1993 ) . a practical method for calculating largest lyapunov exponents from small data sets .physica d : nonlinear phenomena volume 65 , issues 1 - 2 , pages 117 - 134   w. a. broock , j. a. scheinkman , w. d. dechert , b. lebaron(1996 ) . a test for independence based on the correlation dimension .journals in its journal econometric reviews.volume 15 , issue 3 , pages : 197 - 235    theiler , james ; eubank , stephen ; longtin , andr ; galdrikian , bryan ; doyne farmer , j.(1992 ) . testing for nonlinearity in timeseries : the method of surrogate data .physica d : nonlinear phenomena , volume 58 , issue 1 - 4 , p. 77 - 94

Abstract: physicists experimentalists use a large number of observations of a phenomenon , where are the unknown equations that describe it , in order to play the dynamics and obtain information on their future behavior . in this article we study the possibility of reproducing the dynamics of the phenomenon using only a measurement scale . the whitney immersion theorem ideas are presented and generalization of sauer for fractal sets to rebuild the asymptotic behaviour of the phenomena and to investigate , chaotic behavior evidence in the reproduced dynamics . the applications are made in the financial market which are only known stock prices  ,  .   caos ; dinmica no lineal ; sries temporais .