the sensitivity of the modern laser - interferometric gravitational - wave ( gw ) detectors is limited by quantum fluctuations of the probing light over most of the sensitive frequency range .in particular , at higher frequencies their sensitivity is limited by the shot noise ( also known in more general context as the measurement noise ) , created by quantum fluctuations of the phase of the probing light .the resulting sensitivity , about in units of the equivalent displacement noise , is extremely high and has proved to be sufficient for the direct observation of gravitational waves from astrophysical sources .  at the same timethe pair of advanced ligo interferometers , which detected the first gw signals , have not reached yet their design sensitivity , which is planned to provide about a factor three improvement in astrophysical reach .suppression of the shot noise , which is necessary for achieving this goal , will require either an increase of the optical power circulating in the interferometer up to , or the application of squeezed light states , and most probably a combination of both approaches will be used to maximise the sensitivity gain .  due to the heisenberg uncertainty relation, this will lead to the proportional increase of another kind of the quantum noise , namely radiation pressure noise ( also known as the quantum back action noise ) , imposed by the quantum fluctuations of the light power in the interferometer disturbing the test mass positions .the point of balance between the measurement noise and the back action noise is known as the standard quantum limit ( sql ) , and the design sensitivity of the advanced ligo interferometers will touch the sql at one frequency .it has to be emphasized that the sql is not a truly fundamental limit , and several methods have been proposed for overcoming the sql in future gw detectors. a detailed review of these methods can be found _e.g. _ in .one of the most promising approaches for surpassing the sql is based on the _ quantum speed meter _concept , which was first proposed in .the basic idea of this concept is to measure the velocity of the probe mass(es ) instead of their position . in this case , the measurement noise and the back action noise spectral densities depend on the observation frequency in such a way that they can provide cancellation of each other by means of introducing a frequency - independent cross - correlation between them .it can be implemented simply by using a homodyne detector with the properly set homodyne angle .note that in the traditional position - sensitive interferometers , additional long _ filter cavities _ are required for this type of the quantum noises cancellation ( 4-km cavities were proposed in ; it was shown later that much shorter , but still quite long , tens or hundreds of meters , cavities could be used as well , but they could provide only limited sensitivity gain ) .several implementations of the quantum speed meter concept suitable for the gw detectors were proposed , which can be divided into the following two categories : the first one relies on the ordinary michelson interferometer topology of the contemporary gw detectors , but requires an additional long _ sloshing cavity _ and therefore does not provide significant advantages in comparison with the filter cavities based topologies .the second category is based on the zero - area sagnac interferometer topology , which significantly deviates from the standard michelson topology .currently it is a subject of intense r&d efforts .here we propose a new kind of the quantum speed meter , the _ epr speed meter _( from the famous gedanken experiment by einstein , podolsky and rosen ) , which allows to use the michelson interferometer topology and , at the same time , does not require any additional long - baseline filter cavities or other major infrastructure changes .we would like to emphasize that the goal of this short paper is to introduce the concept of this new speed meter type .detailed investigations of the technical implementation , such as the robustness against optical loss , the coupling of laser frequency and amplitude noise , as well as additional add - on techniques , like the injection of squeezed states will be considered in a follow - up article , currently in preparation .this paper is organized as follows . in the next sectionwe reproduce the basic analytical treatment of quantum noise in the position meter and speed meter schemes . in sec.[sec :idea ] we present the concept of the epr speed meter .in sec.[sec : gwdetector ] we consider a possible implementation of our concept in a gw wave detector and provide brief estimates of its sensitivity , using parameters similar to the ones of the envisaged ligo voyager gw detectors .the notations and the parameter values used in this paper are listed in table[tab : params ] ..main notations used in this paper . for the numerical values , we use the ones planned for the next - generation gw detector ligo voyager . [ cols="<,<",options="header " , ]the ( double - sided ) power spectral density of the sum of quantum noise components in a position meter can be presented as follows ( a much more detailed analysis of the quantum noise in interferometers can be found in ) : where is the spectral density of the measurement noise , is the spectral density of the back action force and is the cross - correlation spectral density of these two noise sources ( we assume here that is real in order to avoid subtle but unrelevant to our consideration issues related to the imaginary part of ) . these spectral densities satisfy the heisenberg uncertainty relation in the rest of this section , we assume this relation is saturated and the interferometer is driven by the vacuum and laser fields in the minimum uncertainty quantum state .  in the modern gw detectors , there is no cross - correlation between the shot noise and the radiation pressure noise , because the resonance - tuned configuration is used in these detectors and only the phase quadrature of the outgoing light is measured .hence where is the double - sided sql spectral density .  on the other hand ,if and can be made arbitrarily dependent on frequency , then the spectral density can be minimized , using the exact equality in and setting which gives : in the laser interferometers is proportional to the optical power inside the interferometer .it was shown in that this _ energetic quantum limit _ actually is a general one for all linear stationary interferometric measurements .  the optimized spectral density , in principle ,can be made arbitrarily small simply by increasing this power .however , conditions can only be satisfied in the given frequency band , provided that the spectral densities , , and depend on frequency in a rather specific way which is , sadly , different from the one they acquire due to finite bandwidth of the arm cavities in the existing gw interferometers .therefore , to introduce the desired frequency dependence in a broad band , long additional filter cavities are required .   in the speed meter schemes , the quantum noise has the same general structure , but with the following peculiarities : where is the velocity measurement noise spectral density , is the momentum perturbation noise spectral density , and is the corresponding cross - correlation spectral density .it is important that , , and can be considered as frequency - independent within the interferometer bandwidth .the relation takes the following form : and the sum quantum noise spectral density of the speed meter reads : in the particular case of , similar optimization as for the pm can be made : yielding the quantum noise of the speed meter to follow the sql : note that the corresponding spectral density of the position meter only touches the sql at one given frequency and goes above it elsewhere .therefore , the speed meter provides better sensitivity even in this simple case .  in a more general case of , the following optimization : gives : similar to the position meter case , this spectral density can be arbitrary small , provided that is sufficiently large , which means high enough circulating optical power in the interferometer .contrary to the position meter , no additional elements like filter cavities are required for this .of the same mass ; their output beams are combined by the beamsplitter , forming the `` '' ( position ) and the `` '' ( speed ) outputs .bottom : a more practical collinear version ( tolerant to the angular motion of the mass ) of the same scheme ; in this case , the signs of the beamsplitter reflectivity factors has to be swapped.,title="fig:",scaledwidth=30.0% ] +  of the same mass ; their output beams are combined by the beamsplitter , forming the `` '' ( position ) and the `` '' ( speed ) outputs . bottom : a more practical collinear version ( tolerant to the angular motion of the mass ) of the same scheme ; in this case , the signs of the beamsplitter reflectivity factors has to be swapped.,title="fig:",scaledwidth=42.0% ]  consider now the scheme shown in fig.[fig : sketch_scheme](top ) . here the mass forms a joint movable mirror for two otherwise independent fabri - perot cavities having the same eigenfrequency , the same lengths , but different bandwidths .the cavities are pumped at the frequency and their output fields are combined by the beamsplitter .its two output beams labeled in the picture as `` '' and `` ''are measured by the two homodyne detectors .  using the two - photon amplitudes notations of ,the input / output relations for these cavities can be written as ( see _ e.g. _ ) : where is the cavity number , are the cosine and the sine quadratures of the input field of the cavity , are the corresponding output field quadratures , are the frequency - dependent reflectivities of the cavities for the cavity sideband fields , are the optomechanical transfer functions , are the classical amplitudes of the intracavity fields , normalized as follows : and is the optical power , circulating in the cavity . note that if correspond to the vacuum input fields , then the same is true for .therefore , below we absorb into in order to simplify the equations .the beamsplitter transforms the output fields as follows : where are the new effective input vacuum fields and are the transfer functions for the `` '' and `` '' channels .  in order to create the speed meter type frequency dependence of the optomechanical coupling, we propose to exploit the difference in the frequency dependence of .note that if and then the dc values of and are equal to each other , but the ac values are different . in this case , the leading term of the factor is proportional to , as it is required for the speed measurement . more specifically , in this case  [ calg_pm ]   taking also into account that the `` '' and `` '' noise components in are uncorrelated , it follows from eqs . that the `` '' and `` '' ports of the beamsplitter correspond to two independent meters : the position one for the `` '' port ( note that is flat at low frequencies ) and the speed meter for the `` '' port .  according to the heisenberg uncertainty relation ,each of these meters create the back action force of its own .indeed , it is easy to show that the fluctuational component of the radiation pressure force acting on the mass is equal to ( see again ) : where it is easy to see that the force has the speed meter type frequency dependence .  in principle, a more practical version with the collinear placement of the elements , which is tolerant to the angular motion of the test mass , is also possible , see fig.[fig : sketch_scheme](bottom ) . evidently , in this case the signs of reflectivities of the two beamsplitter surfaces has to be swapped . in all other aspects , the features of this versionare identical to the one considered before .  in general the two outputs of the proposed two - meters setup can be combined in a variety of ways . herewe mainly focus at the simplest pure speed meter case , which can be implemented by `` switching off '' the position - sensitive `` '' channel .this can be done by measuring the amplitude ( cosine ) quadrature of the `` '' output field , which does not contain any information on the mechanical motion , but instead it is correlated with radiation pressure force .therefore , it is entangled with the mechanical motion .the measurement of removes this entanglement in the epr way and projects the mass into the conditional state with effectively eliminated perturbation component created by the `` '' channel .of course , physically this perturbation remains unchanged , but it becomes known to the experimenter and can be eliminated from the output signal by the subsequent appropriate data processing .it could be mentioned that in general case , this filtering can not be done by in real time by using a causal filter . however , for the gw detection this is not an issue because off - line cancellation with an acausal filter can be used in this case .the remaining `` '' channel quantum noise spectral density can be calculated using eqs.([s_pm ] , [ io_pm ] , [ calg_m ] , [ f_fl_pm ] ) .assuming that the output of this channel is measured by the homodyne detector with the homodyne angle , we obtain : where is the kimble factor for the `` '' channel .the three terms in parentheses in originate , respectively , from the shot noise , the cross - correlation between the the shot noise and the radiation pressure noise , and the radiation pressure noise .they directly relate to the corresponding terms in eqs . and .it is instructive to compare with the corresponding factors for the position meter ( michelson ) interferometers : and for the speed meter sagnac interferometer : note that in order to fulfill the speed meter conditions ( [ opt_sm_sql ] , [ opt_sm ] ) , the optomechanical coupling has to be sufficiently strong , , which requires sufficiently high circulating optical power :   more sophisticated frequency dependence of allows to alleviate this problem .indeed has a flat speed meter - like behaviour only up to the smaller of the two bandwidths ; to be specific , we suppose that . at higher frequencies , it can be approximated as and has the position meter type frequency dependence with the bandwidth equal to , compare with .consider therefore the strongly asymmetric case with and with is equal to the required bandwidth of the interferometer . in this casethe speed meter conditions ( [ opt_sm_sql ] , [ opt_sm ] ) give the following requirement for the circulating optical power : which is a factor of weaker than . with the account for this moderate optical power ,the quantum noise at frequencies above the will be dominated by the shot noise , with the position meter - like providing lower shot noise spectral density within the interferometer bandwidth ( ) than in the sagnac speed meter case .the price to pay , however , is the approximately two times lower high - frequency asymptotic of compared to that of the ordinary position meter ( see also fig.[fig : plots_g ] ) .the reason for this is evident : only the part of the total circulating optical power which corresponds to the `` '' channel , that is about half of the total power , participates in the measurement . whether this disadvantage outweighs the advantage of more effective discussed above or not , depends on the specific parameters of the interferometera possible method of implementation of two independent position meters within the standard michelson interferometer topology is to use the two orthogonal polarisations of light .note that polarisation - based schemes of speed meter ( the sagnac interferometer or the michelson interferometer with the additional sloshing cavity ) were discussed already in literature several times .the scheme which we propose here is shown in fig.[fig : setup ] . in this scheme , in order to excite two orthogonal linear polarisations of the carrier light in the interferometer , the polarisation of the pumping laser has to be tilted by the angle defined by the condition :   two strongly different values of the bandwidths can be created by using the signal recycled configuration of the interferometer with a birefringent plate inserted into the signal recycling cavity .this plate has to introduce the phase shift between the two polarisations passing through it ( the quarterwave plate ) . in this case, for one polarisation we obtain a resonant sideband extraction scheme ( large bandwidth ) , while for the other polarisation we obtain a signal recycling scheme ( low bandwidth ) .the bandwidths of the two readouts are given by :    where , are the power reflectivity and transmissivity of the arm cavities input mirrors , and , are the power reflectivity and transmissivity of the signal recycling mirror.  finally , in order to create the output channels `` '' and `` '' , the polarisation beamsplitter tilted by relative to the linear polarisations circulating in the interferometer can be used .  '' and the `` '' channels of the epr speedmeter noise with and ; the michelson / fabry - perot position meter interferometer with ; the sagnac speed meter interferometer with .all other parameters correspond to the ones of the planned ligo voyager gw detector , see table [ tab : params ] . ]  for the sensitivity estimates for this scheme , we use the parameters planned for the next generation gw detector ligo voyager , see table [ tab : params ] . note that due to the increased mirror mass and the changed laser wavelength ( compared to advanced ligo ) and despite the increased circulating optical power , this detector will be even more underpowered than the advanced ligo , with the normalized optical power . therefore , we assume here in most cases that the phase quadrature of the output channel `` '' is measured , which corresponds to homodyne angle , see eq .. in this regime , the speed meter can not overcome the sql , but still has much better low - frequency sensitivity than the position meter , see sec.[sec : sm_vs_pm ] .at the same time , this regime requires smaller optical circulating power than the more advanced sql - beating one with .   and ; the michelson / fabry - perot position meter interferometer with ; the sagnac speed meter interferometer with . in all cases ,the phase quadrature of the output light is supposed to measured ( ) .all other parameters correspond to the ones of the planned ligo voyager gw detector , see table [ tab : params ] . ]figures [ fig : plots_g ] and [ fig : plots_s_1 ] show the optomechanical transfer function and the ( single - sided ) linear quantum noise spectral density , normalised to the equivalent gw signal : respectively . for comparison ,the corresponding plots for the sagnac speed meter interferometer , also for the case of , and for a michelson / fabry - perot ( position meter ) interferometer with the same value of the homodyne angle are also provided .it is easy to see from these plots that the epr speed meter realises the trade - off between the high - frequency and the low - frequency sensitivity , losing discussed above 3db to the position meter in shot noise dominated high- and medium - frequency region , yet providing significantly better sensitivity in the radiation - pressure noise dominated low - frequency region .it is also inferior by to the sagnac speed meter in the medium - frequency range .however , the relative simplicity of implementation of the epr speed meter has the potential to easily outweigh this disadvantage .( the epr speedmeter ) ; ( the two - channel position meter ) ; ( an example of the sql - beating tuning ) ; the michelson / fabry - perot position meter interferometer with . in all cases , , .all other parameters correspond to the ones of the planned ligo voyager gw detector , see table [ tab : params ] . ]an important feature of the of the epr speed meter scheme is the flexibility provided by the two homodyne angles which can be tuned independently . in particular ,if the phase quadratures are measured in both channels , , then this schemes reduces to the position meter , see the corresponding plot in fig.[fig : plots_s_2 ] .the high - frequency sensitivity in this case is the same as in the ordinary michelson positions meter .however , the low - frequency is worse by due to the times larger low - frequency asymptotic of the transfer function , in comparison with ( see fig.[fig : plots_g ] ) , which increases proportionally the radiation - pressure noise .the values of provide sensitivity better than the sql . as fig.[fig : plots_s_2 ] shows , that ( assuming the ligo voyager parameters ) the sql can be surpassed by a factor of up to over a frequency range of half a decade .note that the sub - sql performance will increase with increasing power .the price for this is the sharp increase of the noise in the very low - frequency band due to incomplete cancellation of the `` ''-channel back action .therefore , in any future application the epr speed meter sensitivity can be optimised to obtain maximum reach for any astrophysical source for a given level of other noise sources of the gw detector .we would like to mention three important things here .first , the scheme described in this article is probably one of the simplest ways to realise a quantum speed meter , because it builds to a large extent on the traditional michelson configuration with just minor extensions . note that only a small number of additional optical elements will be needed , such as the additional quarter wave plate in the signal recycling cavity and the polarising beamsplitter at the output port . in additionit has to be ensured that the core interferometer can sustain normal operation for two orthogonal , linear polarisations .secondly , the proposed scheme can be switched back in - situ ( using remote - controlled , in - vacuum waveplates ) to the position meter mode , therefore increasing its medium- and high - frequency sensitivity .this transformation of the measurement mode can likely be obtained within short time - scales and without any replacement of hardware , simply by switching the `` '' output from the amplitude quadrature to the phase quadrature detection .if this switching and the corresponding changes to the control system and calibration system ( e.g. adjusting loop gains and signal transfer functions ) can be done fast enough and without introducing any additional displacement noise , then this possibility could be useful for the detection of gw signals from binary systems mergers , with the waveforms starting from very low frequency and gradually increasing to much higher ones .finally , it is evident that the capabilities of the considered scheme , featuring two outputs , are not limited by the simple configuration considered here .for instance the application of squeezed light states will even further improve the achievable strain sensitivity of our proposed scheme , and such options will be considered in in detail in a follow - up article .this work was supported by the european research council starter grants program erc-2012-stg:307245 and by the royal society international exchanges scheme grant ie160125 .the work of e.k . and f.k .was supported by russian foundation for basic research grants 14 - 02 - 00399 and 16 - 52 - 10069 .the work of f.k . was also supported by ligo nsf grant phy-1305863 .the paper has been assigned ligo document number ligo - p1600351 .we assume that the photocurrents of the two homodyne detectors are combined using the optimal frequency - dependent weight function : -(\omega ) \,.\ ] ] note that this is a software operation and therefore no constrains are applied on the shape of .this gives the following spectral density of the combined noise :    + |1 - \epsilon|^2s_- \,,\ ] ] where , are the sum noise spectral densities at the `` '' and the `` '' outputs , respectively , and is the corresponding cross - correlation spectral density .it can be shown using eqs.([io_pm ] , [ f_fl_sum ] ) , that they are equal to                               , energetic quantum limit in large - scale interferometers , in _ gravitational waves .third edoardo amaldi conference , pasadena , california 12 - 16 july _ , edited by s.meshkov , pages 180189 , melville ny : aip conf .523 , 2000 .

Abstract: we propose a new implementation of a quantum speed meter qnd measurement scheme . it employs two independent optical readouts of the interferometer test masses , featuring strongly different values of the bandwidths and of the optical circulating power , with the special relationship of . the outputs of these two position meters have to be combined by an additional beamsplitter . in this scheme , signals at the common and the differential outputs of the interferometer setup are proportional to the position and the velocity of the test masses , respectively . the influence of the position meter - like back action force associated with the position signal can be cancelled using the epr approach by measuring the amplitude quadrature of the beamsplitter common output correlated with this force . in the standard signal - recycled michelson interferometer topology of the modern gravitational - wave detectors , two independent optical position meters can be implemented by two orthogonal polarisations of the probe light . our analysis shows that the epr speedmeter provides significantly improved sensitivity for all frequencies below compared to an equivalent signal recycled michelson interferometer . we believe the epr speedmeter scheme to be very attractive for future upgrades of gravitational wave detectors , because it requires only minor changes to be implemented in the interferometer hardware and allows to switch between the position meter and the speed meter modes within short time - scales and without any changes to the hardware .


many practical uses of the phenomenon of chaos have been proposed in recent years , as for instance , in communications , in enhancing mixing in chemical processes , in avoiding electromagnetic interferences , in cryptography , in stabilizing plasma fusion , etc . in such applicationsit is necessary to obtain reliable operation of chaotic systems .it is known that most chaotic attractors of smooth systems are embedded with a dense set of periodic windows for any range of parameter values .therefore in practical systems functioning in chaotic mode , a slight fluctuation of a parameter may drive the system out of chaos . on the other hand, it has been shown that some dynamical systems can exhibit robust chaos .a chaotic attractor is said to be robust if , for its parameter values , there exist a neighborhood in the parameter space with absence of periodic windows and the chaotic attractor is unique .robustness is an important property in applications that require reliable operation under chaos in the sense that the chaotic behavior can not be destroyed by arbitrarily small perturbations of the system parameters .for example , robust chaos has efficiently been used in communications schemes .  in this articlewe study a family of singular maps as an example of a simple model of dynamical systems that shows robust chaos on a finite interval of their parameter values . in section 2we introduce this family of maps and investigate their dynamical properties , both analytically and numerically .it is found that the transitions to robust chaos in these systems occur either through the routes of type - i or type - iii intermittency .the region where robust chaos takes place is characterized on the space of parameters of the maps .conclusions are presented in section 3 .as a simple model of a dynamical system displaying robust chaos , we consider the following family of singular maps where , , and is a real parameter .the exponent describes the order of the singularity at the origin that separates two piecewise smooth branches of the map eq .( [ sing_map ] ) .these maps are unbounded , that is , .the schwarzian derivative of the family of maps eq .( [ sing_map ] ) is always positive , i.e. , for .thus maps defined by eq .( [ sing_map ] ) do not belong to the standard universality classes of unimodal maps and do not satisfy singer s theorem . as a consequence , these singular maps do not exhibit a sequence of period - doubling bifurcations .instead , the condition leads to the occurrence of an inverse period - doubling bifurcation , where a stable fixed point on one branch of the singular map losses its stability at some critical value of the parameter to yield robust chaos .it should be noted that robust chaos has also been discovered in smooth , continuous one - dimensional maps .figure [ bif_diag ] shows the bifurcation diagrams of the iterates of map eq .( [ sing_map ] ) as a function of the parameter for two different values of the singularity exponent .figure [ bif_diag ] reveals robust chaos , i.e. , the absence of windows of stable periodic orbits and coexisting attractors , on a well defined interval of the parameter for each value of .  ) as a function of the parameter for two values of the order of the singularity , showing robust chaos .type - i or type - iii intermittencies appear at the boundaries of the robust chaos intervals .( a ) ; ( b ) .,title="fig : " ] + ) as a function of the parameter for two values of the order of the singularity , showing robust chaos .type - i or type - iii intermittencies appear at the boundaries of the robust chaos intervals .( a ) ; ( b ) .,title="fig : " ]  the transition to chaos at the boundaries of the robust chaotic interval occurs by intermittency .intermittent chaos is characterized by the display of long sequences of periodiclike behavior , called the laminar phases , interrupted by comparatively short chaotic bursts .the phenomenon has been extensively studied since the original work of pomeau and manneville classifying type - i , -ii , and -iii instabilities when the floquet multipliers of the local poincar map associated to the system crosses the unit circle .type - i intermittency occurs by a tangent bifurcation when the floquet s multiplier for the poincar map crosses the circle of unitary norm in the complex plane through ; type - ii intermittency is due to a hopf s bifurcation which appears as two complex eigenvalues of the floquet s matrix cross the unitary circle off the real axis ; and type - iii intermittency is associated to an inverse period doubling bifurcation whose floquet s multiplier is .two stable fixed points satisfying and exist for each value of : and , both are seen in figure [ bif_diag ] . for ,the fixed point becomes unstable at the parameter value through an inverse period doubling bifurcation that gives rise to chaos via type - iii intermittency , while the fixed point originates from a tangent bifurcation at the value and the transition to chaos at this value of takes place through type - i intermittency . on the other hand , for behavior of the fixed points is interchanged : experiences a tangent bifurcation at the parameter value and a type - i intermittent transition to chaos occurs ; while the fixed point undergoes an inverse period - doubling bifurcation at the value , setting the scenario for a type - iii intermittent transition to chaos .there exist several unstable period- orbits satisfying and in the chaotic interval $ ] .figure [ unst_orbits ] shows some unstable periodic orbits of the singular map with as a function of the parameter .  , indicated by dotted lines , as a function of .the stable fixed points and are plotted with solid lines . at the parameter value ,the fixed point becomes unstable through an inverse period - doubling bifurcation , giving raise to the unstable fixed point .at the value , a tangent bifurcation takes place and the pair of points ( stable ) and ( unstable ) are born .the period-2 unstable orbit and , satisfying , are shown . ]figure [ crit_bound ] shows the critical boundaries and for the transition to chaos .these boundaries separate the region on the parameter plane where robust chaos takes place from the region where stable fixed points of the maps eq .( [ sing_map ] ) exist .the transition to chaos via type - i intermittency takes place at the parameter boundaries for , and for . on the other hand ,the transition to chaos via type - iii intermittency occurs at the critical parameter values for , and for . the boundaries and on the space of parameters are indicated in figure [ crit_bound ] .the width of the interval for robust chaos on the parameter for a given is   and of the robust chaos region for the singular maps on the space of parameters .the thick , dark line indicates the boundary for the transition to chaos via type - iii intermittency .the thin , light line corresponds to the boundary for the onset of type - i intermittency . ]figure [ lyap_exp ] shows the lyapunov exponent as a function of the parameter for the family of maps eq .( [ sing_map ] ) , for two values of , calculated as with iterates after discarding transients for each parameter value .the boundaries and correspond to the values .the lyapunov exponent is positive on the robust chaos interval .the transition to chaos through type - i intermittency is smooth , as seen in figure [ lyap_exp ] .in contrast , the transition to chaos via type iii intermittency is manifested by a discontinuity of the derivative of the lyapunov exponent at the parameter values corresponding to the critical boundary .this discontinuity is due to the sudden loss of stability of the fixed point associated to the inverse period doubling bifurcation that occurs at the boundary .the lyapunov exponent can be regarded as an order parameter that characterizes the transition to chaos via type - i or type - iii intermittency .this transition can be very abrupt in the case of type - iii intermittency , as seen in figure [ lyap_exp ] .we have introduced a family of singular maps as an example of a simple model of dynamical systems exhibiting robust chaos on a well defined range of parameters .the behavior of these maps has been characterized as a phase diagram in the space of their parameters , showing a region where robust chaos takes place and regions where stable fixed points occur .we have shown that the transitions to robust chaos in these systems occur either through the routes of type - i or type - iii intermittency and have calculated the critical boundaries for each type of transition on the phase diagram of the systems .the simplicity of these singular maps and the robustness of their chaotic dynamics make them useful ingredients in the construction of models and in applications that require the property of chaos .   as a function of the parameter for two values of , calculated over iterations after neglecting iterates representing transient behavior for each value of .( a ) ; ( b ) .,title="fig : " ] +  as a function of the parameter for two values of , calculated over iterations after neglecting iterates representing transient behavior for each value of .( a ) ; ( b ) .,title="fig : " ]  this work was supported by consejo de desarrollo cientfico , humanstico y tecnolgico of the universidad de los andes , mrida , under grant no .c-1396 - 06 - 05-b and by fonacit , venezuela , under grant no . f-2002000426 .

Abstract: we consider a family of singular maps as an example of a simple model of dynamical systems exhibiting the property of robust chaos on a well defined range of parameters . critical boundaries separating the region of robust chaos from the region where stable fixed points exist are calculated on the parameter space of the system . it is shown that the transitions to robust chaos in these systems occur either through the routes of type - i or type - iii intermittency and the critical boundaries for each type of transition have been determined on the phase diagram of the system . the simplicity of these singular maps and the robustness of their chaotic dynamics make them useful ingredients in the construction of models and in applications that require reliable operation under chaos .


gravitational arc systems can be used as a powerful probe of the matter distribution of galaxies and galaxy clusters acting as lenses .further , their abundance can be used to constrain cosmological models .this motivated several arc searches to be carried out , both in wide field surveys ( * ? ? ?* ; * ? ? ?* ; * ? ? ?* ; * ? ? ?* ; * ? ? ?* ; * ? ? ?* ; * ? ? ?* ; * ? ? ?* ; * ? ? ?* ; * ? ? ?* ; * ? ? ?* erben et al .2012 , in prep . ) , as well as in images targeting know clusters . upcoming wide field imaging surveys , such as the dark energy survey ( des ; * ? ? ?* ; * ? ? ?* ) , which started operations in 2012 , are expected to detect of the order of strong lensing systems , about an order of magnitude increase with respect to the current largest surveys .two primary approaches have been followed in practical applications of gravitational arc systems . on the one hand ,_ inverse modelling _ attempts to `` deproject '' the arcs in individual lens systems to determine lens and source properties . on the other hand , _ arc statistics _ aims at counting the number of arcs in cluster samples and comparing with the predictions from cosmological models .both approaches require the lens equation to be solved numerically for finite sources numerous times .the inverse modelling typically needs arc images obtained from a multidimensional space of source positions and lens parameters scanned during the minimization process to find the best solution for sources and lenses ( e.g. , * ? ? ?for this reason analyses using the inverse modelling are often restricted to simple lens models , in particular models with elliptic lens potentials ( so - called pseudo - elliptical ) and/or to point sources , for example , considering bright spots in arcs as multiple images of point sources .  for arc statistics , predictions for the arc cross sectionmust be derived as a function of source and lens properties and the cosmological model , again by obtaining a large set of arc images .the cross section is then convolved with the distribution of lens properties expected in a given cosmology and convolved with the source distribution .another approach is to use directly high resolution n - body simulations obtaining arc images by ray - tracing through the mass distribution for a large number of sources .it is therefore useful to develop approximate methods for obtaining gravitational arcs , which will be particularly useful given the increase of strong lensing systems to be discovered by the next generation wide - field surveys . a most promising technique for this purpose is given by the perturbative approach , which provides an approximate solution for the lens equation close to the einstein ring , leading to analytic solutions for arcs .the power of this approach is that it can be applied , in principle , to generic lens models , including those arising from simulations .the method is suitable for large tangential arcs , since the solutions are accurate for images located close to the einstein ring corresponding to the circularly averaged lensing potential .another important feature of the method is that it naturally reproduces arcs resulting from the merger of multiple images , which can not be accounted for with other approximate methods for arcs proposed in the literature ( e.g. , * ? ? ?* ; * ? ? ?such merger arcs are key for lens inversion methods and also play an important role in the arc cross section .the perturbative approach has already been used for inverse modelling in .given that it reproduces arc contours that can be associated to isophotes , it could also be used to simulate the brightness distribution of arcs , in a similar way to what was implemented in for arc shaped contours .an important issue for practical applications of this approach is the determination of its domain of validity .this topic is discussed in , comparisons with arc simulations are presented in , and a recent work by has investigated arcs in several configurations for a pseudo - elliptical model in this approach . however a systematic study of its limit of applicability has not yet been carried out . in this paperwe make a first attempt to determine a domain of validity of the method in terms of the parameter space of the lens model .we will restrict to the simple case of pseudo - elliptical models , which are nevertheless widely used for the inverse modelling .moreover , for simplicity , we will restrict the comparisons with the exact solution for three quantities connected to arcs , but which do not involve the lensing of finite sources .we expect that the limits here obtained can be connected to the domain of validity for arcs and extended for more general models , but this is left to subsequent explorations .  in this work, our purpose is twofold .the first is to explore the application of the perturbative approach to determine quantities arising from the local lens mapping , such as the arc cross section for infinitesimal circular sources ( deformation cross section ) .the second is to determine a domain of validity such that the critical curves , caustics , and deformation cross section are accurately obtained .this study is performed for the pseudo - elliptical navarro frenk white model ( pnfw ) , determining regions of its parameter space where the perturbative approach provides a good approximation for these quantities .we also consider the singular isothermal elliptic potential ( siep ) model and show that the solution of the perturbative approach is exact in this case .the outline of this paper is as follows : in sec .[ bstgl ] we present a few basic results of gravitational lensing theory , introduce the radial lens models to be used in this work , and discuss models with elliptic lensing potentials . in sec .[ pert_appro_sec ] we review the perturbative approach , present its application to the computation of the deformation cross section , and discuss its implementation to pseudo - elliptical models . in sec .[ limits_validity ] we establish a metric for the comparison between the perturbative approach and the exact solution for critical curves and caustics and determine a domain of validity for the perturbative approach . in sec .[ conclud ] we summarize the results and present concluding remarks .in this section we present a brief review of the lensing theory to set up the notation and to define the quantities associated with pseudo - elliptical models . for a more detailed descriptionsee , e.g. , chap .8 of , chap .6 of and chap . 3 of .the _ lens equation _ relates the two - dimensional position ( with respect to the optical axis ) of the observed images to those of the sources .we may choose a a length scale and define and , with , where are the angular diameter distances from the observer to the lens and source respectively . using these definitions the lens equationis written as where is the `` dimensionless '' deflection angle and is the `` dimensionless '' _ lensing potential _ ,  the local distortion in the lens plane is described by the jacobian matrix of eq .( [ lens_eq_ad ] ) the two eigenvalues of this matrix are written as and , where and are the convergence and the shear given below .points satisfying the conditions define the radial and tangential critical curves respectively .mapping these curves onto the source plane , we obtain the caustics .  for axially symmetric models the deflection angle ,convergence and shear are given by , \label{conv_axial } \\\gamma(x)&=&\frac{1}{2}\left [ \frac{\alpha(x)}{x}-\frac{d\alpha(x)}{d  x}\right].\label{shear_axial}\end{aligned}\ ] ] where is the mean surface density within a radius and is the critical surface density .  in this work , one model we will make use of is the singular isothermal sphere ( sis ) , which is useful to model lenses at the galactic scale .its dimensionless lensing potential , deflection angle , convergence and shear are given by  where we choose the einstein radius to be the characteristic scale where is the velocity dispersion .from this potential analytic solutions of the lens equation can be obtained for finite sources .we will also make use of the navarro frenk white model ( * ? ? ?* ; * ? ? ?* hereafter nfw ) , often used to represent lenses in the galaxy to galaxy cluster mass scales .this model has two independent parameters and . by fixing and defining the characteristic convergence as the lensing potentialis given by  which is a function of the parameter alone .models with elliptical potentials ( the so - called pseudo - elliptical models ) provide simple analytical solutions for some lensing quantities .they have been widely used in lens inversion problems and are implemented in several public codes for lens inversion such as _ gravlens _ , _ lensview _ , _ lenstool _ , and _ glafic _they have also been used for arc simulations .pseudo - elliptical models , with potential , are built from a given axially symmetric potential by replacing the radial coordinate by where such that the ellipticity of the lensing potential is where the orientation was chosen such that the major axis of the ellipse is along the axis ( i.e. , ) .the deflection angle , convergence , and shear can be written as combinations of the lensing functions of the corresponding axially symmetric model for any choice of and  .the siep and pnfw models are obtained by following this procedure for the potentials given in eqs .( [ pot_sis ] ) and ( [ pot_nfw ] ) , respectively .for a given lens model , the perturbative approach allows one to obtain analytic solutions for arcs as perturbations of the einstein ring solution . in this workwe investigate the limits of applicability of the perturbative approach , by considering simple non - axially symmetric models and by looking at local properties of the lens mapping , instead of lensed finite sources .in this section we briefly review the perturbative approach and use it for the derivation of the caustics and critical curves , the deformation cross section and quantities needed for its computation .the method is also applied to models with elliptical lensing potentials .the gist of the perturbative approach for gravitational arcs developed by is to obtain an analytic solution for the lens equation considering the lens as a perturbation of an axially symmetric configuration and the source position as a small deviation from the optical axis ( i.e. , positioned transversely away from perfect observer lens source alignment ) . in other words ,the arcs are found as perturbations of the einstein ring configuration . in this workwe will consider the thin lens and the single lens plane approximations , which imply a unique solution for the einstein ring .the einstein ring is the image of a source aligned with an axially symmetric lens ( with lensing potential ) .its radius is obtained by solving the at the centre of the source plane , i.e. arcs can be obtained by perturbing the equation above either by shifting the position of the source away from the optical axis and/or by adding a non - circular perturbation to the lensing potential .these perturbations are described by these perturbations are assumed to be of the same order in ( the strength of the perturbation ) throughout the following calculations , such that the response to such perturbations is given by the displacement of the radial coordinate in the lens plane was used as a characteristic scale .this choice is equivalent to setting in our equations . in this workwe have made the choice of keeping explicitly in the equations for more generality , allowing us , for example , to choose another characteristic scale of the problem .] , i.e. , where we also assume the same order in such that .  to find we solve eq .( [ lens_eq_ad ] ) by expanding the solution around . expanding the lensing potential in a taylor series around , we have (\epsilon\overline{x})^n , \label{pot_expand}\ ] ] where {x = x_{\rm e } } , \nonumber \\ \label{fn - cn_def } \\f_n(\phi ) & \equiv & \frac{1}{n!}\left[\frac{\partial^n\delta \psi}{\partial\,x^n } \right]_{x = x_{\rm e}}=\frac{\epsilon}{n!}\left[\frac{\partial^n\overline{\varphi}}{\partial\,x^n } \right]_{x = x_{\rm e}}. \nonumber\end{aligned}\ ] ]  inserting and ( [ pot_expand ] ) into eq .( [ lens_eq_ad ] ) , we find that the resulting equation at zeroth order in is which is the einstein ring equation .using the relations above and , the resulting equation at the first order in is given by where . from eqs .( [ angle_axial])([shear_axial ] ) we have and therefore can be expressed as   eq . ( [ lens_eq_pa_1 ] ) is the lens equation in the perturbative approach .it can be solved for for each angular position of the source , given a perturbation described by . to obtain the images of a finite source , we must first parametrize its boundary . then , by varying from to , each point of that boundary is mapped to the lens plane through eqs .( [ lens_eq_pa_1 ] ) . as a result ,a new equation with separated radial and angular components is formed , whose solution is obtained straightforwardly   it is important emphasize that the solutions ( ) of eq .( [ lens_eq_pa_1 ] ) are valid only to first order in the perturbations in eq .( [ pert_approx_1 ] ) , i.e. only for points near the einstein ring . for pointsfar from this curve , the solutions are not expected to be highly accurate . for this reason, the perturbative approach is particularly useful for applications involving tangential arcs . in this work , instead of using finite sources , we focus on the potential applicability of this method to quantities based on the local mapping as a first step to quantify the differences with the exact solutions .the jacobian matrix for the lens mapping is where is the jacobian of the transformation from the lens plane to the source plane in polar coordinates from eq .( [ lens_eq_pa_1 ] ) and is the standard jacobian matrix from polar to cartesian coordinates .the calculation of the eigenvalues of the lens mapping is then straightforward from the equation above and they are given by   , \\\lambda_{r}=\kappa_2 . \end{array}\ ] ]  therefore , the radial coordinate of the tangential critical curve is and the parametric equations of the critical curve are simply inserting in eq .( [ lens_eq_pa_1 ] ) , the parametric equations of the tangential caustic are found to be    for infinitesimal circular sources , the length - to - width ratio of arcs can be approximated by the ratio of the eigenvalues of the lens mapping jacobian matrix   under this approximation , it is possible to define a region where gravitational arcs are expected to form by fixing a value for the threshold length to width ratio .such region is limited by the curves ( constant distortion curves ) .although the condition ( [ lw ] ) does not hold for merger arcs , nor for large or elliptical sources , the curves defined above still provide a typical scale for the region of arc formation . in this work ,we adopt the common choice ( unless explicitly stated otherwise ) .we denote the radial coordinates of these curves as .they are obtained by solving , with and given in the perturbative approach by eq .( [ eigv_1 - 2_pa ] ) .it follows that   the constant distortion curves in the lens plane are therefore self - similar to the tangential critical curve .the mapping of these curves to the source plane is done by substituting in eq .( [ lens_eq_pa_1 ] ) .for instance , the curve has the following parametric equations ,\\  \\y_{2\lambda}=\frac{f_1+\kappa_2 x_{\rm e } } { r_{\rm th}-1}\sin{\phi}+\frac{1}{x_{\rm e}}\left [ \left(\frac{r_{\rm th}}{r_{\rm th}-1}\right)\frac{d^2f_0}{d\phi}\sin{\phi}-\frac{d f_0}{d\phi}\cos{\phi}\right ] . \end{array}\ ] ]  the parametric equations of the curve are given by the expressions above with the substitution .there is no self - similarity between these curves and the tangential caustics .as mentioned in the introduction , the _ arc cross section _ is a key ingredient in arc statistics calculations .is is defined as the effective area in the source plane such that sources within it will be mapped into images with .the definition of this area must take into account the image multiplicity given the source position ( i.e. multiply - imaged regions are counted multiple times , see e.g. , * ? ? ?* ) . the computation of the arc cross section in general demands ray - tracing simulations , which are computationally expensive .an alternative is to use the infinitesimal circular source approximation , eq .( [ lw ] ) , which allows the computation of the arc cross section to be carried out directly from the local mapping from lens to source plane . in this case, is computed in the lens plane by ( see , e.g. , * ? ? ?* ; * ? ? ?* ; * ? ? ?* ) , where is the magnification and the integral is performed over the region of arc formation above the chosen threshold .the quantity is known as the dimensionless deformation cross section .  in the perturbative approach , the magnification can be written from eq .( [ eigv_1 - 2_pa ] ) as where is given in eq .( [ xcc_pm ] ) . inserting the equation above in eq .( [ dcs_dim ] ) and integrating the radial coordinate within the lower and upper limits given in eq .( [ x_l - mper ] ) , it is straightforward to obtain   note that for , as expected from the behaviour of the deformation cross section with  .  for axially symmetric models ( ) the cross section is given simply by   the expression above is exact for the sis model .for other axially symmetric models this expression is still an approximation , since the curves are obtained approximatively .we write the elliptical potential as ,\ ] ] such that the perturbed potential becomes from the definitions ( [ fn - cn_def ] ) and using the identities ( [ angle_axial])([shear_axial ] ) , it follows that ^ 2 , \nonumber\end{aligned}\ ] ] where and are the deflection angle and shear of the corresponding axially symmetric lens .these expressions hold for any parametrization of the lensing potential ellipticity and for any pseudo - elliptical lens .  for small values of the lensing potential ellipticity , eqs .( [ dfn_pe ] ) reduce to from eq .( [ xcc_pm ] ) and the expressions above , we have ,\label{xt_approx}\ ] ] and inserting this into eq ( [ sec - choq - pert - app2 ] ) we get .\ ] ] thus , for small ellipticities , the deviation with respect to the axially symmetric case is quadratic .  instead of using and is more intuitive to express the results in terms of the ellipticity of the potential .several parameterizations have been used to define the ellipticity in this context . from now on , we adopt the convention  where is the potential ellipticity parameter .the connection to the ellipticity of the mass distribution depends on the model . for the siep to first order in  . for the pnfw model this relation depends on and expressions for provided in .[ pnfw_ccurves_mp - se ] shows the comparison for caustics and critical curves between the perturbative approach and the exact solution for the pnfw model for different values of and .one of the simplest and most often used lens models is given by the siep .for this model , using expressions ( [ pot_sis ] ) in eq .( [ dfn_pe ] ) , the perturbative functions are where is given in eq .( [ deltaphi ] ) . when substituted into eqs .( [ lens_eq_pa_1 ] ) the expressions above lead to which are the components of the lens equation of this model without any approximation .hence , the solution of the perturbative approach is exact in the case of lensing by the siep model .the same conclusion does not hold for the pnfw model .we will thus investigate the domain of validity for this model in the next section .previous attempts to quantify the differences between exact and perturbative solutions were carried out in the literature . proposed a method based on the relative importance of the third - order term in the taylor series of the gravitational potential . performed a qualitative analysis of a particular arc configuration , varying some of the system parameters and establishing criteria based on the position and multiplicity of the images .however , they did not define a metric to compare the solutions nor carry on the analysis for more general configurations .investigating the domain of validity of the perturbative approach with finite sources would require a large parameter space to be probed , including the lens and source parameters and their relative positions .on the other hand , as a starting point , we may look at quantities that are dependent only on the lens , such as the tangential caustic and critical curve and the deformation cross section ( the latter will depend also on the choice of ) . besides reducing the parameter space for example , for and in the pnfw case it is simpler to define metrics to quantify the deviation of the perturbed solution from the exact one .we expect that the constraints on the domain of validity determined from the quantities above can be connected to those arising from the images of finite sources .thus , exploring the simplest case before may provide guidance to the determination of the domain of validity of the method finite sources in the future . setting a domain of validity from the lens model alonemay provide a rapid method to adjudicate validity of the perturbative method _ a priori _ , just from the lensing potential , without the need of obtaining images of the sources .  in this section , we shall attempt to quantify the deviation of critical curves and caustics using a figure - of - merit akin to the one proposed in .we will then compare the deformation cross sections and , finally , combine the results to obtain limits that define a region in the parameter space of pnfw models where the perturbative approach can be used to accurately obtain local properties of a given lens system .   to quantify the deviation of the solution of the perturbative approach from the exact one for critical curves and caustics we use a figure - of - merit defined asthe mean weighted squared fractional radial difference between the curves , i.e. ) is formally equal to the one proposed in , where it was used to compare an isocontour of to an ellipse . herethe same expression is used to compare two solutions for caustics or critical curves . ] ^ 2 }  { \sum_{i=1}^n w_i\ , x_{\rm es}^2(\phi_{i } ) } ,  \label{d2_pm}\ ] ] where and are the radial coordinates of the tangential curves ( either critical curves or caustics ) obtained from the exact solution and with the perturbative approach , respectively .these are computed on a discrete set of points defined by the polar angle .further , is a weight to account for a possible non - uniform distribution of points in .  choosing a cut - off value for , we can define a range in for which the curves obtained with both the exact and perturbative solutions will be similar enough to each other .the cut - off value is then chosen by visually comparing the exact and perturbative solutions for the critical curves and caustics associated with several values of , for combinations of the pnfw lens parameters .  before presenting the results, we should stress a technical point . in the particular case of caustics ,calculating the two functions in the same polar angle becomes a non - trivial issue .this is because in general , the source plane points ( ) are not equally distributed in angle , as they are obtained scanning angular values in the lens plane which map nonlinearly to angular values in the source plane .thus in general , a source plane angle does not correspond to the same lens plane angle . yet , to compute for the caustics , it is necessary that both and be calculated at the same polar angle position in the source plane .thus , to enforce this last point , we first determine the polar angle corresponding to each point ( ) obtained with the exact solution , i.e. , and obtain the corresponding radial coordinate . in the same way , we compute the polar angle of the tangential caustic obtained with the perturbative approach ( which we denote by ) , i.e., where and are given in eq .( [ ca_pm ] ) and .we then vary the angle ( only the interval is needed , for symmetry reasons ) such that for each radial position , the angles and are chosen to have the same value at step . finally , having determined ( ) for the exact solution and we proceed to compute as in eq .( [ d2_pm ] ) .[ d2_pnfw ] shows as a function of for some values of . ] . in the left panel ,the results for critical curves are shown .since the perturbation increases with , also increases with , as we might expect .in addition , decreases as increases , at least for . in the right panel, we show the results for caustics .the behaviour of is qualitatively similar to that of critical curves , except for at the highest , where the behaviours are reversed .however , the values of computed for caustics are higher than the corresponding ones for critical curves , for a given .this means that imposing cut - off values of for matching caustics , we will match the corresponding critical curves automatically .we found by visual inspection that for there is a very good match for the caustic curves . in fig .[ pnfw_ccurves_mp - se ] we show the values of calculated for each example , demonstrating visually the validity of this diagnostic measure . in particular, we have checked that cut - off values of higher but close to our chosen limit of are not suited for matching caustic curves well .  to estimate the validity of the perturbative approach , introduced the parameter , where corresponds to the difference between the _ arc contours _ obtained in the perturbative approach and the einstein radius , and is the third - order term in the taylor expansion of the gravitational potential ( see eq .[ pot_expand ] ) . in order for the perturbative approach to be accurate, should be small . for models based on the sis profile ,this condition is always true , since ( which is consistent with the fact that the perturbative method is exact in this case ) . for other pseudo - elliptical models , usually .here we adapt the definition of to be used for critical curves , such that is now the radial deviation of these curves with respect to .we associate a unique value of to the tangential critical curve , using its maximum value over this curve , which corresponds to where is given in eq .( [ xcc_pm ] ) and . following alard s criterion ( i.e. ), it would be expected that the critical curves and caustics obtained with the perturbative approach would be close to the ones obtained in the exact case when both and are small .we compute for the curves shown in fig .[ pnfw_ccurves_mp - se ] , obtaining , and from left to right panels .contrary to expectations , when increases , the curves obtained with the perturbative approach become more similar to the exact solutions .therefore , the criterion does not reflect the validity of the perturbative approach for these cases .moreover , is not scale - invariant ( i.e. , where is the length scale of the pnfw model ) .these considerations show that this measure is not well - suited to assess the limit of validity of the method for caustics and critical curves .this result emphasizes the relevance of our definition of as a measure for the validity of the perturbative approach for critical curves and caustics .  for the application of our criterion, we define , for a given , as the ellipticity threshold giving .this will be used as a measure of the limit of applicability for the perturbative approach for critical curves and caustics .[ emax_d2_pnfw ] shows the maximum values of as a function of for the pnfw model , for some cut - off values of .the function shown in this figure is well - fitted by a pad approximant of the form with and .   in this section, we compare the exact and perturbative solutions for the deformation cross section in order to establish limits of validity for the approximation of this quantity .we then contrast these limits to those obtained for caustics and critical curves as done in sec .[ sec - curvelimits ] ( i.e. by imposing for each ) .if within this regime the perturbative approach and the exact solution of the deformation cross section do not agree well , this can impose additional limits to the applicability of the perturbative approach .  to quantify the difference between the deformation cross sections , we compute their relative difference where the subscripts and refer to the exact and perturbative calculations , respectively .  in fig .[ dcs_pnfw - e - ks ] we show as a function of for some values of . in the left panelwe compare the exact solution with the expansion for low ellipticities in the perturbative approach ,( [ cross_section - approx ] ) , while in the right panel we compare with the general expression , eq .( [ sec - choq - pert - app2 ] ) . of the deviation from the axially symmetric case , eq .( [ cross_section - approx ] ) is an excellent approximation for eq .( [ sec - choq - pert - app2 ] ) for low ellipticities .thus the left panel of fig .[ dcs_pnfw - e - ks ] would remain essentially unchanged if we used eq .( [ sec - choq - pert - app2 ] ) instead of eq .( [ cross_section - approx ] ) there . ] the perturbative calculation for the axially symmetric nfw model ( , eq .[ sigma_mp - axial ] ) is a good approximation in this case , since for the entire allowed range of . for values of perturbative approach is a good approximation only for . as increases, the difference is larger at smaller values of .however , the perturbative calculation is accurate to within about for up to ( see the right panel of fig .[ dcs_pnfw - e - ks ] ) .  additionally , we computed as a function of the threshold .we find that can exceed 50% at values of , since for these values of , the _ constant distortion curves _ are far from the tangential critical curves , meaning that the premises of the perturbative approach do not apply .however , as increases , the relative deviations among the deformation cross sections decrease .in particular , we found that for and , these relative deviations do not depend on .  in fig .[ dcs_pnfw - space ] we show isocontours of , for the exact and perturbative calculations , in the parameter space together with the curve .we see that the constraints imposed by and are complementary , meaning that for the constraint obtained with caustics and critical curves is the strongest , while the opposite is true for if we impose that the maximum fractional deviation for the cross section is 10% .we may then combine the constraints to define a region limited approximately by the curves within this region the perturbative approach can replace the exact computation of critical curves , caustics , and deformation cross section with high accuracy .the perturbative approach provides analytical solutions for gravitational arcs by solving the lens equation linearised around the einstein ring solution .this method has a wide range of potential applications , from the inverse problem in strong lensing to fast arc simulations .this technique goes beyond other analytical approximations in the literature in that it may be used for generic lens models ( including mass distributions arising from n - body simulations ) and for finite sources .a key aspect for practical applications of the method that has not been systematically addressed before is the determination of its limit of validity .motivated by this issue , in this paper we aimed to determine the accuracy of the perturbative approach for caustics and critical curves and for the deformation arc cross section .although these quantities do not involve arcs ( i.e. the lensing of finite sources ) they allow one to obtain limits on the accuracy of the linearized mapping from the perturbative approach . also , the parameter space to be probed is significantly decreased , since these quantities depend basically on the lens properties and not the source ones .we have considered a restricted set of lens models , more specifically those with elliptical lens potentials , and in particular the pnfw and siep models , which are nevertheless widely used in strong lensing applications , specially for the inverse modelling .whenever possible , we sought to derive analytic expressions for the quantities involved in the calculations , many of which are new .some are valid for the perturbative approach in general , others apply to pseudo - elliptic lens models .the main results of the paper are summarized below .we obtained analytic expressions for the constant distortion curves in the perturbative approach ( eqs .[ x_l - mper ] and [ x_l_src_mper ] ) , which , in the lens plane , are found to be self - similar to the tangential critical curve .we derived an analytic formula for the deformation cross section ( eq . [ sec - choq - pert - app2 ] ) , which reproduces the scaling of the arc cross section with obtained numerically in previous works . foraxially symmetric models the cross section is obtained in closed form ( eq . [ sigma_mp - axial ] ) .we have obtained simple analytic expressions for the perturbative functions for pseudo - elliptical models , which are valid for any choice of the ellipticity parametrization ( eq . [ dfn_pe ] ) .these expressions generalize those given in and in .we derive approximate solutions to the tangential critical curve ( eq .[ xt_approx ] ) and for the deformation cross section ( eq . [ cross_section - approx ] ) for low ellipticities in pseudo - elliptical models .we show that the deviation of the cross section with respect to the axially symmetric case is quadratic in the ellipticity .we have considered the siep and the pnfw models to represent lenses at galaxy and galaxy cluster mass scales .we have shown that the perturbative approach provides the exact solution for the siep model .for the pnfw model , we compared the critical curves and caustics obtained with this approach with those obtained with the exact solution for a wide range of values of and .we show that the criterion proposed by extended to be applied the tangential critical curve ( eq . [ dmax_pnfw ] ) , is not adequate to set a limit of validity for these cases . to this end, we use a figure - of - merit , ( eq .[ d2_pm ] ) to quantify the deviation of the perturbative approach from the exact solution for caustics and critical curves .we verify that provides a quantitative description of the deviation among both solutions .in particular , decreases with ( as can be drawn from fig .[ pnfw_ccurves_mp - se ] ) and increases with ( as expected from the increasing of the perturbation to the lensing potential with ) . since the deviation between the exact and perturbative solutions for caustics is higher than the deviation for critical curves , it is sufficient to set a limit on for caustics to ensure a small deviation for critical curves .  by setting a threshold on computed at caustics ,a maximum value of is determined for each , such that a good matching for caustics and also for critical curves is ensured .we determine these maximum values by choosing .this defines a domain of applicability of the perturbative approach for the pnfw model in the range of being considered .we provide a fitting function for ( eq . [ emax_pm_fit ] ) . for ,the perturbative approach is limited to .however , for it is possible to use this approach even up to for these cases .another limit on the pfnw model parameters is obtained from the comparison of the deformation cross section for both exact and perturbative calculations .the fractional deviation is less than 10% ( fig .[ dcs_pnfw - space ] ) for and ( corresponding to ) .we may use these results to set further constraints on the ellipticity parameter of the pnfw model , by requiring an agreement with the exact , besides the condition .this ensures that caustics , critical curves , and the local mapping are well reproduced by the perturbative approach for the pnfw model .the combined restriction , imposing the matching for caustics and an agreement to about 10% for deformation cross sections , is given in eq .( [ region_lt-10 ] ) .  in this paper we provided a first systematic attempt to set limits on the domain of applicability of the perturbative approach for strong lensing in terms of the parameters of a given lens model , more specifically for the pnfw model .the limits are imposed so that the caustics , critical curves and deformation cross section match the exact solutions with a given accuracy .although these quantities are useful for strong lensing applications , it is important to determine a domain of validity for arcs / finite sources .for example , investigated the domain of validity of the perturbative approach for extended circular sources .it is argued that perturbative approach can be used for sources with radius up to .this result should be extended for generic configurations probing the space of the source and lens parameters and their relative position .we expect that the limits here obtained can be connected to the domain of validity for arcs providing guidance to the exploration of this wider parameter space .the systematic application to arcs and connection to the current results is left for a subsequent work .it is also important to check whether the criterion established here for the threshold can be applied to other lens models , so that we have an _ a priori _ criterium for the domain of validity of the perturbative approach regardless of the specific model .the usefulness of the perturbative approach justifies the search for a determination its accuracy and limit of applicability .once this is established we will be able to safely use this promising technique in a number of applications , within its domain of validity .hsdm is funded by the brazilian agencies faperj ( e-26/101.784/2010 ) , cnpq ( pdj/162989/2011 - 3 ) , and the pci / mcti program at cbpf ( 301.860/2011 - 4 ) .gbc is funded by cnpq and capes .bm is supported by faperj .mm is partially supported by cnpq ( grant 312876/2009 - 2 ) and faperj ( grant e-26/110.516/2012 ) . mssg acknowledges the hospitality of the centro brasileiro de pesquisas fsicas ( cbpf ) , where part of this work was performed , and the pci / mcti program ( 170.524/2006 - 0 ) .we also thank the support of the laboratrio interinstitucional de e - astronomia ( linea ) operated jointly by cbpf , the laboratrio nacional de computao cientfica ( lncc ) , and the observatrio nacional ( on ) and funded by the ministry of science , technology and innovation ( mcti ) .mm acknowledges c. alard for useful discussions regarding the perturbative approach for arcs .we thank marcos lima for useful discussions .

Abstract: the perturbative approach ( pa ) introduced by provides analytic solutions for gravitational arcs by solving the lens equation linearized around the einstein ring solution . this is a powerful method for lens inversion and simulations in that it can be used , in principle , for generic lens models . in this paper we aim to quantify the domain of validity of this method for three quantities derived from the linearized mapping : caustics , critical curves , and the deformation cross section ( i.e. the arc cross section in the infinitesimal circular source approximation ) . we consider lens models with elliptical potentials , in particular the singular isothermal elliptic potential and pseudo - elliptical navarro  frenk white models . we show that the pa is exact for this first model . for the second , we obtain constraints on the model parameter space ( given by the potential ellipticity parameter and characteristic convergence ) such that the pa is accurate for the aforementioned quantities . in this process we obtain analytic expressions for several lensing functions , which are valid for the pa in general . the determination of this domain of validity could have significant implications for the use of the pa , but it still needs to be probed with extended sources .   [ firstpage ]  galaxies : cluster : general , cosmology : dark matter , gravitational lensing : strong .