Melnikov distance

In mathematics, the Melnikov method is a tool to identify the existence of chaos in a class of dynamical systems under periodic perturbation.
Introduction.
The Melnikov method is used in many cases to predict the occurrence of chaotic orbits in non-autonomous smooth nonlinear systems under periodic perturbation. According to the method is possible to construct a function called "Melnikov function", and hence to predict either regular or chaotic behavior of a studied dynamical system. Thus, the Melnikov function will be used to determine a measure of distance between stable and unstable manifolds in the Poincaré map. Moreover, when this measure is equal to zero, by the method, those manifolds crossed each other transversally and from that crossing the system will become chaotic.
This method appeared in 1890 by H. Poincaré and by V. Melnikov in 1963 and could be called the "Poincaré-Melnikov Method". Moreover, it was described by several textbooks as Guckenheimer & Holmes, Kuznetsov, S. Wiggins, Awrejcewicz & Holicke and others. There are many applications for Melnikov distance as it can be used to predict chaotic vibrations. In this method, critical amplitude is found by setting the distance between homoclinic orbits and stable manifolds equal to zero. Just like in Guckenheimer & Holmes where they were the first who based on the KAM theorem, determined a set of parameters of relatively weak perturbed Hamiltonian systems of two-degrees-of-freedom, at which homoclinic bifurcation occurred.
The Melnikov distance.
Consider the following class of systems given by
formula_1or in vector formformula_2
where formula_3, formula_4, formula_5 and
formula_6
Assume that system (1) is smooth on the region of interest, formula_7 is a small perturbation parameter and formula_8 is a periodic vector function in formula_9with the period formula_10.
If formula_11, then there is an unperturbed system
formula_12
From this system (3), looking at the phase space in Figure 1, consider the following assumptions
To obtain the Melnikov function, some tricks have to be used, for example, to get rid of the time dependence and to gain geometrical advantages new coordinate has to be used formula_20 that is cyclic type given by formula_21Then, the system (1) could be rewritten in vector form as follows
formula_22
Hence, looking at Figure 2, the three-dimensional phase space formula_23where formula_24and formula_25has the hyperbolic fixed point formula_26of the unperturbed system becoming a periodic orbit formula_27 The two-dimensional stable and unstable manifolds of formula_28by formula_29and formula_30 are denoted, respectively. By the assumption formula_31 formula_29 and formula_30coincide along a two-dimensional homoclinic manifold. This is denoted by formula_34where formula_35 is the time of flight from a point formula_36to the point formula_37on the homoclinic connection.
In the Figure 3, for any point formula_38 a vector is constructed formula_39, normal to the formula_40as follows formula_41Thus varying formula_35and formula_43serve to move formula_39to every point on formula_45
Splitting of stable and unstable manifolds.
If formula_46 is sufficiently small, which is the system (2), then formula_47 becomes formula_48 formula_40 becomes formula_50 and the stable and unstable manifolds become different from each other. Furthermore, for this sufficiently small formula_7in a neighborhood formula_52 the periodic orbit formula_47of the unperturbed vector field (3) persists as a periodic orbit, formula_54 Moreover, formula_55 and formula_56 are formula_57 formula_7-close to formula_59 and formula_60 respectively.
Consider the following cross-section of the phase space formula_61 then formula_62 and formula_63 are the trajectories of the
unperturbed and perturbed vector fields, respectively. The projections of these trajectories onto formula_64are given by formula_65 and formula_66 Looking at the Figure 4, splitting of formula_67 and formula_68 is defined hence, consider the points that intersect formula_39 transversely as formula_70 and formula_71, respectively. Therefore, it is natural to define the distance between formula_67 and formula_73 at the point formula_74 denoted by formula_75and it can be rewritten as formula_76 Since formula_70and formula_71lie on formula_79and formula_80 and then formula_81 can be rewritten by 
formula_22
The manifolds formula_67 and formula_73 may intersect formula_39 in more than one point as shown in Figure 5. For it to be possible, after every intersection, for formula_7 sufficiently small, the trajectory must pass through formula_87 again.
Deduction of the Melnikov function.
Expanding in Taylor series the eq. (5) about formula_88 gives us formula_89 where formula_90 and formula_91
When formula_92 then the Melnikov function is defined to be
formula_93
since formula_94is not zero on formula_36, considering formula_35finite and formula_97 
Using eq. (6) it will require knowing the solution to the perturbed problem. To avoid this, Melnikov defined a time dependent Melnikov function 
formula_98 
Where formula_99 and formula_100 are the trajectories starting at formula_101 and formula_102 respectively. Taking the time-derivative of this function allows for some simplifications. The time-derivative of one of the terms in eq. (7) isformula_2
From the equation of motion, formula_104
then
formula_105Plugging equations (2) and (9) back into (8) gives
formula_106
The first two terms on the right hand side can be verified to cancel by explicitly evaluating the matrix multiplications and dot products. formula_107 has been reparameterized to formula_108.
Integrating the remaining term, the expression for the original terms does not depend on the solution of the perturbed problem.
formula_98
The lower integration bound has been chosen to be the time where formula_110, so that formula_111 and therefore the boundary terms are zero. 
Combining these terms and setting formula_112 the final form for the Melnikov distance is obtained by
formula_98
Then, using this equation, the following theorem
Theorem 1: Suppose there is a point formula_114such that
Then, for formula_7 sufficiently small, formula_67 and formula_73 intersect transversely at formula_120 Moreover, if formula_121 for all formula_122, then formula_123
Simple zeros of the Melnikov function imply chaos.
From theorem 1 when there is a simple zero of the Melnikov function implies in transversal intersections of the stable formula_67and formula_73manifolds that results in a homoclinic tangle. Such tangle is a very complicated structure with the stable and unstable manifolds intersecting an infinite number of times.
Consider a small element of phase volume, departing from the neighborhood of a point near the transversal intersection, along the unstable manifold of a fixed point. Clearly, when this volume element approaches the hyperbolic fixed point it will be distorted considerably, due to the repetitive infinite intersections and stretching (and folding) associated with the relevant invariant sets. Therefore, it is reasonably expect that the volume element will undergo an infinite sequence of stretch and fold transformations as the horseshoe map. Then, this intuitive expectation is rigorously confirmed by a theorem stated as follows
Theorem 2: Suppose that a diffeomorphism formula_126where formula_127is an n-dimensional manifold, has a hyperbolic fixed point formula_128with a stable formula_129and formula_130unstable manifold that intersect transversely at some point formula_131, formula_132where formula_133 Then, formula_127contains a hyperbolic set formula_135 invariant under formula_136on which formula_137is topologically conjugate to a shift on finitely many symbols.
Thus, according to the theorem 2, it implies that the dynamics with a transverse homoclinic point is topologically similar to the horseshoe map and it has the property of sensitivity to initial conditions and hence when the Melnikov distance (10) has a simple zero, it implies that the system is chaotic.