Homoclinic bifurcations

Consider an autonomous system of ordinary differential equations depending on a parameter

(a1)

where is smooth. Denote by the flow (continuous-time dynamical system) corresponding to (a1). Let be an equilibrium of the system at . An orbit starting at a point is called homoclinic to the equilibrium point of (a1) at if as . Generically, presence of a homoclinic orbit at implies a global codimension-one bifurcation of (a1), since the homoclinic orbit disappears for all sufficiently small . Moreover, the disappearance of a homoclinic orbit leads to the creation or destruction of one (or more) limit cycle nearby. When such a cycle approaches the homoclinic orbit as , its period tends to infinity. In some cases, there are infinitely many limit cycles in a neighbourhood of for sufficiently small , since the Poincaré return map near the homoclinic orbit demonstrates Smale's horseshoes [a14] and their associated shift dynamics.

First, consider the case when is an hyperbolic equilibrium, i.e. the Jacobian matrix has no eigenvalues on the imaginary axis. Suppose that has eigenvalues with positive real part

and eigenvalues with negative real part

(). The equilibrium has unstable and stable invariant manifolds and composed by outgoing and incoming orbits, respectively; .

The eigenvalues with positive (negative) real part that are closest to the imaginary axis are called the unstable (stable) leading eigenvalues, while the corresponding eigenspaces are called the unstable (stable) leading eigenspaces. Almost all orbits on the stable and unstable manifolds tend to the equilibrium as () along the corresponding leading eigenspace. Exceptional orbits form a non-leading manifold tangent to the eigenspace corresponding to the non-leading eigenvalues.

The saddle quantity of a hyperbolic equilibrium is the sum of the real parts of its leading eigenvalues: , where is a leading unstable eigenvalue and is a leading stable eigenvalue. Generically, leading eigenspaces are either one- or two-dimensional. In the first case, an eigenspace corresponds to a simple real eigenvalue, while in the second case it corresponds to a simple pair of complex-conjugate eigenvalues. Reversing the time direction, if necessary, one has only three typical configurations of the leading eigenvalues:

a) (saddle) the leading eigenvalues are real and simple: ;

b) (saddle-focus) the stable leading eigenvalues are non-real and simple: , , while the unstable leading eigenvalue is real and simple;

c) (focus-focus) the leading eigenvalues are non-real and simple:

The following theorems by A.A. Andronov and E.A. Leontovich [a1] (in the saddle case when ) and L.P. Shil'nikov (otherwise) [a11], [a13] are valid (see also [a15], [a2], [a5]).

(Saddle) For any generic one-parameter system (a1) having a saddle equilibrium point with a homoclinic orbit at , there exists a neighbourhood of in which a unique limit cycle bifurcates from as passes through zero. Moreover, if , and if .

(Saddle-focus) For any generic one-parameter system (a1) having a saddle-focus equilibrium point with a homoclinic orbit at , there exists a neighbourhood of such that one of the following alternatives holds:

a) if , a unique limit cycle bifurcates from in as passes through zero, ;

b) if , the system has an infinite number of saddle limit cycles in for all sufficiently small .

(Focus-focus) For any generic one-parameter system (a1) having a focus-focus equilibrium point with a homoclinic orbit at , there exists a neighbourhood of in which the system has an infinite number of saddle limit cycles in for all sufficiently small .

The genericity conditions mentioned above have some common parts:

1) the leading eigenspaces are either one- or two-dimensional and ;

2) tends to as along the leading eigenspaces;

3) the intersection of the tangent spaces to and at each point on is one-dimensional;

4) and split by an distance as moves away from zero, where is the continuation of for small .

There is also a case-dependent non-degeneracy condition dealing with the global topology of and around at . The exact formulation of this condition can be found in [a2]. In the planar case (), only conditions 1) and 4) are relevant.

Suppose now that is a non-hyperbolic equilibrium of (a1) at , having a homoclinic orbit . Only the case when has a simple zero eigenvalue and no other eigenvalues on the imaginary axis (i.e., is a saddle-node, cf. Saddle node) appears in generic one-parameter families (has codimension-one). If the saddle-node has a single homoclinic orbit , then, generically, a unique limit cycle bifurcates from , when the saddle-node disappears via the fold bifurcation. The cycle can be either attracting/repelling or saddle type, depending on the location of the non-zero eigenvalues of on the complex plane. If the saddle-node has more than two homoclinic orbits, , then, generically, infinitely many saddle limit cycles appear from , when the equilibrium disappears. The genericity conditions include the non-degeneracy of the underlying fold bifurcation, as well as the requirement that departs and returns to the saddle-node along the null-vector of the Jacobian matrix evaluated at for . The two-dimensional case has been treated in [a1]. The cases with were considered by Shil'nikov [a9], [a12] and presented in [a2], [a5].

In generic discrete-time dynamical systems defined by iterations of diffeomorphisms, orbits which are homoclinic to a hyperbolic fixed point persist under small parameter variations. Stable and unstable manifolds of the fixed point intersect transversally along the homoclinic orbits, implying the existence of the Poincaré homoclinic structure with infinitely many saddle periodic orbits [a14], [a7], [a10], [a8], [a6]. The homoclinic structure appears/disappears via a non-transversal homoclinic bifurcation, when the stable and the unstable manifolds of the fixed point become tangent along the homoclinic orbit [a3], [a4], [a16].

References

[a1]	A.A. Andronov, E.A. Leontovich, I.I. Gordon, A.G. Maier, "Theory of bifurcations of dynamical systems on a plane" , Israel Program Sci. Transl. (1971) (In Russian)
[a2]	V.I. Arnol'd, V.S. Afraimovich, Yu.S. Il'yashenko, L.P. Shil'nikov, "Bifurcation theory" V.I. Arnol'd (ed.) , Dynamical Systems V , Encycl. Math. Sci. , Springer (1994) (In Russian)
[a3]	N.K. Gavrilov, L.P. Shilnikov, "On three-dimensional systems close to systems with a structurally unstable homoclinic curve: I" Mat. USSR Sb. , 17 (1972) pp. 467–485 (In Russian)
[a4]	N.K. Gavrilov, L.P. Shilnikov, "On three-dimensional systems close to systems with a structurally unstable homoclinic curve: II" Mat. USSR Sb. , 19 (1973) pp. 139–156 (In Russian)
[a5]	Yu.A. Kuznetsov, "Elements of applied bifurcation theory" , Springer (1995)
[a6]	J. Moser, "Stable and random motions in dynamical systems" , Princeton Univ. Press (1973)
[a7]	Yu.I. Neimark, "On motions close to a bi-asymptotic motion" Dokl. AKad. Nauk SSSR , 142 (1967) pp. 1021–1024 (In Russian)
[a8]	Z. Nitecki, "Differentiable dynamics" , MIT (1971)
[a9]	L.P. Shil'nikov, "On the generation of a periodic motion from a trajectory which leaves and re-enters a saddle-saddle state of equilibrium" Soviet Math. Dokl. , 7 (1966) pp. 1155–1158 (In Russian)
[a10]	L.P. Shil'nikov, "On a Poincaré–Birkhoff problem" Mat. USSR Sb. , 3 (1967) pp. 353–371 (In Russian)
[a11]	L.P. Shil'nikov, "On the generation of periodic motion from trajectories doubly asymptotic to an equilibrium state of saddle type" Mat. USSR Sb. , 6 (1968) pp. 427–437 (In Russian)
[a12]	L.P. Shil'nikov, "On a new type of bifurcation of multidimensional dynamical systems" Soviet Math. Dokl. , 10 (1969) pp. 1368–1371 (In Russian)
[a13]	L.P. Shil'nikov, "A contribution to the problem of the structure of an extended neighborhood of a rough equilibrium state of saddle-focus type" Mat. USSR Sb. , 10 (1970) pp. 91–102 (In Russian)
[a14]	S. Smale, "Differentiable dynamical systems" Bull. Amer. Math. Soc. , 73 (1967) pp. 747–817
[a15]	S. Wiggins, "Global bifurcations and chaos" , Springer (1988)
[a16]	S. Wiggins, "Introduction to applied non-linear dynamical systems and chaos" , Springer (1990)