Homoclinic bifurcations

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

$$ \tag{a1 } {\dot{x} } = f ( x, \alpha ) , \quad x \in \mathbf R ^ {n} , \alpha \in \mathbf R ^ {1} , $$

where $ f $ is smooth. Denote by $ \varphi _ \alpha ^ {t} $ the flow (continuous-time dynamical system) corresponding to (a1). Let $ x _ {0} $ be an equilibrium of the system at $ \alpha = 0 $. An orbit $ \Gamma _ {0} $ starting at a point $ x \in \mathbf R ^ {n} $ is called homoclinic to the equilibrium point $ x _ {0} $ of (a1) at $ \alpha = 0 $ if $ \varphi _ {0} x ^ {t} \rightarrow x _ {0} $ as $ t \rightarrow \pm \infty $. Generically, presence of a homoclinic orbit at $ \alpha = 0 $ implies a global codimension-one bifurcation of (a1), since the homoclinic orbit disappears for all sufficiently small $ | \alpha | > 0 $. 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 $ \Gamma _ {0} $ as $ | \alpha | \rightarrow 0 $, its period tends to infinity. In some cases, there are infinitely many limit cycles in a neighbourhood of $ \Gamma _ {0} $ for sufficiently small $ | \alpha | $, since the Poincaré return map near the homoclinic orbit demonstrates Smale's horseshoes [a14] and their associated shift dynamics.

First, consider the case when $ x _ {0} $ is an hyperbolic equilibrium, i.e. the Jacobian matrix $ A = f _ {x} ( x _ {0} ,0 ) $ has no eigenvalues on the imaginary axis. Suppose that $ A $ has $ n _ {u} $ eigenvalues with positive real part

$$ 0 < { \mathop{\rm Re} } \lambda _ {1} \leq \dots \leq { \mathop{\rm Re} } \lambda _ {n _ {u} } $$

and $ n _ {s} $ eigenvalues with negative real part

$$ { \mathop{\rm Re} } \mu _ {n _ {s} } \leq \dots \leq { \mathop{\rm Re} } \mu _ {1} < 0 $$

( $ n _ {s} + n _ {u} = n $). The equilibrium $ x _ {0} $ has unstable and stable invariant manifolds $ W ^ {u} ( x _ {0} ) $ and $ W ^ {s} ( x _ {0} ) $ composed by outgoing and incoming orbits, respectively; $ { \mathop{\rm dim} } {W ^ {u,s } ( x _ {0} ) } = n _ {u,s } $.

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 $ x _ {0} $ as $ t \rightarrow - \infty $( $ t \rightarrow + \infty $) 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 $ \sigma _ {0} $ of a hyperbolic equilibrium is the sum of the real parts of its leading eigenvalues: $ \sigma _ {0} = { \mathop{\rm Re} } \lambda _ {1} + { \mathop{\rm Re} } \mu _ {1} $, where $ \lambda _ {1} $ is a leading unstable eigenvalue and $ \mu _ {1} $ 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: $ \mu _ {1} < 0 < \lambda _ {1} $;

b) (saddle-focus) the stable leading eigenvalues are non-real and simple: $ \mu _ {1} = {\overline \mu \; } _ {2} $, $ { \mathop{\rm Re} } \mu _ {1,2 } < 0 $, while the unstable leading eigenvalue $ \lambda _ {1} > 0 $ is real and simple;

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

$$ \lambda _ {1} = {\overline \lambda \; } _ {2} , \mu _ {1} = {\overline \mu \; } _ {2} , { \mathop{\rm Re} } \mu _ {1,2 } < 0 < { \mathop{\rm Re} } \lambda _ {1,2 } . $$

The following theorems by A.A. Andronov and E.A. Leontovich [a1] (in the saddle case when $ n = 2 $) 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 $ x _ {0} $ with a homoclinic orbit $ \Gamma _ {0} $ at $ \alpha = 0 $, there exists a neighbourhood $ U _ {0} $ of $ \Gamma _ {0} \cup x _ {0} $ in which a unique limit cycle $ L _ \alpha $ bifurcates from $ \Gamma _ {0} $ as $ \alpha $ passes through zero. Moreover, $ { \mathop{\rm dim} } {W ^ {s} ( L _ \alpha ) } = n _ {s} + 1 $ if $ \sigma _ {0} < 0 $, and $ { \mathop{\rm dim} } {W ^ {s} ( L _ \alpha ) } = n _ {s} $ if $ \sigma _ {0} > 0 $.

(Saddle-focus) For any generic one-parameter system (a1) having a saddle-focus equilibrium point $ x _ {0} $ with a homoclinic orbit $ \Gamma _ {0} $ at $ \alpha = 0 $, there exists a neighbourhood $ U _ {0} $ of $ \Gamma _ {0} \cup x _ {0} $ such that one of the following alternatives holds:

a) if $ \sigma _ {0} < 0 $, a unique limit cycle $ L _ \alpha $ bifurcates from $ \Gamma _ {0} $ in $ U _ {0} $ as $ \alpha $ passes through zero, $ { \mathop{\rm dim} } {W ^ {s} ( L _ \alpha ) } = n _ {s} + 1 $;

b) if $ \sigma _ {0} > 0 $, the system has an infinite number of saddle limit cycles in $ U _ {0} $ for all sufficiently small $ | \alpha | $.

(Focus-focus) For any generic one-parameter system (a1) having a focus-focus equilibrium point $ x _ {0} $ with a homoclinic orbit $ \Gamma _ {0} $ at $ \alpha = 0 $, there exists a neighbourhood $ U _ {0} $ of $ \Gamma _ {0} \cup x _ {0} $ in which the system has an infinite number of saddle limit cycles in $ U _ {0} $ for all sufficiently small $ | \alpha | $.

The genericity conditions mentioned above have some common parts:

1) the leading eigenspaces are either one- or two-dimensional and $ \sigma _ {0} \neq 0 $;

2) $ \Gamma _ {0} $ tends to $ x _ {0} $ as $ t \rightarrow \pm \infty $ along the leading eigenspaces;

3) the intersection of the tangent spaces to $ W ^ {s} ( x _ {0} ) $ and $ W ^ {u} ( x _ {0} ) $ at each point on $ \Gamma _ {0} $ is one-dimensional;

4) $ W ^ {s} ( x _ \alpha ) $ and $ W ^ {u} ( x _ \alpha ) $ split by an $ O ( \alpha ) $ distance as $ \alpha $ moves away from zero, where $ x _ \alpha $ is the continuation of $ x _ {0} $ for small $ | \alpha | > 0 $.

There is also a case-dependent non-degeneracy condition dealing with the global topology of $ W ^ {s} ( x _ {0} ) $ and $ W ^ {u} ( x _ {0} ) $ around $ \Gamma _ {0} $ at $ \alpha = 0 $. The exact formulation of this condition can be found in [a2]. In the planar case ( $ n = 2 $), only conditions 1) and 4) are relevant.

Suppose now that $ x _ {0} $ is a non-hyperbolic equilibrium of (a1) at $ \alpha = 0 $, having a homoclinic orbit $ \Gamma _ {0} $. Only the case when $ x _ {0} $ has a simple zero eigenvalue and no other eigenvalues on the imaginary axis (i.e., $ x _ {0} $ 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 $ \Gamma _ {0} $, then, generically, a unique limit cycle bifurcates from $ \Gamma _ {0} $, 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 $ x _ {0} $ on the complex plane. If the saddle-node has more than two homoclinic orbits, $ \Gamma _ {1} \dots \Gamma _ {m} $, then, generically, infinitely many saddle limit cycles appear from $ \Gamma _ {1} \cup \dots \cup \Gamma _ {m} $, when the equilibrium $ x _ {0} $ disappears. The genericity conditions include the non-degeneracy of the underlying fold bifurcation, as well as the requirement that $ \Gamma _ {0} $ departs and returns to the saddle-node along the null-vector of the Jacobian matrix evaluated at $ x _ {0} $ for $ \alpha = 0 $. The two-dimensional case has been treated in [a1]. The cases with $ n > 2 $ 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].

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