Homoclinic orbit

Homoclinic orbit with hyperbolic fixed point

In the mathematics of dynamic systems ( autonomous differential equation systems ), a homoclinic orbit is a trajectory (orbit) that starts from a hyperbolic fixed point (saddle point) and leads back to it. While homoclinic orbits lead back from a fixed point to this, heteroclinic orbits run between two different fixed points, which can also be saddle points.

At a hyperbolic fixed point there are associated stable manifolds whose orbits converge towards it, and unstable manifolds in which the points of an orbit approach the fixed point. The dimension of these manifolds is based on the number of positive and negative real parts of eigenvalues when the differential equation is linearized around the fixed point. In two dimensions you have a curve as an orbit, which runs from the hyperbolic fixed point to this; in more dimensions you also consider families of solution curves and the fixed point can e.g. B. also be a closed path. With the Poincaré cut ${\ displaystyle t \ to \ infty}$ ${\ displaystyle t \ to - \ infty}$ (the intersection of the orbits with a surface perpendicular to the phase space flow) this can be reduced to a two-dimensional view: the periodic orbit (period ) is a fixed point in the Poincaré section, which asymptotically approaching the periodic orbit in its vicinity approach in the Poincare - Intersection of the fixed point at the stable manifold when considering the flow (the iteration of the Poincaré mapping in the distance ) for and at the unstable manifold for . ${\ displaystyle T}$ ${\ displaystyle T}$ ${\ displaystyle t \ to \ infty}$ ${\ displaystyle t \ to - \ infty}$

Formally, a homoclinic orbit can be defined in this way. Let f be a diffeomorphism of a compact, unbounded manifold M and let p be a fixed point of f ( ). The orbit of a point is homoclinic if . If one approaches two different fixed points for and , one speaks of a heteroclinic orbit. If one considers two dimensions and let p be a hyperbolic fixed point, then the points that move away from p while iterating f lie on an invariant curve (unstable manifold) and the points approaching p (stable manifold) and homoclinic points q lie on both curves ( ). The homoclinic point q is called transversal if it intersects transversely in q. With q is also homoclinic. ${\ displaystyle f (p) = p}$ ${\ displaystyle f ^ {n} (q)}$ ${\ displaystyle q \ neq p}$ ${\ displaystyle lim_ {n \ to \ pm \ infty} f ^ {n} (q) = p}$ ${\ displaystyle n \ to \ infty}$ ${\ displaystyle n \ to - \ infty}$ ${\ displaystyle W ^ {-} (p)}$ ${\ displaystyle W ^ {+} (p)}$ ${\ displaystyle q \ in W ^ {+} (p) \ cap W ^ {-} (p) - \ {p \}}$ ${\ displaystyle W ^ {\ pm} (p)}$ ${\ displaystyle f ^ {n} (q)}$

Henri Poincaré found in 1888 that the dynamic system can show a very complex, chaotic behavior in the vicinity of the fixed point, if unstable and stable manifolds intersect transversely on the homoclinic orbit during disturbances (i.e. the stable and unstable orbit does not meet there tangentially ). He had previously assumed that the homoclinic orbits of unstable and stable manifolds would form a connected manifold, thus finding an integration invariant and proving the stability of the simplified model of the three-body problem that he was investigating. When asked by the appraiser of the award work he had submitted in Sweden, however, he found that he had overlooked the possibility of transversal cutting. The picture was now completely different: the unstable manifold intersected the stable one near the fixed point infinitely often and near the intersection point this led to very chaotic behavior. The same applied to the other branch of the homoclinic orbit, where the stable manifold intersected the unstable one infinitely often. Poincaré called this a homoclinic network (English: homoclinic tangle, in the physics literature stochastic layers) and described it using the Poincaré cut near the fixed point.

The transverse intersection of the stable and unstable manifold is called the transverse homoclinic point (according to the theorem of Kupka and Smale, this is the typical case). Since stable and unstable manifolds of the hyperbolic point are invariant under forward and backward iteration with a Poincaré map, if there is a homoclinic point, there are infinitely many. In the forward and backward iteration, one lies on both the stable and the unstable manifold and that infinitely often. On the other hand, the unstable manifold cannot intersect itself (and analogously the stable one) because of the uniqueness of the solution of the differential equation for a given initial condition, which results in a very complex dynamic.

The complex behavior of the system in the homoclinic network according to Poincaré also becomes clear as a consequence of Jacob Palis' lemma: Let the stable and the unstable manifold be at the hyperbolic fixed point p (where the dimension of u is) and let D be a u-dimensional disk transversal to . Then the iterates converge to . If one considers two dimensions, a surrounding interval D of the transversal homoclinic point q, which lies on the unstable manifold, will finally be iterated in an arbitrarily small neighborhood of an interval of the unstable manifold around the fixed point p. ${\ displaystyle \ lambda}$ ${\ displaystyle W ^ {+} (p)}$ ${\ displaystyle W ^ {-} (p)}$ ${\ displaystyle W ^ {-} (p)}$ ${\ displaystyle W ^ {+} (p)}$ ${\ displaystyle f ^ {n} (D)}$ ${\ displaystyle W ^ {-} (p)}$

In the early 1960s Stephen Smale found a simple geometric structure, the horseshoe map , which explains the chaotic behavior in the homoclinic network. This is the subject of the Birkhoff and Smale theorem , which says that such a horseshoe exists in an arbitrary neighborhood of a hyperbolic fixed point p of a diffeomorphism f for one iteration of f if a transverse homoclinic point q exists. In horseshoe mapping, a square is mapped onto itself by stretching it and bending it back like a horseshoe (this corresponds to the alternating expansion and compression with alternating movement on the stable and unstable manifold).

Birkhoff confirmed Poincaré's conjectures around 1935 by showing (in two dimensions) that near homoclinic orbits there are a very complex network of periodic points: in an arbitrarily small neighborhood of a transverse homoclinic point there are periodic points. In the 1940s, Mary Cartwright and John Edensor Littlewood (and, in the US, Norman Levinson , whose work was the inspiration for Smale's work on the horseshoe) studied homoclinic orbits in the Van der Pol equation, which describes forced oscillations in vacuum tubes. Typical of the van der Pol oscillator was the occurrence of a periodic orbit with a frequency much higher than that of the excitation frequency and an alternating stable periodic (with one frequency) and bistable chaotic behavior (with two frequencies) depending on the size of the excitation amplitude (more precisely by Mark Levi 1981 declared).

Heteroclinic orbit in the phase space of a pendulum

The term homoclinic and heteroclinic point was introduced by Poincaré in the third volume of his Méthodes Nouvelles de la Mécanique Celeste (1899, Chapter 33) (originally he called them doubly asymptotic solutions).

literature

Boris Hasselblatt, Anatole Katok: First course in dynamics. Cambridge University Press, 2003.
Boris Hasselblatt, Anatole Katok: Introduction to the modern theory of dynamical systems. Cambridge University Press, 1995.
Jacob Palis, Floris Takens: Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations. Cambridge University Press, 1993.
Jürgen Moser: Stable and random motions in dynamical systems. Princeton University Press, 1973.

Web links

Quiudong Wang, On the homoclinic tangles of Henri Poincaré (pdf)

Individual evidence

↑ Guckenheimer, Holmes, Nonlinear oscillations, p. 222
^ Hasselblatt, Katok, Introduction to the modern theory of dynamical systems, p. 292
↑ Palis, de Melo, Geometric theory of dynamical systems, Springer 1982
^ Hasselblatt, Katok, First Course in Dynamics, p. 322
↑ Birkhoff, Novelles recherches sur les systèmes dynamiques, Mem. Pontific. Acad. Sci. Novi Lyncaei, 1, 1935, pp. 85-216
↑ This is how Smale and Birkhoff's theorem is formulated, e.g. B. Mrowka, A short proof of the Birkhoff-Smale theorem, Proc. AMS, Volume 93, 1985, p. 377. It follows as a corollary from the Smale and Birkhoff Theorem on the Existence of Horseshoes, Smale, Differentiable Dynamical Systems, Bull. AMS, Volume 73, 1967, p. 775, online , and was the formulation in which Birkhoff proved the theorem in two dimensions.
^ Levi, Qualitative analysis of the periodically forced relaxation oscillations, Memoirs AMS 1981

[1] Guckenheimer, Holmes, Nonlinear oscillations, p. 222

[2] Hasselblatt, Katok, Introduction to the modern theory of dynamical systems, p. 292

[3] Palis, de Melo, Geometric theory of dynamical systems, Springer 1982

[4] Hasselblatt, Katok, First Course in Dynamics, p. 322

[5] Birkhoff, Novelles recherches sur les systèmes dynamiques, Mem. Pontific. Acad. Sci. Novi Lyncaei, 1, 1935, pp. 85-216

[6] This is how Smale and Birkhoff's theorem is formulated, e.g. B. Mrowka, A short proof of the Birkhoff-Smale theorem, Proc. AMS, Volume 93, 1985, p. 377. It follows as a corollary from the Smale and Birkhoff Theorem on the Existence of Horseshoes, Smale, Differentiable Dynamical Systems, Bull. AMS, Volume 73, 1967, p. 775, online , and was the formulation in which Birkhoff proved the theorem in two dimensions.

[7] Levi, Qualitative analysis of the periodically forced relaxation oscillations, Memoirs AMS 1981