User:Eugene M. Izhikevich/Proposed/Global bifurcations

• Yulij Ilyashenko, Cornell University, Ithaca, New York

Bifurcations are abrupt changes of the topology of phase portraits of parameter depending vector fields when the parameter (which is supposed of dimension one for simplicity) passes a special critical value. We assume that at least on one side of this value the phase portraits are topologically equivalent; the corresponding vector fields are structurally stable. Most studied are bifurcations that occur on the boundary of the set of the Morse-Smale systems. These systems have a nonwondering set that consists of a finite number of singular points and periodic orbits, all hyperbolic and their stable and unstable manifolds intersect transversally. When any of these assumptions is violated, a bifurcation occurs. These bifurcations may be considered in the vicinity of singular points and periodic orbits. This gives rise to the theory of local bifurcations. This theory is especially rich in case of many parameters. Bifurcations that occur for the same degeneracies in the total phase space, are called nonlocal or global bifurcations. This theory was originated by Shil'nikov, later accompanied by his students, see Chapter 3 in survey [AAIS], and later but independently by Newhouse, Palis and Takens, see [NPT].

Homoclinic orbits of nonhyperbolic singular points

Figure 1: A homoclinic orbit of a planar saddlenode (A) and its bifurcations (B)

Figure 2: a) Two homoclinic orbits of a saddlenode singular points in \({\mathbb{R}}^3\ .\) b) The Poincaré map \(P\) for a postcritical parameter value.

When a singular point in the plane becomes nonhyperbolic, a homoclinic orbit may occur without increasing the codimension of degeneracy, see Figure 1 for dimension two. In this case a periodic orbit may be born under the bifurcation. This figure, as well as Figure 1 b, Figure 6 a,b, Figure 7 b, is borrowed from [IL]. In higher dimension multiple homoclinic orbits of a non-hyperbolic fixed point may coexist. Their bifurcation produces a hyperbolic set, like a Smale horseshoe, see Figure 2.

Homoclinic orbits of nonhyperbolic cycles

Bifurcations that accompany occurrence of nonhyperbolic periodic orbits (also called cycles) are even more complicated. Various surfaces filled by homoclinic orbits of these cycles may occur. They are a) tori or Klein bottles, b) multiple tori and/or Klein bottles, c) snakes, d) twisted tori, e) nonsmooth tori and many others. These surfaces are shown on Figure 3 a,b,c,d,e. Their bifurcations give rise to:

• invariant torus or a Klein bottle in case a).
• a partially hyperbolic invariant set with the dynamics of a skew product over the Bernoulli shift in the base, and the fiber a circle, case b).
• a suspension over the Smale-Williams solenoid, case d), Figure 4.
• a suspension over an endomorphism of a circle, case e).

Figure 3: Various types of homoclinic surfaces of a saddlenode cycle	Figure 4: The Smale-Williams solenoid; the Poincaré map \(P\) for the bifurcation of the twisted torus is the solenoid map
	Figure 5: a) Planar separatrix loop b) Its bifurcation

Nontransversal intersections of stable and unstable manifolds of hyperbolic singular points

Such intersections may occur in any dimensions beginning with \(2\ .\) For invariant manifolds of a hyperbolic saddle nontransversal intersection is simply nonempty one. For a hyperbolic saddle in the plane such an intersection implies coincidence of separatrixes, see Figure 5. For a hyperbolic saddle in \({\mathbb{R}}^3\ ,\) \(n > 3\ ,\) one homoclinic curve corresponds to the critical parameter value. The cases of all real (one real and two complex eigenvalues) are shown in Figure 6 a (Figure 6 b respectively). Under the perturbation, the first case gives rise to a periodic orbit; the second one to a hyperbolic invariant set. The corresponding Poincaré maps for a perturbed system are shown in Figure 7, cases a) and b).

Figure 6: a) A homoclinic loop of a hyperbolic fixed point with all real eigenvalues b) The same for one real and two complex eigenvalues.

Lorentz attractor

The famous Lorentz attractor was explained [ABS], [GW] by means of the global bifurcations theory. The chaotic behavior of its orbits may be explained by means of the Poincaré map that resembles duplicated Figure 7 a, see Figure 8. This map has a vague similarity with the Smale horseshoe map shown on the Figure 2 b.

Nontransversal intersections of stable and unstable manifolds of hyperbolic periodic orbits

A transversal intersection of stable and unstable manifolds of a periodic orbit is possible because the sum of dimensions of these manifolds is greater by \(1\) than the dimension of the phase space. A nontransvrsal intersection of such manifolds is tangency. A similar dynamical system with discrete time is a map of a planar domain with a hyperbolic fixed point whose stable and unstable manifolds are tangent, see Figure 9.

The bifurcations of a map with the homoclinic tangency of stable and unstable manifolds of a hyperbolic fixed point gives rise to the so called Newhouse phenomenon. Namely, there exists a neighborhood of a system with the homoclinic tangency, and a residual subset of this neighborhood such that any map from this subset has a countable number of attracting periodic orbits. The above neighborhood is called a Newhouse domain. Some other striking phenomena that occur in this domain are described in [PT].

A concise presentation of a large part of the global bifurcation theory may be found on [IL]. The presentation is based on the theory of normal forms for local families [IYa] and hyperbolic theory.

Figure 7: a) A hyperbolic fixed point of the Poincaré for a homoclinic loop of a hyperbolic saddle with all real eigenvalues b) The same for one real and two complex eigenvalues. A countable number of Smale horseshoes occurs if the complex eigenvalues are closer to the imaginary axis than the real one

Figure 8: The Poincaré for the Lorentz system

Figure 9: Homoclinic tangency

References

• [AAIS] V. Arnold, V. Afraimovich, Yu. Ilyashenko, L. Shil'nikov, Bifurcation theory. In Encyclopedia of Mathematical Sciences, v 5, Moscow 1986, Springer 1994.
• [ABS] V. Afraimovich, Bykov, L. Shil'nikov, On attracting structurally unstable limit sets of Lorenz attractor type, (Russian) Trudy Moskov. Mat. Obshch. 44 (1982), 150--212; Trans. of the Moscow Math. Soc. 44(1983)153-216.
• [GW] J. Guckenheimer, R.F. Williams, Structural stability of Lorenz attractors. Inst. Hautes Études Sci. Publ. Math. No. 50 (1979), 59--72.
• [IL] Yu. Ilyashenko, Weigu Li, Nonlocal Bifurcations, a Monograph, published by AMS, ser. Mathematical surveys and Monographs, 1998, vol.66.
• [IYa] Yu. Ilyashenko, S. Yakovenko, Smooth normal forms for local families of diffeomorphisms and vector fields, Russian Math. Surveys, 1991, v.46, N 1, p.3-39.
• [K]Yu.A. Kuznetsov [2004] Elements of Applied Bifurcation Theory, Springer, 3rd edition.
• [NPT] S. Newhause, J. Palis, F. Takens, Bifurcations and stability of families of diffeomorphisms, Inst. Hautes Études Sci. Publ. Math. No. 57 (1983), 5--71.
• [PT] J. Palis, F. Takens, Hyperbolicity and sensitive chaotic dynamics at homoclinic bifurcations. Fractal dimensions and infinitely many attractors, Cambridge Studies in Advanced Mathematics, 35, Cambridge University Press, Cambridge, 1993. x+234 pp.
• [SSTC1] Shilnikov L.P., Shilnikov A., Turaev D. and Chua, L. [1998] Methods of Qualitative Theory in Nonlinear Dynamics. Part I. World Scientific.
• [SSTC2] Shilnikov L.P., Shilnikov A., Turaev D. and Chua, L. [2001] Methods of Qualitative Theory in Nonlinear Dynamics. Part II.World Scientific.

External Links

• Yulij Ilyashenko's website

See also

Bifurcation, Morse-Smale systems, Smale horseshoe, Hyperbolic dynamics, Attractor