Two-Dimensional Manifolds of Modified Chen System with Time Delay

Two-dimensional unstable manifolds of the modified Chen system are constructed at equilibrium solution by “expanding up” along the unstable eigen-direction, hence it is tangent to the unstable eigenspace. In general, unstable manifold expands to the attraction basin of the corresponding limit cycle or attractor. With the introduction of time delay, the two-dimensional unstable manifold of an unstable focus is simulated via expanding solution orbits with restriction condition on the associated foliations. The simulated unstable manifold coincides with the attraction basin of the limit cycle of the delay differential equations.

Multi-pass stable periodic points of diffeomorphism of a plane with a homoclinic orbit

A diffeomorphism of a plane into itself with a fixed hyperbolic point and a nontransversal point homoclinic to it is studied. There are various ways of touching a stable and unstable manifold at a homoclinic point. Periodic points whose trajectories do not leave the vicinity of the trajectory of a homoclinic point are divided into a countable set of types. Periodic points of the same type are called n-pass periodic points if their trajectories have n turns that lie outside a sufficiently small neighborhood of the hyperbolic point. Earlier in the articles of Sh. Newhouse, L. P. Shil’nikov, B. F. Ivanov and other authors, diffeomorphisms of the plane with a nontransversal homoclinic point were studied, it was assumed that this point is a tangency point of finite order. In these papers, it was shown that in a neighborhood of a homoclinic point there can be infinite sets of stable two-pass and three-pass periodic points. The presence of such sets depends on the properties of the hyperbolic point. In this paper, it is assumed that a homoclinic point is not a point with a finite order of tangency of a stable and unstable manifold. It is shown in the paper that for any fixed natural number n, a neighborhood of a nontransversal homolinic point can contain an infinite set of stable n-pass periodic points with characteristic exponents separated from zero.

Different types of stable periodic points of diffeomorphism of a plane with a homoclinic orbit

A diffeomorphism of the plane into itself with a fixed hyperbolic point is considered; the presence of a nontransverse homoclinic point is assumed. Stable and unstable manifolds touch each other at a homoclinic point; there are various ways of touching a stable and unstable manifold. In the works of Sh. Newhouse, L. P. Shilnikov and other authors, studied diffeomorphisms of the plane with a nontranverse homoclinic point, under the assumption that this point is a tangency point of finite order. It follows from the works of these authors that an infinite set of stable periodic points can lie in a neighborhood of a homoclinic point; the presence of such a set depends on the properties of the hyperbolic point. In this paper, it is assumed that a homoclinic point is not a point at which the tangency of a stable and unstable manifold is a tangency of finite order. Allocate a countable number of types of periodic points lying in the vicinity of a homoclinic point; points belonging to the same type are called n-pass (multi-pass), where n is a natural number. In the present paper, it is shown that if the tangency is not a tangency of finite order, the neighborhood of a nontransverse homolinic point can contain an infinite set of stable single-pass, double-pass, or three-pass periodic points with characteristic exponents separated from zero.

On 4-dimensional flows with wildly embedded invariant manifolds of a periodic orbit

In the present paper we construct an example of 4-dimensional flows on 𝕊3 × 𝕊1 whose saddle periodic orbit has a wildly embedded 2-dimensional unstable manifold. We prove that such a property has every suspension under a non-trivial Pixton's diffeomorphism. Moreover we give a complete topological classification of these suspensions.

Heteroclinic path to spatially localized chaos in pipe flow

In shear flows at transitional Reynolds numbers, localized patches of turbulence, known as puffs, coexist with the laminar flow. Recently, Avila et al. (Phys. Rev. Lett., vol. 110, 2013, 224502) discovered two spatially localized relative periodic solutions for pipe flow, which appeared in a saddle-node bifurcation at low Reynolds number. Combining slicing methods for continuous symmetry reduction with Poincaré sections for the first time in a shear flow setting, we compute and visualize the unstable manifold of the lower-branch solution and show that it extends towards the neighbourhood of the upper-branch solution. Surprisingly, this connection even persists far above the bifurcation point and appears to mediate the first stage of the puff generation: amplification of streamwise localized fluctuations. When the state-space trajectories on the unstable manifold reach the vicinity of the upper branch, corresponding fluctuations expand in space and eventually take the usual shape of a puff.

Invariant Spectra in N-Coupled Standard Maps

This paper examines the evolution of the dynamical spectra of stretching numbers in a system of [Formula: see text]-coupled Standard Maps. It is found that the convergence rate of the spectra to their invariant forms is independent of the dimensionality 2[Formula: see text] and the nonlinearity/coupling parameters. This rate is impressively faster than one could predict on the basis of ergodicity. This effect is probably associated with the manifold of the maximum Lyapunov exponent along which the main stretching occurs. It seems that dynamical spectra depend mainly on the dramatically smaller subspace defined by this unstable manifold.