An important problem in the dynamics of surface homeomorphisms is determining the forcing relation between orbits. The forcing relation between periodic orbits can be computed using standard algorithms, though this does not give much information on the structure of the forcing relation. Here we consider forcing relations between homoclinic orbits, and their relationships with periodic orbits. We outline a general procedure for computing the forcing relation, and apply this to compute the equivalence and forcing relations for homoclinic orbits of the Smale horseshoe map. We also outline a method for determining forcing relations from graph maps, which allows us to compute the forcing relation between the so-called star homoclinic orbits.
We prove that for every three-dimensional vector field, either it can be accumulated by Morse-Smale ones, or it can be accumulated by ones with a transverse homoclinic intersection of some hyperbolic periodic orbit in the $C^1$ topology.
Bifurcations of dynamical systems, described by a second order differential equations and by an impact condition are studied. It is shown that the variation of parameters when the number of impacts of a periodic solution increases, leads to the occurrence of a hyperbolic chaotic invariant set.
The paper contains a rigorous proof of existence of symbolic dynamics chaos in the generalized Hénon map's 4th iterate $H^4$, which was conjectured in the paper \textit{A 3D Smale Horseshoe in a Hyperchaotic Discrete-Time System} of Li and Yang, 2007. We prove also the uniform hyperbolicity of the invariant set with symbolic dynamics. The proofs are computer-assisted with the use of C++ library \textit{CAPD} for interval arithmetic, differentiation and integration.
The set of transverse homoclinic intersections for a saddle-focus equilibrium in the planar equilateral restricted four-body problem admit certain simple homoclinic orbits which form the skeleton of the complete homoclinic intersection -- or homoclinic web. In the present work, the planar restricted four-body problem is viewed as an invariant subsystem of the spatial problem, and the influence of this planar homoclinic skeleton on the spatial dynamics is studied from a numerical point of view. Starting from the vertical Lyapunov families emanating from saddle focus equilibria, we compute the s…
The complexity of a dynamical system exhibiting a homoclinic orbit is given by the orbits that it forces. In this work we present a method, based in pruning theory, to determine the dynamical core of a homoclinic orbit of a Smale diffeomorphism on the 2-disk. Due to Cantwell and Conlon, this set is uniquely determined in the isotopy class of the orbit, up a topological conjugacy, so it contains the dynamics forced by the homoclinic orbit. Moreover we apply the method for finding the orbits forced by certain infinite families of homoclinic horseshoe orbits and propose its generalization to an a…
We show that the homoclinic C*-algebras of mixing Smale spaces are classifiable by the Elliott invariant. To obtain this result, we prove that the stable, unstable, and homoclinic C*-algebras associated to such Smale spaces have finite nuclear dimension. Our proof of finite nuclear dimension relies on Guentner, Willett, and Yu's notion of dynamic asymptotic dimension.
Group actions on a Smale space and the actions induced on the C*-algebras associated to such a dynamical system are studied. We show that an effective action of a discrete group on a mixing Smale space produces a strongly outer action on the homoclinic algebra. We then show that for irreducible Smale spaces, the property of finite Rokhlin dimension passes from the induced action on the homoclinic algbera to the induced actions on the stable and unstable C*-algebras. In each of these cases, we discuss the preservation of properties---such as finite nuclear dimension, Z-stability, and classifica…
A new geometric criterion is derived for the existence of chaos in continuous-time autonomous systems in three-dimensional Euclidean spaces, where a type of Smale horseshoe in a subshift of finite type exists, but the intersection of stable and unstable manifolds of two points on a hyperbolic periodic orbit does not imply the existence of a Smale horseshoe of the same type on cross-sections of these two points. This criterion is based on the existence of a hyperbolic periodic orbit, differing from the classical equilibrium-based Shilnikov criterion and the condition of transversal homoclinic o…
In this paper using approach of 1-D auxiliary maps we prove the existence of trapping domains containing attractors of the multidimensional Henon-like maps. For both of quadratic and cubic nonlinearities we obtain sufficient conditions of topological Smale horseshoes existence. The complex structure of attractors is discussed in the case of small coupling parameter.
The stable and unstable manifolds of a saddle fixed point (SFP) of the Bonhoeffer-van der Pol oscillator are numerically studied. A correspondence between the existence of homoclinic tangencies (whic are related to the creation or destruction of Smale horseshoes) and the chaos observed in the bifurcation diagram is described. It is observed that in the non-chaotic zones of the bifurcation diagram, there may or may not be Smale horseshoes, but there are no homoclinic tangencies.
We study a homoclinic flip bifurcation of case~\textbf{C}, where a homoclinic orbit to a saddle equilibrium with real eigenvalues changes from being orientable to nonorientable. This bifurcation is of codimension two, and it is the lowest codimension for a homoclinic bifurcation of a real saddle to generate chaotic behavior in the form of (suspended) Smale horseshoes and strange attractors. We present a detailed numerical case study of how global stable and unstable manifolds of the saddle equilibrium and of bifurcating periodic orbits interact close to such bifurcation. This is a step forward…
In this paper, we consider the scalar reaction-diffusion equations $\partial_t u=Δu + f(x,u,\nabla u)$ on a bounded domain $Ω\subset\mathbb{R}^d$ of class $C^2$. We show that the heteroclinic and homoclinic orbits connecting hyperbolic equilibria and hyperbolic periodic orbits are transverse, generically with respect to f. One of the main ingredients of the proof is an accurate study of the singular nodal set of solutions of linear parabolic equations. Our main result is a first step for proving the genericity of Kupka-Smale property, the generic hyperbolicity of periodic orbits remaining unpr…
A classical construction due to Newhouse creates horseshoes from hyperbolic periodic orbits with large period and weak domination through local $C^1$-perturbations. Our main theorem shows that, when one works in the $C^1$ topology, the entropy of such horseshoes can be made arbitrarily close to an upper bound deriving from Ruelle's inequality, i.e., the sum of the positive Lyapunov exponents (or the same for the inverse diffeomorphism, whichever is smaller). Adapting classical techniques, we use perturbations that are local and can be chosen to preserve volume or symplectic form or a homoclini…
In this paper, we show that, for scalar reaction-diffusion equations $u_t=u_{xx}+f(x,u,u_x)$ on the circle $S^1$, the Morse-Smale property is generic with respect to the non-linearity $f$. In \cite{CR}, Czaja and Rocha have proved that any connecting orbit, which connects two hyperbolic periodic orbits, is transverse and that there does not exist any homoclinic orbit, connecting a hyperbolic periodic orbit to itself. In \cite{JR}, we have shown that, generically with respect to the non-linearity $f$, all the equilibria and periodic orbits are hyperbolic. Here we complete these results by showi…
D. Ruelle considered a general setting where he is able to characterize equilibrium states for Hölder potentials based on properties of conjugating homeomorphism in the so called Smale spaces. On this setting he also shows a relation of KMS states of $C^*$-algebras and equilibrium probabilities of Thermodynamic Formalism. A later paper by N. Haydn and D. Ruelle presents a shorter proof of this equivalence. Here we consider similar problems but now on the symbolic space $Ω= \{1,2,...,d\}^{\mathbb{Z} - \{ 0 \} }$ and the dynamics will be given by the shift $τ$. In the case of potentials dependin…
