Shil’nikov chaos in the 4D Lorenz–Stenflo system modeling the time evolution of nonlinear acoustic-gravity waves in a rotating atmosphere

The Lorenz–Stenflo system serves as a model of the time evolution of nonlinear acoustic-gravity waves in a rotating atmosphere. In the present paper, we study the Shil’nikov chaos which arises in the 4D Lorenz–Stenflo system. The analytical and numerical results constitute an application of the Shil’nikov theorems to a 4D system (whereas most results present in the literature deal with applying the Shil’nikov theorems to 3D systems), which allows for the study of chaos along homoclinic and heteroclinic orbits arising as solutions to the Lorenz–Stenflo system. We verify the observed chaos via competitive modes analysis—a diagnostic for chaotic systems. We give an analytical test, completely in terms of the model parameters, for the Smale Horseshoe chaos near homoclinic orbits of the origin, as well as for the case of specific heteroclinic orbits. Numerical results are shown for other cases in which the general analytical method becomes too complicated to apply. These results can be extended to more complicated higher-dimensional systems governing plasmas, and, in particular, may be used to shed light on period-doubling and Smale Horseshoe chaos that arises in such models.

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We study a twoparameter family of threedimensional vector elds that are small perturbations of an integrable system possessing a line of degenerate saddle points connected by a manifold of homoclinic loops Under perturbation this manifold splits and undergoes a quadratic homoclinic tangency Perturbation methods followed by geometrical analysis reveal the presence of countablyinnite sets of homoclinic orbits to and a nonwandering set topologically conjugate to a shift on two symbols a Smale Horseshoe We use the symbolic description to identify and partially order bifurcation sequences in which the homoclinic orbits appear and we formally derive an explicit twodimensional Poincare return map to further illustrate our results The problem was motivated by the search for traveling structures such as fronts and domain walls in partial dierential equations In this paper we continue the study of homoclinic and heteroclinic orbits in nearintegrabl

Homoclinic Saddle‐Node Bifurcations and Subshifts in a Three‐Dimensional Flow

We study a two‐parameter family of three‐dimensional vector fields that are small perturbations of an integrable system possessing a line Γ of degenerate saddle points connected by a manifold of homoclinic loops. Under perturbation, this manifold splits and undergoes a quadratic homoclinic tangency. Perturbation methods followed by geometrical analyses reveal the presence of countably‐infinite sets of homoclinic orbits to Γ and a non‐wandering set topologically conjugate to a shift on two symbols (a Smale Horseshoe). We use the symbolic description to identify and partially order bifurcation sequences in which the homoclinic orbits appear, and we formally derive an explicit two‐dimensional Poincaré return map to further illustrate our results. The problem was motivated by the search for travelling ‘structures’ such as fronts and domain walls in partial differential equations.

Chaos in a mapping describing elastoplastic oscillations

We study the local and global dynamical behavior of a two dimensional piecewise linear map which describes the asymptotic motions of a single degree of freedom, parametrically excited, elastoplastic oscillator after it has settled down to purely elastic oscillations. We give existence and stability conditions for periodic orbits and prove that chaos, in the form of a Smale Horseshoe, exists at specific, but representative, parameter values. We interpret simulations of the elastoplastic oscillator itself in the light of these results.

Melnikov’s method in String Theory

Melnikov’s method is an analytical way to show the existence of classical chaos generated by a Smale Horseshoe. It is a powerful technique, though its applicability is somewhat limited. In this paper, we present a solution of type IIB supergravity to which Melnikov’s method is applicable. This is a brane-wave type deformation of the AdS_5×S^5 background. By employing two reduction ansätze, we study two types of coupled pendulum-oscillator systems. Then the Melnikov function is computed for each of the systems by following the standard way of Holmes and Marsden and the existence of chaos is shown analytically.

Nonsymmetric Saddle-Node Pairs for the Reversible Smale Horseshoe Map

In the reversible Smale Horseshoe, we introduce a new symbol sequence other than the conventional one made of symbols 0 and 1. This system is based on subregionsE,F,S ,a ndD of resonance regions of rotation number between 0 and 1/2. We call E(p/q),F(p/q),S(p/q), and D(p/q )f or 0

nonsymmetric saddle node pairs for the reversible Smale Horseshoe map

In the reversible Smale Horseshoe, we introduce a new symbol sequence other than the conventional one made of symbols 0 and 1. This system is based on subregionsE,F,S ,a ndD of resonance regions of rotation number between 0 and 1/2. We call E(p/q),F(p/q),S(p/q), and D(p/q )f or 0

saddle-node pairs. We propose a procedure to find the pair component for a given nonsymmetric periodic orbit of the general type. If it turns out that there is no pair component, we suggest that the periodic orbit has been born through equiperiod bifurcation or period-doubling bifurcation. Subject Index: 030

Topological Horseshoes and Computer Assisted Verification of Chaotic Dynamics

In this tutorial paper, we present a history of Smale Horseshoe and an overview of the progress of topological Horseshoe theory. Then we offer a pedagogical exposition of elements of topological Horseshoe theory with a lot of examples. Finally we demonstrate some typical applications of topological Horseshoe theory to practical dynamical systems.

A 3D Smale Horseshoe in a Hyperchaotic Discrete-Time System

This paper presents a three-dimensional topological Horseshoe in the hyperchaotic generalized Henon map. It looks like a planar Smale Horseshoe with an additional vertical expansion, so we call it 3D Smale Horseshoe. In this way, a computer assisted verification of existence of hyperchaos is provided by means of interval analysis.