Chaotic behavior analysis based on sliding bifurcations (original) (raw)

Chaotic behavior analysis based on corner bifurcations

Nonlinear Analysis: Hybrid Systems, 2009

In this paper, a mathematical analysis in order to generate a chaotic behavior for piecewise smooth systems submitted to one of it's specific bifurcations, namely the corner one, is proposed. This study is based on period doubling method.

Bifurcations of Chaotic Attractors in One-Dimensional Piecewise Smooth Maps

International Journal of Bifurcation and Chaos, 2014

In this work, we classify the bifurcations of chaotic attractors in 1D piecewise smooth maps from the point of view of underlying homoclinic bifurcations of repelling cycles which are located before the bifurcation at the boundary of the immediate basin of the chaotic attractor.

On the occurrence of chaos via different routes to chaos: period doubling and border-collision bifurcations

Journal of Mathematical Sciences, 2009

This paper introduces a new 2D piecewise smooth discrete-time chaotic mapping with rarely observed phenomenon – the occurrence of the same chaotic attractor via different and distinguishable routes to chaos: period doubling and border-collision bifurcations as typical futures. This phenomenon is justified by the location of system equilibria of the proposed mapping, and the possible bifurcation types in smooth dissipative systems.

Chapter 11 Bifurcation Theory of Dynamical Chaos

2018

The purpose of the present chapter is once again to show on concrete new examples that chaos in one-dimensional unimodal mappings, dynamical chaos in systems of ordinary differential equations, diffusion chaos in systems of the equations with partial derivatives and chaos in Hamiltonian and conservative systems are generated by cascades of bifurcations under universal bifurcation Feigenbaum-Sharkovsky-Magnitskii (FShM) scenario. And all irregular attractors of all such dissipative systems born during realization of such scenario are exclusively singular attractors that are the nonperiodic limited trajectories in finite dimensional or infinitely dimensional phase space any neighborhood of which contains the infinite number of unstable periodic trajectories.

Bifurcations from phase-locked dynamics to chaos in a piecewise-linear map

2011

Recent work has shown that torus formation in piecewise-smooth maps can take place through a special type of border-collision bifurcation in which a pair of complex conjugate multipliers for a stable cycle abruptly jump out of the unit circle. Transitions from an ergodic to a resonant torus take place via border-collision fold bifurcations. The purpose of the present paper is to examine the transition to chaos through torus destruction in such maps. Considering a piecewise-linear normal-form map we show that this transition, by virtue of the interplay of bordercollision bifurcations with period-doubling and homoclinic bifurcations, can involve mechanisms that differ qualitatively from those described by Afraimovich and Shilnikov.

Another New Chaotic System: Bifurcation and Chaos Control

International Journal of Bifurcation and Chaos, 2020

We propose a new simple three-dimensional continuous autonomous model with two nonlinear terms and observe the dynamical behavior with respect to system parameters. This system changes the stability of fixed point via Hopf bifurcation and then undergoes a cascade of period-doubling route to chaos. We analytically derive the first Lyapunov coefficient to investigate the nature of Hopf bifurcation. We investigate well-separated regions for different kinds of attractors in the two-dimensional parameter space. Next, we introduce a timescale ratio parameter and calculate the slow manifold using geometric singular perturbation theory. Finally, the chaotic state annihilates by decreasing the value of the timescale ratio parameter.

Bifurcations and Periodic Orbits in Chaotic Maps

Open Systems & Information Dynamics (OSID), 2001

A hierarchy of universalities in families of 1-D maps is discussed. Breakdown of universalities in families of 3-D maps is shown on selected examples of such families.

Period-Doubling Scenario Without Flip Bifurcations in a One-Dimensional Map

International Journal of Bifurcation and Chaos, 2005

In this work a one-dimensional piecewise-smooth dynamical system, representing a Poincaré return map for dynamical systems of the Lorenz type, is investigated. The system shows a bifurcation scenario similar to the classical period-doubling one, but which is influenced by so-called border collision phenomena and denoted as border collision period-doubling bifurcation scenario. This scenario is formed by a sequence of pairs of bifurcations, whereby each pair consists of a border collision bifurcation and a pitchfork bifurcation. The mechanism leading to this scenario and its characteristic properties, like symmetry-breaking and symmetry-recovering as well as emergence of coexisting attractors, are investigated.

Chaotic behavior of driven, second-order, piecewise linear systems

Chaos Solitons & Fractals, 2017

In this paper the chaotic behavior of second-order, discontinuous systems with a pseudo-equilibrium point on a discontinuity surface is analyzed. The discontinuous system is piecewise linear and approximated to a non-smooth continuous system. The discontinuous term is represented by a sign function that is replaced by a saturation function with high slope. Some of the conditions that determine the chaotic behavior of the approximate system are formally established. Besides, the convergence of its chaotic solutions to those of the discontinuous system is shown. Several bifurcation diagrams of both systems show the similarity of their dynamical behavior in a wide parameter range, and particularly for a parameter region determined from the application of the Melnikov technique to non-smooth systems, where a chaotic behavior can be displayed.

Bifurcation Analysis and Fractal Dimensions of a NonLinear Map

In this paper a two dimensional non-linear map is taken, whose various dynamic behavior is analyzed. Some useful numerical algorithms to obtain fixed points and bifurcation values of period n 2 , n  0,1,2... . have been discussed. It has shown how the ratio of three successive period doubling bifurcation points ultimately converge to the Feigenbaum constant. This ascertains that the map follows the period doubling route to chaos. The parameter value where chaos starts is verified by lyapunov exponent. Further various fractal dimensions like Correlation dimension, Box-counting and Information dimension have been calculated to verify the geometry of the strange attractor.