Corresponding Basins Of Attraction Research Articles

Algorithms for computing basins of attraction associated with a rational self-map of the Hopf fibration based on Lyapunov exponents

The objective of this work is twofold. One, we develop a theory of iteration for complex rational functions so that the problems of overflow and indeterminacy caused by null, or almost null, denominators can be avoided when developing implementations. Second, we present easily implementable methods that allow the calculation of attracting cycles as well as the graphical representation of their basins of attraction.In order to deal with our first goal we work with homogeneous complex coordinates and we take the complex projective line as a model, which is expressed as a quotient of the 3-sphere through the Hopf fibration. An irreducible representation of a rational function can now be presented as a Hopf fibration endomorphism. As well as our second goal is concerned, we use Lyapunov functions and exponents to calculate the cycles associated with endomorphisms and to graphically represent the corresponding basins of attraction. We point out that our algorithms are based on the calculation of a finite set of non-negative real constants and their calculation does not depend on the previous determination of the fixed points or attracting cycles.

Intermingled attractors in an asymmetrically driven modified Chua oscillator.

Understanding the asymptotic behavior of a dynamical system when system parameters are varied remains a key challenge in nonlinear dynamics. We explore the dynamics of a multistable dynamical system (the response) coupled unidirectionally to a chaotic drive. In the absence of coupling, the dynamics of the response system consists of simple attractors, namely, fixed points and periodic orbits, and there could be chaotic motion depending on system parameters. Importantly, the boundaries of the basins of attraction for these attractors are all smooth. When the drive is coupled to the response, the entire dynamics becomes chaotic: distinct multistable chaos and bistable chaos are observed. In both cases, we observe a mixture of synchronous and desynchronous states and a mixture of synchronous states only. The response system displays a much richer, complex dynamics. We describe and analyze the corresponding basins of attraction using the required criteria. Riddled and intermingled structures are revealed.

On the Dynamics of a Biomimetic Model of a Sympodial Tree: From Bifurcations Diagrams and 6D Basins of Attraction to Dynamical Integrity and Robustness

Abstract This work is concerned with a mechanical model of a sympodial tree with first-level branches, which has been shown to exhibit certain properties potentially suitable for biomimetic applications. To investigate these potential benefits further from the viewpoint of the system nonlinear behavior under external periodic excitation, modern numerical tools related to the concept of dynamical integrity are either adjusted or newly developed for this system for the first time. First, multistable regions of interest are isolated from bifurcation diagrams and the effect of damping is investigated. Then, in order to obtain the corresponding basins of attraction of this highly dimensional model, an original computational procedure is developed that includes cell mapping with 406 cells, where each cell represents an initial condition required to construct the map. Full 6D basins are computed, and they are reported for various values of the damping parameter and the excitation frequency. Those basins are then used to calculate the dynamic integrity factors so that the dominant steady-state can be determined. Finally, the integrity profiles are reported to illustrate how the robustness varies by changing the system parameters.

Attractors and transient in sequential dynamical systems

ABSTRACTIn this article, we show how to determine attractors in sequential dynamical systems on maxterm and minterm Boolean functions and their corresponding basins of attraction. Furthermore, we make possible to know when an attractor is globally attractive. As the main result of this work, upper bounds for the transient in such models are provided. In order to do that, we distinguish two possible scenarios: fixed point sequential dynamical systems and periodic orbit sequential dynamical systems.

Comparing the basins of attraction for several methods in the circular Sitnikov problem with spheroid primaries

The circular Sitnikov problem, where the two primary bodies are prolate or oblate spheroids, is numerically investigated. In particular, the basins of convergence on the complex plane are revealed by using a large collection of numerical methods of several order. We consider four cases, regarding the value of the oblateness coefficient which determines the nature of the roots (attractors) of the system. For all cases we use the iterative schemes for performing a thorough and systematic classification of the nodes on the complex plane. The distribution of the iterations as well as the probability and their correlations with the corresponding basins of convergence are also discussed. Our numerical computations indicate that most of the iterative schemes provide relatively similar convergence structures on the complex plane. However, there are some numerical methods for which the corresponding basins of attraction are extremely complicated with highly fractal basin boundaries. Moreover, it is proved that the efficiency strongly varies between the numerical methods.

Phases in a General Chaotic Three-Coupled-Laser Array

Phases in a general chaotic three-coupled-laser array are numerically investigated. Phase attractors are firstly found to exist within corresponding basins of attraction on the projection plane of phase differences of the three-coupled-laser array. Whether the chaos appears or not is related not only to the coupling strength but also to the initial phase differences. For a large coupling strength new phase attractors can occur. With the increase of coupling strength, the three-coupled-laser array has a great chance of resulting in a quiescent to chaotic state. Based on these results, we present the method to reach phase-locking when the coupling strength is strong.

Multistability and Self‐Similarity in the Parameter‐space of a Vibro‐impact System

The dynamics of a dissipative vibro‐impact system called impact‐pair is investigated. This system is similar to Fermi‐Ulam accelerator model and consists of an oscillating one‐dimensional box containing a point mass moving freely between successive inelastic collisions with the rigid walls of the box. In our numerical simulations, we observed multistable regimes, for which the corresponding basins of attraction present a quite complicated structure with smooth boundary. In addition, we characterize the system in a two‐dimensional parameter space by using the largest Lyapunov exponents, identifying self‐similar periodic sets.