THE GLOBAL BIFURCATIONS THAT LEAD TO TRANSIENT TUMBLING CHAOS IN A PARAMETRICALLY DRIVEN PENDULUM

2000 ◽  
Vol 10 (09) ◽  
pp. 2161-2175 ◽  
Author(s):  
W. SZEMPLIŃSKA-STUPNICKA ◽  
E. TYRKIEL ◽  
A. ZUBRZYCKI
Keyword(s):  
Crucial Role ◽  
Phase Plane ◽  
Fractal Structure ◽  
Basins Of Attraction ◽  
Oscillatory Motion ◽  
Global Bifurcations ◽  
Coexisting Attractors ◽  
Computer Aided ◽  
Tumbling Motion ◽  
Basin Boundaries

Criteria for occurrence of transient tumbling chaos in the parametrically driven pendulum are examined by computer aided methods. It is shown that stable manifolds of the saddles associated with the rotating attractors play a crucial role in separating the oscillating and rotating attractors in the phase-plane. A sequence of global bifurcations related to the saddles generates the fractal structure of the basins of attraction of the coexisting attractors. The fractal structure of the basin boundaries implies occurrence of the transient tumbling motion, i.e. the transients that involve rotations with changing directions and oscillatory motion.

Sequences of Global Bifurcations and the Related Outcomes after Crisis of the Resonant Attractor in a Nonlinear Oscillator

1997 ◽  
Vol 07 (11) ◽  
pp. 2437-2457 ◽  
Author(s):  
W. Szemplińska-Stupnicka ◽  
E. Tyrkiel
Keyword(s):  
Nonlinear Oscillator ◽  
Duffing Oscillator ◽  
Nonlinear Resonance ◽  
Basins Of Attraction ◽  
Global Bifurcations ◽  
System Behavior ◽  
Coexisting Attractors ◽  
System Parameters

The problem of the system behavior after annihilation of the resonant attractor in the region of the nonlinear resonance hysteresis is considered. The sequences of global bifurcations, in connection with the associated metamorphoses of basins of attraction of coexisting attractors, are examined. The study allows one to reveal the mechanism that governs the phenomenon of the post crisis ensuing transient trajectory to settle onto one or another remote attractor. The problem is studied in detail for the twin-well potential Duffing oscillator. The boundary which splits the considered region of system parameters into two subdomains, where the outcome is unique or the two outcomes are possible, is defined.

BISTABILITY AND DYNAMICAL TRANSITIONS IN A PHASE-MODULATED DELAYED SYSTEM

2009 ◽  
Vol 23 (23) ◽  
pp. 2733-2743 ◽  
Author(s):  
YONGXIANG ZHANG ◽  
GUIQIN KONG ◽  
JIANNING YU
Keyword(s):  
Lyapunov Exponents ◽  
Parameter Space ◽  
Chaotic Dynamics ◽  
Basins Of Attraction ◽  
Parameter Study ◽  
Global Bifurcations ◽  
Coexisting Attractors ◽  
System Parameters ◽  
Different Types ◽  
Delayed System

We study a delayed system with feedback modulation of the nonlinear parameter. Study of the system as a function of nonlinearity and modulation parameters reveals complex dynamical phenomena: different types of coexisting attractors, local or global bifurcations and transitions. Bistability and dynamical attractors can be distinguished in some parameter-space regions, which may be useful to drive chaotic dynamics, unstable attractors or bistability towards regular dynamics. At the bifurcation to bistability, two striking features are that they lead to fundamentally unpredictable behavior of orbits and crisis of attractors as system parameters are varied slowly through the critical curve. Unstable attractors are also investigated in bistable regions, which are easily mistaken for true multi-periodic orbits judging merely from zero measure local basins. Lyapunov exponents and basins of attraction are also used to characterize the phenomenon observed.

MULTISTABILITY IN A BUTTERFLY FLOW

2013 ◽  
Vol 23 (12) ◽  
pp. 1350199 ◽  
Author(s):  
CHUNBIAO LI ◽  
J. C. SPROTT
Keyword(s):  
Limit Cycle ◽  
Limit Cycles ◽  
Fractal Structure ◽  
Equilibrium Points ◽  
Symmetric Pair ◽  
Coexisting Attractors ◽  
Quadratic Nonlinearities ◽  
Stable Equilibria ◽  
Basin Boundaries ◽  
Special Parameter

A dynamical system with four quadratic nonlinearities is found to display a butterfly strange attractor. In a relatively large region of parameter space the system has coexisting point attractors and limit cycles. At some special parameter combinations, there are five coexisting attractors, where a limit cycle coexists with two equilibrium points and two strange attractors in different attractor basins. The basin boundaries have a symmetric fractal structure. In addition, the system has other multistable regimes where a pair of point attractors coexist with a single limit cycle or a symmetric pair of limit cycles and where a symmetric pair of limit cycles coexist without any stable equilibria.

BASIN FRACTALIZATIONS GENERATED BY A TWO-DIMENSIONAL FAMILY OF (Z1–Z3–Z1) MAPS

2006 ◽  
Vol 16 (03) ◽  
pp. 647-669 ◽  
Author(s):  
GIAN-ITALO BISCHI ◽  
LAURA GARDINI ◽  
CHRISTIAN MIRA
Keyword(s):  
Fractal Structure ◽  
Basins Of Attraction ◽  
Real Rank ◽  
Two Dimensional ◽  
Global Bifurcations ◽  
Rank One

Two-dimensional (Z1–Z3–Z1) maps are such that the plane is divided into three unbounded open regions: a region Z3, whose points generate three real rank-one preimages, bordered by two regions Z1, whose points generate only one real rank-one preimage. This paper is essentially devoted to the study of the structures, and the global bifurcations, of the basins of attraction generated by such maps. In particular, the cases of fractal structure of such basins are considered. For the class of maps considered in this paper, a large variety of dynamic situations is shown, and the bifurcations leading to their occurrence are explained.

THE OSCILLATION–ROTATION ATTRACTORS IN THE FORCED PENDULUM AND THEIR PECULIAR PROPERTIES

2002 ◽  
Vol 12 (01) ◽  
pp. 159-168 ◽  
Author(s):  
WANDA SZEMPLIŃSKA-STUPNICKA ◽  
ELŻBIETA TYRKIEL
Keyword(s):  
Control Parameter ◽  
Fractal Structure ◽  
Basins Of Attraction ◽  
Parameter Plane ◽  
System Control ◽  
Periodic Force ◽  
Forced Pendulum ◽  
Insight Into

The paper is aimed at exploration of the properties of the oscillation–rotation attractors in the dissipative pendulum driven by external periodic force. The study of regions of existence of the orbits in the system control parameter plane, coexistence with other attractors, fractal structure of their basins of attraction, and the role they play in the onset of the tumbling chaos, give an insight into some peculiar features of the oscillation–rotation attractors and their bifurcational structures.

THE EFFECT OF NONLINEAR DAMPING ON THE UNIVERSAL ESCAPE OSCILLATOR

1999 ◽  
Vol 09 (04) ◽  
pp. 735-744 ◽  
Author(s):  
MIGUEL A. F. SANJUÁN
Keyword(s):  
Energy Dissipation ◽  
Period Doubling ◽  
Nonlinear Damping ◽  
Basins Of Attraction ◽  
Period Doubling Bifurcation ◽  
Damping Coefficient ◽  
Nonlinear Dissipation ◽  
Fractal Basin Boundaries ◽  
Basin Boundaries

This paper analyzes the role of nonlinear dissipation on the universal escape oscillator. Nonlinear damping terms proportional to the power of the velocity are assumed and an investigation on its effects on the dynamics of the oscillator, such as the threshold of period-doubling bifurcation, fractal basin boundaries and how the basins of attraction are destroyed, is carried out. The results suggest that increasing the power of the nonlinear damping, has similar effects as of decreasing the damping coefficient for a linearly damped case, showing the very importance of the level or amount of energy dissipation.

Fractal Basin Boundaries and Chaotic Motions in Spring-Pendulum System

2003 ◽  
Author(s):  
Aria Alasty ◽  
Rasool Shabani
Keyword(s):  
Numerical Simulations ◽  
Multiple Scales ◽  
Basins Of Attraction ◽  
Chaotic Attractors ◽  
Global Behavior ◽  
Chaotic Motions ◽  
Pendulum System ◽  
Fractal Basin Boundaries ◽  
System Parameters ◽  
Basin Boundaries

This study investigates chaotic response in the spring-pendulum system. In this system beside of strange attractors, multiple regular attractors may coexist for some values of system parameters, where it is important to study the global behavior of the system using the basin boundaries of the attractors. Multiple scales method is used to distinguish the regions of stable and unstable attractors. In unstable regions, bifurcation diagram and poincare´ maps are used to study the existence of quasi-periodic and chaotic attractors. Results show that the jumping phenomena may occur when multiple regular attractors exist and for this case fractal basins of attraction are developed using numerical simulations.

scholarly journals Stability and Multistability of a Bounded Rational Mixed Duopoly Model with Homogeneous Product

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-16 ◽  
Author(s):  
Wei Zhou ◽  
Na Zhao ◽  
Tong Chu ◽  
Ying-Xiang Chang
Keyword(s):  
Joint Venture ◽  
Fractal Structure ◽  
Initial Conditions ◽  
Equilibrium Points ◽  
Basins Of Attraction ◽  
Mixed Duopoly ◽  
Formation Mechanisms ◽  
Multiple Attractors ◽  
Homogeneous Product ◽  
The Stability

In this paper, a mixed duopoly dynamic model with bounded rationality is built, where a public-private joint venture and a private enterprise produce homogeneous products and compete in the same market. The purpose of this research is to study the stability and the multistability of the established model. The local stability of all the equilibrium points is discussed by using Jury condition, and the stability region of the Nash equilibrium point has been given. A special fractal structure called “hub of periodicity” has been found in the two-parameter space by numerical simulation. In addition, the phenomena of multistability (also called coexistence of multiple attractors) are also studied using basins of attraction and 1-D bifurcation diagrams with adiabatic initial conditions. We find that there are two different coexistences of multiple attractors. And, the fractal structure of the attracting basin is also analyzed, and the formation mechanisms of “holes” and “contact” bifurcation have been revealed. At last, the long-term profits of the enterprises are studied. We find that some enterprises can even make more profits under a chaotic circumstance.

scholarly journals Multistability in a Fractional-Order Centrifugal Flywheel Governor System and Its Adaptive Control

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Bo Yan ◽  
Shaobo He ◽  
Shaojie Wang
Keyword(s):  
Fractional Order ◽  
Lyapunov Stability Theory ◽  
Adaptive Controller ◽  
Basins Of Attraction ◽  
Entropy Measure ◽  
Coexisting Attractors ◽  
Minimum Order ◽  
System Parameters ◽  
Multiple Coexisting Attractors ◽  
Derivative Order

In this paper, a 4D fractional-order centrifugal flywheel governor system is proposed. Dynamics including the multistability of the system with the variation of system parameters and the derivative order are investigated by Lyapunov exponents (LEs), bifurcation diagram, phase portrait, entropy measure, and basins of attraction, numerically. It shows that the minimum order for chaos of the fractional-order centrifugal flywheel governor system is q = 0.97, and the system has rich dynamics and produces multiple coexisting attractors. Moreover, the system is controlled by introducing the adaptive controller which is proved by the Lyapunov stability theory. Numerical analysis results verify the effectiveness of the proposed method.

Antimonotonicity, Chaos and Multiple Attractors in a Novel Autonomous Jerk Circuit

2017 ◽  
Vol 27 (07) ◽  
pp. 1750100 ◽  
Author(s):  
J. Kengne ◽  
A. Nguomkam Negou ◽  
Z. T. Njitacke
Keyword(s):  
Three Dimensional ◽  
Period Doubling ◽  
Basins Of Attraction ◽  
Electrical Circuit ◽  
Semiconductor Diode ◽  
Chaotic Attractors ◽  
Coexisting Attractors ◽  
Multiple Attractors ◽  
Systematic Analysis ◽  
The Stability

We perform a systematic analysis of a system consisting of a novel jerk circuit obtained by replacing the single semiconductor diode of the original jerk circuit described in [Sprott, 2011a] with a pair of semiconductor diodes connected in antiparallel. The model is described by a continuous time three-dimensional autonomous system with hyperbolic sine nonlinearity, and may be viewed as a control system with nonlinear velocity feedback. The stability of the (unique) fixed point, the local bifurcations, and the discrete symmetries of the model equations are discussed. The complex behavior of the system is categorized in terms of its parameters by using bifurcation diagrams, Lyapunov exponents, time series, Poincaré sections, and basins of attraction. Antimonotonicity, period doubling bifurcation, symmetry restoring crises, chaos, and coexisting bifurcations are reported. More interestingly, one of the key contributions of this work is the finding of various regions in the parameters’ space in which the proposed (“elegant”) jerk circuit experiences the unusual phenomenon of multiple competing attractors (i.e. coexistence of four disconnected periodic and chaotic attractors). The basins of attraction of various coexisting attractors display complexity (i.e. fractal basins boundaries), thus suggesting possible jumps between coexisting attractors in experiment. Results of theoretical analyses are perfectly traced by laboratory experimental measurements. To the best of the authors’ knowledge, the jerk circuit/system introduced in this work represents the simplest electrical circuit (only a quadruple op amplifier chip without any analog multiplier chip) reported to date capable of four disconnected periodic and chaotic attractors for the same parameters setting.

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