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.

Coexisting Hidden Attractors in a 4-D Simplified Lorenz System

2014 ◽  
Vol 24 (03) ◽  
pp. 1450034 ◽  
Author(s):  
Chunbiao Li ◽  
J. C. Sprott
Keyword(s):  
Parameter Space ◽  
Limit Cycles ◽  
Symmetric Pair ◽  
Hidden Attractors ◽  
Ode System ◽  
Quadratic Nonlinearities ◽  
Arnold Tongues ◽  
Two Parameters ◽  
Basin Boundaries ◽  
Simplified Lorenz System

A new simple four-dimensional equilibrium-free autonomous ODE system is described. The system has seven terms, two quadratic nonlinearities, and only two parameters. Its Jacobian matrix everywhere has rank less than 4. It is hyperchaotic in some regions of parameter space, while in other regions it has an attracting torus that coexists with either a symmetric pair of strange attractors or with a symmetric pair of limit cycles whose basin boundaries have an intricate fractal structure. In other regions of parameter space, it has three coexisting limit cycles and Arnold tongues. Since there are no equilibria, all the attractors are hidden. This combination of features has not been previously reported in any other system, especially one as simple as this.

Limit Cycles in a Class of Quartic Kolmogorov Model with Three Positive Equilibrium Points

2015 ◽  
Vol 25 (06) ◽  
pp. 1550080 ◽  
Author(s):  
Chaoxiong Du ◽  
Yirong Liu ◽  
Qi Zhang
Keyword(s):  
Hopf Bifurcation ◽  
Limit Cycle ◽  
Limit Cycles ◽  
Equilibrium Points ◽  
Positive Equilibrium ◽  
Bifurcation Problem ◽  
Kolmogorov Model ◽  
Limit Cycle Bifurcation ◽  
Perturbed Model

Limit cycle bifurcation problem of Kolmogorov model is interesting and significant both in theory and applications. In this paper, we will focus on investigating limit cycles for a class of quartic Kolmogorov model with three positive equilibrium points. Perturbed model can bifurcate three small limit cycles near (1, 2) or (2, 1) under a certain condition and can bifurcate one limit cycle near (1, 1). In addition, we have given some examples of simultaneous Hopf bifurcation and the structure of limit cycles bifurcated from three positive equilibrium points. The limit cycle bifurcation problem for Kolmogorov model with several positive equilibrium points are less seen in published references. Our result is good and interesting.

scholarly journals Coexisting Oscillation and Extreme Multistability for a Memcapacitor-Based Circuit

2017 ◽  
Vol 2017 ◽  
pp. 1-13 ◽  
Author(s):  
Guangyi Wang ◽  
Chuanbao Shi ◽  
Xiaowei Wang ◽  
Fang Yuan
Keyword(s):  
Limit Cycle ◽  
Infinite Number ◽  
Chaotic Attractor ◽  
Electronic Circuit ◽  
Equilibrium Points ◽  
Experimental Result ◽  
Coexisting Attractors ◽  
Extreme Multistability ◽  
Fix Point ◽  
The Stability

The coexisting oscillations are observed with a memcapacitor-based circuit that consists of two linear inductors, two linear resistors, and an active nonlinear charge-controlled memcapacitor. We analyze the dynamics of this circuit and find that it owns an infinite number of equilibrium points and coexisting attractors, which means extreme multistability arises. Furthermore, we also show the stability of the infinite many equilibria and analyze the coexistence of fix point, limit cycle, and chaotic attractor in detail. Finally, an experimental result of the proposed oscillator via an analog electronic circuit is given.

scholarly journals Limit Cycles in Planar Piecewise Linear Hamiltonian Systems with Three Zones Without Equilibrium Points

2020 ◽  
Vol 30 (11) ◽  
pp. 2050157
Author(s):  
Alexander Fernandes da Fonseca ◽  
Jaume Llibre ◽  
Luis Fernando Mello
Keyword(s):  
Limit Cycle ◽  
Hamiltonian Systems ◽  
Limit Cycles ◽  
Piecewise Linear ◽  
Equilibrium Points ◽  
Linear Hamiltonian Systems

We study the existence of limit cycles in planar piecewise linear Hamiltonian systems with three zones without equilibrium points. In this scenario, we have shown that such systems have at most one crossing limit cycle.

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.

Bistable and Coexisting Attractors in Current Modulated Edge Emitting Semiconductor Laser: Control and Microcontroller-Based Design

2021 ◽  
Author(s):  
Nasr Saeed ◽  
Serdar Çiçek ◽  
André Cheage Chamgoué ◽  
Sifeu Takougang Kingni ◽  
Zhouchao Wei
Keyword(s):  
Numerical Analysis ◽  
Semiconductor Laser ◽  
Limit Cycle ◽  
Chaotic Behavior ◽  
Equilibrium Points ◽  
Period Doubling ◽  
Chaotic Attractors ◽  
Coexisting Attractors ◽  
Laser Control ◽  
The Stability

Abstract This paper reports on the numerical analysis, control of coexisting attractors and microcontroller-based design of current modulated edge emitting semiconductor laser (CMEESL). The stability of equilibrium points of solitary edge emitting semiconductor laser found is investigated. By varying the amplitude of modulation current density, CMEESL displays periodic behaviors, period-doubling to chaotic behavior, bistability and coexistence between limit cycle and chaotic attractors. The coexistence between chaotic and limit cycle attractors is destroyed and controlled to a desired monostable trajectory by means of the linear augmentation method. In addition, a microcontroller-based circuit is also designed to indicate that CMEESL can be used in real applications. Microcontroller-based circuit outputs and numerical analysis results confirm each other.

Rotor Balancing Via an Enhanced Automatic Dynamic Balancer With Inductively Coupled Shunt Circuit

2019 ◽  
Vol 141 (3) ◽  
Author(s):  
Xiaowen Su ◽  
Hans A. DeSmidt
Keyword(s):  
Limit Cycle ◽  
Limit Cycles ◽  
Equilibrium Points ◽  
Lagrangian Description ◽  
Widespread Application ◽  
Inductively Coupled ◽  
Rlc Circuit ◽  
Automatic Balancing ◽  
Rotor Balancing ◽  
Balancing Principle

Despite the elegant nature of the automatic balancing principle for passive imbalance vibration control, the co-existence of undesired whirling limit-cycles is a major impediment to the more widespread application of automatic dynamic balancing devices also called automatic dynamic balancer (ADB) in industry. To enlarge the region of stable perfect balancing and to eliminate whirling limit-cycles, we develop an innovative enhanced ADB system. This new idea harnesses the automatic balancing principle via moving permanent magnet balancer masses which are inductively coupled to a parallel resistor–inductor–capacitor (RLC) circuit. It is found that the circuit parameters can be adjusted properly to suppress the whirling limit-cycle to enlarge the perfect balancing region. We start from a Lagrangian description of the system and get nonlinear autonomous equations-of-motion. We then solve two dominant steady-state solutions for the enhanced ADB system. One solution is for the perfect balancing equilibrium points (EPs), which can be solved analytically. While the other solution is for the whirling limit-cycle which is solved via a harmonic balance method. The stability of these solutions is then evaluated through eigenvalue analysis and Floquet theory. The newly involved electrical parameters, such as coupling coefficient, equivalent capacitance, and equivalent resistance, are designed via an arc-length continuation method to destabilize the limit-cycle solutions to then guarantee stable rotor balancing.

Limit Cycle Bifurcations for Piecewise Smooth Hamiltonian Systems with a Generalized Eye-Figure Loop

2016 ◽  
Vol 26 (12) ◽  
pp. 1650204 ◽  
Author(s):  
Jihua Yang ◽  
Liqin Zhao
Keyword(s):  
Lower Bound ◽  
Limit Cycle ◽  
Hamiltonian Systems ◽  
Limit Cycles ◽  
Upper Bound ◽  
Melnikov Function ◽  
Piecewise Smooth ◽  
First Order ◽  
Period Annulus

This paper deals with the limit cycle bifurcations for piecewise smooth Hamiltonian systems. By using the first order Melnikov function of piecewise near-Hamiltonian systems given in [Liu & Han, 2010], we give a lower bound and an upper bound of the number of limit cycles that bifurcate from the period annulus between the center and the generalized eye-figure loop up to the first order of Melnikov function.

scholarly journals ON PERIODIC SOLUTIONS OF LIÉNARD TYPE EQUATIONS

2013 ◽  
Vol 18 (5) ◽  
pp. 708-716 ◽  
Author(s):  
Svetlana Atslega ◽  
Felix Sadyrbaev
Keyword(s):  
Periodic Solutions ◽  
Limit Cycle ◽  
Limit Cycles ◽  
Type Equation ◽  
Convex Shape ◽  
Period Annulus ◽  
Conservative Equation ◽  
Liénard Type Equation

The Liénard type equation x'' + f(x, x')x' + g(x) = 0 (i) is considered. We claim that if the associated conservative equation x'' + g(x) = 0 has period annuli then a dissipation f(x, x') exists such that a limit cycle of equation (i) exists in a selected period annulus. Moreover, it is possible to define f(x, x') so that limit cycles appear in all period annuli. Examples are given. A particular example presents two limit cycles of non-convex shape in two disjoint period annuli.

3D Printing — The Basins of Tristability in the Lorenz System

2017 ◽  
Vol 27 (08) ◽  
pp. 1750128 ◽  
Author(s):  
Anda Xiong ◽  
Julien C. Sprott ◽  
Jingxuan Lyu ◽  
Xilu Wang
Keyword(s):  
Lorenz System ◽  
Initial Conditions ◽  
Equilibrium Points ◽  
Basins Of Attraction ◽  
Symmetric Pair ◽  
State Space Approach ◽  
3D Printers ◽  
The Lorenz System ◽  
Almost All ◽  
Space Approach

The famous Lorenz system is studied and analyzed for a particular set of parameters originally proposed by Lorenz. With those parameters, the system has a single globally attracting strange attractor, meaning that almost all initial conditions in its 3D state space approach the attractor as time advances. However, with a slight change in one of the parameters, the chaotic attractor coexists with a symmetric pair of stable equilibrium points, and the resulting tri-stable system has three intertwined basins of attraction. The advent of 3D printers now makes it possible to visualize the topology of such basins of attraction as the results presented here illustrate.

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