Bifurcations and Chaos in the Duffing Equation with Parametric Excitation and Single External Forcing

2017 ◽  
Vol 27 (08) ◽  
pp. 1750125 ◽  
Author(s):  
Tao Jiang ◽  
Zhiyan Yang ◽  
Zhujun Jing
Keyword(s):  
Parametric Excitation ◽  
Chaotic Motion ◽  
Period Doubling ◽  
Melnikov Method ◽  
Original System ◽  
Duffing Equation ◽  
Periodic Perturbation ◽  
Phase Portraits ◽  
External Forcing ◽  
Dynamical Behaviors

We study the Duffing equation with parametric excitation and single external forcing and obtain abundant dynamical behaviors of bifurcations and chaos. The criteria of chaos of the Duffing equation under periodic perturbation are obtained through the Melnikov method. And the existence of chaos of the averaged system of the Duffing equation under the quasi-periodic perturbation [Formula: see text] (where [Formula: see text] is not rational relative to [Formula: see text]) and [Formula: see text] is shown, but the existence of chaos of averaged system of the Duffing equation cannot be proved when [Formula: see text],[Formula: see text]7–15, whereas the occurrence of chaos in the original system can be shown by numerical simulation. Numerical simulations not only show the correctness of the theoretical analysis but also exhibit some new complex dynamical behaviors, including homoclinic or heteroclinic bifurcation surfaces, bifurcation diagrams, Lyapunov exponent diagrams, phase portraits and Poincaré maps. We find a large chaotic region with some solitary period parameter points, a large period and quasi-period region with some solitary chaotic parameter points, period-doubling to chaos and chaos to inverse period-doubling, nondense curvilinear chaotic attractor, nonattracting chaotic motion, nonchaotic attracting set, fragmental chaotic attractors. Almost chaotic motion and almost nonchaotic motion appear through adjusting the parameters of the Duffing system, which can be taken as a strategy of chaotic control or a strategy of chaotic motion to nonchaotic motion.

Bifurcations and Chaos of Duffing–van der Pol Equation with Nonsymmetric Nonlinear Restoring and Two External Forcing Terms

2014 ◽  
Vol 24 (03) ◽  
pp. 1430011 ◽  
Author(s):  
Zhiyan Yang ◽  
Tao Jiang ◽  
Zhujun Jing
Keyword(s):  
Numerical Simulation ◽  
Periodic Orbits ◽  
Period Doubling ◽  
Original System ◽  
Periodic Perturbation ◽  
Phase Portraits ◽  
External Forcing ◽  
Van Der Pol Equation ◽  
Van Der Pol ◽  
Period Doubling Bifurcations

Bifurcations and chaos of Duffing–van der Pol equation with nonsymmetric nonlinear restoring and two external forcing terms are investigated. The threshold values of the existence of chaotic motion are obtained under periodic perturbation. By the second-order averaging method, we prove the criteria of the existence of chaos in an averaged system under quasi-periodic perturbation for ω2 = nω1 + εσ, n = 1, 2, 3, 5, and cannot prove the criterion of existence of chaos in an averaged system under quasi-periodic perturbation for ω2 = nω1 + εσ, n = 4, 6, 7, …, where σ is not rational to ω1, but can show the occurrence of chaos in the original system by numerical simulation. Numerical simulation including homoclinic or heteroclinic bifurcation surfaces, bifurcation diagrams, maximal Lyapunov exponents, phase portraits and Poincaré maps, not only show the consistence with the theoretical analysis but also exhibit more new complex dynamical behaviors. We show that cascades of interlocking period-doubling and reverse period-doubling bifurcations lead to interleaving occurrence of chaotic behaviors and quasi-periodic orbits, symmetry-breaking of periodic orbits in chaotic regions, onset of chaos occurring more than once, chaos suddenly disappearing to periodic orbits, strange nonchaotic attractor, nonattracting chaotic set and nice chaotic attractors.

Subharmonic Bifurcations and Transition to Chaos in a Pipe Conveying Fluid under Harmonic Excitation

2013 ◽  
Vol 444-445 ◽  
pp. 791-795
Author(s):  
Yi Xiang Geng ◽  
Han Ze Liu
Keyword(s):  
Chaotic Behavior ◽  
Melnikov Method ◽  
Harmonic Excitation ◽  
Phase Portraits ◽  
Pipe Conveying Fluid ◽  
Bifurcation Diagrams ◽  
Dynamical Behaviors ◽  
Transition To Chaos ◽  
Subharmonic Bifurcations ◽  
Conveying Fluid

The subharmonic and chaotic behavior of a two end-fixed fluid conveying pipe whose base is subjected to a harmonic excitation are investigated. Melnikov method is applied for the system, and Melnikov criterions for subharmonic and homoclinic bifurcations are obtained analytically. The numerical simulations (including bifurcation diagrams, maximal Lyapunov exponents, phase portraits and Poincare map) confirm the analytical predictions and exhibit the complicated dynamical behaviors.

COMPLEX DYNAMICS IN PENDULUM EQUATION WITH PARAMETRIC AND EXTERNAL EXCITATIONS I

2006 ◽  
Vol 16 (10) ◽  
pp. 2887-2902 ◽  
Author(s):  
ZHUJUN JING ◽  
JIANPING YANG
Keyword(s):  
Bifurcation Diagram ◽  
Periodic Orbits ◽  
Complex Dynamics ◽  
Period Doubling ◽  
Periodic Perturbation ◽  
Phase Portraits ◽  
Chaotic Attractors ◽  
Maximum Lyapunov Exponent ◽  
Pendulum Equation ◽  
Parametric And External Excitations

Pendulum equation with parametric and external excitations is investigated in (I) and (II). In (I), by applying Melnikov's method, we prove the criterion of existence of chaos under periodic perturbation. The numerical simulations, including bifurcation diagram of fixed points, bifurcation diagram of system in three- and two-dimensional space, homoclinic and heteroclinic bifurcation surface, Maximum Lyapunov exponent, phase portraits, Poincaré map, are plotted to illustrate theoretical analysis, and to expose the complex dynamical behaviors including the period-n (n = 2 to 6, 10, 15 and 20) orbits in different chaotic regions, interlocking periodic orbits, symmetry-breaking of periodic orbit, cascade of period-doubling bifurcations from period-5 and -10 orbits, reverse period-doubling bifurcation, onset of chaos which occurs more than once for a given external frequency or parametric frequency and chaos suddenly converting to periodic orbits, sudden jump in the size of attractors which is associated with the transverse intersection of stable and unstable manifolds of perturbed saddle, hopping behavior of chaos, transient chaos with complex periodic windows and interior crisis, varied chaotic attractors including the more than three-band and eight-band chaotic attractors, chaotic attractor after strange nonchaotic attractor. In particular, we observe that the system can leave chaotic region to periodic motion by adjusting damping δ, spring constant α and frequency Ω of parametric excitation which can be considered as a control strategy. In (II), we will investigate the complex dynamics under quasi-periodic perturbation.

COMPLEX DYNAMICS IN PENDULUM EQUATION WITH PARAMETRIC AND EXTERNAL EXCITATIONS II

2006 ◽  
Vol 16 (10) ◽  
pp. 3053-3078 ◽  
Author(s):  
ZHUJUN JING ◽  
JIANPING YANG
Keyword(s):  
Complex Dynamics ◽  
Periodic Perturbation ◽  
Phase Portraits ◽  
Maximum Lyapunov Exponent ◽  
Melnikov's Method ◽  
Melnikov’S Method ◽  
Pendulum Equation ◽  
Dynamical Behaviors ◽  
Periodic Sets ◽  
Parametric And External Excitations

This paper (II) is a continuation of "Complex dynamics in pendulum equation with parametric and external excitations (I)." By applying second-order averaging method and Melnikov's method, we obtain the criterion of existence of chaos in an averaged system under quasi-periodic perturbation for Ω = nω + ∊ν, n = 1, 2, 4 and cannot prove the criterion of existence of chaos in averaged system under quasi-periodic perturbation for Ω = nω + ∊ν, n = 3, 5–15 by Melnikov's method, where ν is not rational to ω. However, we show the occurrence of chaos in the averaged and original systems under quasi-periodic perturbation for Ω = nω + ∊ν, n = 3, 5 by numerical simulation. The numerical simulations, include the bifurcation diagram of fixed points, bifurcation diagrams in three- and two-dimensional spaces, homoclinic bifurcation surface, maximum Lyapunov exponent, phase portraits, Poincaré map, are plotted to illustrate theoretical analysis, and to expose the complex dynamical behaviors, including period-3 orbits in different chaotic regions, interleaving occurrence of chaotic behaviors and quasi-periodic behaviors, a different kind of interior crisis, jumping behavior of quasi-periodic sets, different nice quasi-periodic attractors, nonchaotic attractors and chaotic attractors, coexistence of three quasi-periodic sets, onset of chaos which occurs more than once for a given external frequency or amplitudes, and quasi-periodic route to chaos. We do not find the period-doubling cascade. The dynamical behaviors under quasi-periodic perturbation are different from that of periodic perturbation.

scholarly journals Circuit Implementation of a Modified Chaotic System with Hyperbolic Sine Nonlinearities Using Bi-Color LED

Technologies ◽  
2021 ◽  
Vol 9 (1) ◽  
pp. 15
Author(s):  
Christos K. Volos ◽  
Lazaros Moysis ◽  
George D. Roumelas ◽  
Aggelos Giakoumis ◽  
Hector E. Nistazakis ◽  
...  
Keyword(s):  
Lyapunov Exponents ◽  
Chaotic Behavior ◽  
Period Doubling ◽  
Original System ◽  
Dynamical Analysis ◽  
Phase Portraits ◽  
Circuit Realization ◽  
Hyperbolic Sine ◽  
Route To Chaos ◽  
Sine Functions

In this paper, a chaotic three dimansional dynamical system is proposed, that is a modification of the system in Volos et al. (2017). The new system has two hyperbolic sine nonlinear terms, as opposed to the original system that only included one, in order to optimize system’s chaotic behavior, which is confirmed by the calculation of the maximal Lyapunov exponents and Kaplan-Yorke dimension. The system is experimentally realized, using Bi-color LEDs to emulate the hyperbolic sine functions. An extended dynamical analysis is then performed, by computing numerically the system’s bifurcation and continuation diagrams, Lyapunov exponents and phase portraits, and comparing the numerical simulations with the circuit simulations. A series of interesting phenomena are unmasked, like period doubling route to chaos, coexisting attractors and antimonotonicity, which are all verified from the circuit realization of the system. Hence, the circuit setup accurately emulates the chaotic dynamics of the proposed system.

BIFURCATION AND CHAOS IN THE TINKERBELL MAP

2011 ◽  
Vol 21 (11) ◽  
pp. 3137-3156 ◽  
Author(s):  
SHAOLIANG YUAN ◽  
TAO JIANG ◽  
ZHUJUN JING
Keyword(s):  
Three Dimensional ◽  
Period Doubling ◽  
Phase Portraits ◽  
Transient Chaos ◽  
Flip Bifurcation ◽  
Maximum Lyapunov Exponent ◽  
Invariant Circle ◽  
Dynamical Behaviors ◽  
Fold Bifurcation ◽  
Period Doubling Bifurcations

In this paper, the dynamical behaviors of the Tinkerbell map are investigated in detail. Conditions for the existence of fold bifurcation, flip bifurcation and Hopf bifurcation are derived, and chaos in the sense of Marotto is verified by both analytical and numerical methods. Numerical simulations include bifurcation diagrams in two- and three-dimensional spaces, phase portraits, and the maximum Lyapunov exponent and fractal dimension, as well as the distribution of dynamics in the parameter plane, which exhibit new and interesting dynamical behaviors. More specifically, this paper reports the findings of chaos in the sense of Marotto, a route from an invariant circle to transient chaos with a great abundance of periodic windows, including period-2, 7, 8, 9, 10, 13, 17, 19, 23, 26 and so on, and suddenly appearing or disappearing chaos, convergence of an invariant circle to a period-one orbit, symmetry-breaking of periodic orbits, interlocking period-doubling bifurcations in chaotic regions, interior crisis, chaotic attractors, coexisting (2, 10, 13) chaotic sets, two coexisting invariant circles, two attracting chaotic sets coexisting with a non-attracting chaotic set, and so on, all in the Tinkerbell map. In particular, it is found that there is no obvious road from period-doubling bifurcations to chaos, but there is a route from a period-one orbit to an invariant circle and then to transient chaos as the parameters are varied. Combining the existing results in the current literature with the new results reported in this paper, a more complete understanding of the Tinkerbell map is obtained.

COMPLEX DYNAMICS IN A PENDULUM EQUATION WITH A PHASE SHIFT

2012 ◽  
Vol 22 (12) ◽  
pp. 1250307 ◽  
Author(s):  
XIANWEI CHEN ◽  
XIANGLING FU ◽  
ZHUJUN JING
Keyword(s):  
Phase Shift ◽  
Periodic Orbits ◽  
Period Doubling ◽  
Periodic Perturbation ◽  
Chaotic Attractors ◽  
Melnikov's Method ◽  
Periodic Perturbations ◽  
Melnikov’S Method ◽  
Dynamical Behaviors ◽  
Period Doubling Bifurcations

Pendulum equation with a phase shift, parametric and external excitations is investigated in detail. By applying Melnikov's method, we prove the criteria of existence of chaos under periodic perturbation. Numerical simulations, including bifurcation diagrams of fixed points, bifurcation diagrams of the system in three- and two-dimensional spaces, homoclinic and heteroclinic bifurcation surfaces, Maximum Lyapunov exponents (ML), Fractal Dimension (FD), phase portraits, Poincaré maps are plotted to illustrate the theoretical analysis, and to expose the complex dynamical behaviors including the onset of chaos, sudden conversion of chaos to period orbits, interior crisis, periodic orbits, the symmetry-breaking of periodic orbits, jumping behaviors of periodic orbits, new chaotic attractors including two-three-four-five-six-eight-band chaotic attractors, nonchaotic attractors, period-doubling bifurcations from period-1, 2, 3 and 5 to chaos, reverse period-doubling bifurcations from period-3 and 5 to chaos, and so on.By applying the second-order averaging method and Melnikov's method, we obtain the criteria of existence of chaos in an averaged system under quasi-periodic perturbation for Ω = nω + ϵν, n = 1, 2, 4, but cannot prove the criteria of existence of chaos in the averaged system under quasi-periodic perturbation for Ω = nω + ϵν, n = 3, 5 – 15, by Melnikov's method, where ν is not rational to ω. By using numerical simulation, we have verified our theoretical analysis and studied the effect of parameters of the original system on the dynamical behaviors generated under quasi-periodic perturbations, such as the onset of chaos, jumping behaviors of quasi-periodic orbits, interleaving occurrence of chaotic behaviors and nonchaotic behaviors, interior crisis, quasi-periodic orbits to chaotic attractors, sudden conversion of chaos to quasi-periodic behaviors, nonchaotic attractors, and so on. However, we did not find period-doubling and reverse period-doubling bifurcations. We found that the dynamical behaviors under quasi-periodic perturbations are different from that under periodic perturbations, and the dynamics with a phase shift are different from the dynamics without phase shift.

scholarly journals Heteroclinic Bifurcation Behaviors of a Duffing Oscillator with Delayed Feedback

2018 ◽  
Vol 2018 ◽  
pp. 1-12 ◽  
Author(s):  
Shao-Fang Wen ◽  
Ju-Feng Chen ◽  
Shu-Qi Guo
Keyword(s):  
Necessary Conditions ◽  
Duffing Oscillator ◽  
Period Doubling ◽  
Melnikov Method ◽  
Threshold Value ◽  
Phase Portraits ◽  
Heteroclinic Bifurcation ◽  
Velocity Feedback ◽  
Time Histories ◽  
New Phenomena

The heteroclinic bifurcation and chaos of a Duffing oscillator with forcing excitation under both delayed displacement feedback and delayed velocity feedback are studied by Melnikov method. The Melnikov function is analytically established to detect the necessary conditions for generating chaos. Through the analysis of the analytical necessary conditions, we find that the influences of the delayed displacement feedback and delayed velocity feedback are separable. Then the influences of the displacement and velocity feedback parameters on heteroclinic bifurcation and threshold value of chaotic motion are investigated individually. In order to verify the correctness of the analytical conditions, the Duffing oscillator is also investigated by numerical iterative method. The bifurcation curves and the largest Lyapunov exponents are provided and compared. From the analysis of the numerical simulation results, it could be found that two types of period-doubling bifurcations occur in the Duffing oscillator, so that there are two paths leading to the chaos in this oscillator. The typical dynamical responses, including time histories, phase portraits, and Poincare maps, are all carried out to verify the conclusions. The results reveal some new phenomena, which is useful to design or control this kind of system.

Behavior of a Self-Sustained Electromechanical Transducer and Routes to Chaos

2005 ◽  
Vol 128 (3) ◽  
pp. 282-293 ◽  
Author(s):  
J. C. Chedjou ◽  
K. Kyamakya ◽  
I. Moussa ◽  
H.-P. Kuchenbecker ◽  
W. Mathis
Keyword(s):  
Electronic Circuit ◽  
Initial Conditions ◽  
Dynamical Behavior ◽  
Period Doubling ◽  
Electromechanical System ◽  
Phase Portraits ◽  
Coupling Coefficients ◽  
Oscillatory Solutions ◽  
Electromechanical Transducer ◽  
Design Engineers

This paper studies the dynamics of a self-sustained electromechanical transducer. The stability of fixed points in the linear response is examined. Their local bifurcations are investigated and different types of bifurcation likely to occur are found. Conditions for the occurrence of Hopf bifurcations are derived. Harmonic oscillatory solutions are obtained in both nonresonant and resonant cases. Their stability is analyzed in the resonant case. Various bifurcation diagrams associated to the largest one-dimensional (1-D) numerical Lyapunov exponent are obtained, and it is found that chaos can appear suddenly, through period doubling, period adding, or torus breakdown. The extreme sensitivity of the electromechanical system to both initial conditions and tiny variations of the coupling coefficients is also outlined. The experimental study of̱the electromechanical system is carried out. An appropriate electronic circuit (analog simulator) is proposed for the investigation of the dynamical behavior of the electromechanical system. Correspondences are established between the coefficients of the electromechanical system model and the components of the electronic circuit. Harmonic oscillatory solutions and phase portraits are obtained experimentally. One of the most important contributions of this work is to provide a set of reliable analytical expressions (formulas) describing the electromechanical system behavior. These formulas are of great importance for design engineers as they can be used to predict the states of the electromechanical systems and respectively to avoid their destruction. The reliability of the analytical formulas is demonstrated by the very good agreement with the results obtained by both the numeric and the experimental analysis.

scholarly journals Chaos and Control in Coronary Artery System

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Yanxiang Shi
Keyword(s):  
Coronary Artery ◽  
Feedback Control ◽  
Numerical Simulations ◽  
Melnikov Method ◽  
Phase Portraits ◽  
Bifurcation Diagrams ◽  
Nonlinear Dynamical ◽  
The Road ◽  
Two Systems ◽  
And Control

Two types of coronary artery system N-type and S-type, are investigated. The threshold conditions for the occurrence of Smale horseshoe chaos are obtained by using Melnikov method. Numerical simulations including phase portraits, potential diagram, homoclinic bifurcation curve diagrams, bifurcation diagrams, and Poincaré maps not only prove the correctness of theoretical analysis but also show the interesting bifurcation diagrams and the more new complex dynamical behaviors. Numerical simulations are used to investigate the nonlinear dynamical characteristics and complexity of the two systems, revealing bifurcation forms and the road leading to chaotic motion. Finally the chaotic states of the two systems are effectively controlled by two control methods: variable feedback control and coupled feedback control.

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