THE MECHANISM OF A CONTROLLED PENDULUM SYNCHRONIZING WITH PERIODIC MOTIONS IN A PERIODICALLY FORCED, DAMPED DUFFING OSCILLATOR

2011 ◽  
Vol 21 (07) ◽  
pp. 1813-1829 ◽  
Author(s):  
ALBERT C. J. LUO ◽  
FUHONG MIN
Keyword(s):  
Dynamical Systems ◽  
Chaotic Motion ◽  
Control Parameter ◽  
Duffing Oscillator ◽  
Periodic Motions ◽  
Chaotic Motions ◽  
Invariant Domain

In this paper, the analytical conditions for the controlled pendulum synchronizing with periodic motions in the Duffing oscillator are developed using the theory of discontinuous dynamical systems. From the analytical conditions, the synchronization invariant domain is obtained. The partial and full synchronizations of the controlled pendulum with periodic motions in the Duffing oscillator are discussed. The control parameter map for the synchronization is achieved from the analytical conditions, and numerical illustrations of the partial and full synchronizations are carried out to illustrate the analytical conditions. This synchronization is different from the controlled Duffing oscillator synchronizing with chaotic motion in the periodically excited pendulum. Because the periodically forced, damped Duffing oscillator possesses periodic and chaotic motions, further investigation on the controlled pendulum synchronizing with complicated periodic and chaotic motions in the Duffing oscillator will be accomplished in sequel.

The Synchronization of a Periodically Driven Pendulum With Periodic Motions in a Periodically Forced, Damped Duffing Oscillator

2010 ◽  
Author(s):  
Albert C. J. Luo ◽  
Fuhong Min
Keyword(s):  
Dynamical Systems ◽  
Duffing Oscillator ◽  
Periodic Motions ◽  
Invariant Domain ◽  
Periodically Driven

In this paper, the analytical conditions for the controlled pendulum synchronizing with periodic motions in Duffing oscillator is developed through the theory of discontinuous dynamical systems. The conditions for the synchronization invariant domain are obtained, and the partial and full synchronizations are illustrated for the analytical conditions.

ANALYTICAL DYNAMICS OF PERIOD-m FLOWS AND CHAOS IN NONLINEAR SYSTEMS

2012 ◽  
Vol 22 (04) ◽  
pp. 1250093 ◽  
Author(s):  
ALBERT C. J. LUO ◽  
JIANZHE HUANG
Keyword(s):  
Dynamical Systems ◽  
Analytical Solutions ◽  
Nonlinear Dynamical Systems ◽  
Duffing Oscillator ◽  
Harmonic Balance Method ◽  
Period Doubling ◽  
Nonlinear Damping ◽  
Periodic Motions ◽  
Chaotic Motions ◽  
Nonlinear Dynamical

In this paper, the analytical solutions for period-m flows and chaos in nonlinear dynamical systems are presented through the generalized harmonic balance method. The nonlinear damping, periodically forced, Duffing oscillator was investigated as an example to demonstrate the analytical solutions of periodic motions and chaos. Through this investigation, the mechanism for a period-m motion jumping to another period-n motion in numerical computation is found. In this problem, the Hopf bifurcation of periodic motions is equivalent to the period-doubling bifurcation via Poincare mappings of dynamical systems. The stable and unstable period-m motions can be obtained analytically. Even more, the stable and unstable chaotic motions can be achieved analytically. The methodology presented in this paper can be applied to other nonlinear vibration systems, which is independent of small parameters.

Bifurcation Analysis of a Fermi-Acceleration Oscillator Under Different Excitations

2011 ◽  
Author(s):  
Albert C. J. Luo ◽  
Yu Guo
Keyword(s):  
Dynamical Systems ◽  
Bifurcation Analysis ◽  
Fermi Acceleration ◽  
Periodic Motions ◽  
Chaotic Motions

In this paper, switchability and bifurcation of motions in a double excited Fermi acceleration oscillator is discussed using the theory of discontinuous dynamical systems. The two oscillators are chosen to have different excitation and parameters. The analytical conditions for motion switching in such a Fermi-oscillator are presented. Bifurcation scenario for periodic and chaotic motions is presented, and the analytical predictions of periodic motions are presented. Finally, different motions in such an oscillator are illustrated.

Analytical Dynamics of a Ball Bouncing on a Vibrating Table

2012 ◽  
Author(s):  
Yu Guo ◽  
Albert C. J. Luo
Keyword(s):  
Dynamical Systems ◽  
Numerical Simulations ◽  
Analytical Dynamics ◽  
Periodic Motions ◽  
Chaotic Motions ◽  
Theory Of Flow ◽  
Mapping Structures ◽  
Ball Bouncing ◽  
Vibrating Table

In this paper, the theory of flow switchability for discontinuous dynamical systems is applied. Domains and boundaries for such a discontinuous problem are defined and analytical conditions for motion switching are developed. The conditions explain the important role of switching phase on the motion switchability in such a system. To describe different motions, the generic mappings and mapping structures are introduced. Bifurcation scenarios for periodic and chaotic motions are presented for different motions and switchability. Numerical simulations are provided for periodic motions with impacts only and with impact chatter to stick in the system.

scholarly journals The damping term makes the Smale-horseshoe heteroclinic chaotic motion easier

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Huijing Sun ◽  
Hongjun Cao
Keyword(s):  
Chaotic Motion ◽  
Analytical Techniques ◽  
Nonlinear Damping ◽  
Deterministic System ◽  
Maximum Lyapunov Exponent ◽  
Periodic Motions ◽  
Damping Term ◽  
Chaotic Motions ◽  
Parametrically Excited ◽  
Smale Horseshoe

<p style='text-indent:20px;'>The nonlinear Rayleigh damping term that is introduced to the classical parametrically excited pendulum makes the parametrically excited pendulum more complex and interesting. The effect of the nonlinear damping term on the new excitable systems is investigated based on analytical techniques such as Melnikov theory. The threshold conditions for the occurrence of Smale-horseshoe chaos of this deterministic system are obtained. Compared with the existing conclusion, i.e. the smaller the damping term is, the easier the chaotic motions become when the damping term is linear, our analysis, however, finds that the smaller or the larger the damping term is, the easier the Smale-horseshoe heteroclinic chaotic motions become. Moreover, the bifurcation diagram and the patterns of attractors in Poincaré map are studied carefully. The results demonstrate the new system exhibits rich dynamical phenomena: periodic motions, quasi-periodic motions and even chaotic motions. Importantly, according to the property of transitive as well as the fractal layers for a chaotic attractor, we can verify whether a attractor is a quasi-periodic one or a chaotic one when the maximum lyapunov exponent method is difficult to distinguish. Numerical simulations confirm the analytical predictions and show that the transition from regular to chaotic motion.</p>

Switchability and Bifurcation of Motions in a Double-Excited Fermi-Acceleration Oscillator

2010 ◽  
Author(s):  
Albert C. J. Luo ◽  
Yu Guo
Keyword(s):  
Dynamical Systems ◽  
Fermi Acceleration ◽  
Periodic Motions ◽  
Chaotic Motions

In this paper, switchability and bifurcation of motions in a double excited Fermi acceleration oscillator is discussed using the theory of discontinuous dynamical systems. The analytical conditions for motion switching in such a Fermi-oscillator are presented. Bifurcation scenario for periodic and chaotic motions is presented, and the analytical predictions of periodic motions are presented. Finally, periodic and chaotic motions in such an oscillator are illustrated.

On Global Transversality and Chaos in Two-Dimensional Nonlinear Dynamical Systems

2007 ◽  
Vol 2 (3) ◽  
pp. 242-248 ◽  
Author(s):  
Albert C. J. Luo
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Dynamical Systems ◽  
Duffing Oscillator ◽  
Approximate Expression ◽  
Melnikov Function ◽  
Two Dimensional ◽  
Chaotic Motions ◽  
Nonlinear Dynamical ◽  
Energy Increment ◽  
Exact Energy

In this paper, the global transversality and tangency in two-dimensional nonlinear dynamical systems are discussed, and the exact energy increment function (L-function) for such nonlinear dynamical systems is presented. The Melnikov function is an approximate expression of the exact energy increment. A periodically forced, damped Duffing oscillator with a separatrix is investigated as a sampled problem. The corresponding analytical conditions for the global transversality and tangency to the separatrix are derived. Numerical simulations are carried out for illustrations of the analytical conditions. From analytical and numerical results, the simple zero of the energy increment (or the Melnikov function) may not imply that chaos exists. The conditions for the global transversality and tangency to the separatrix may be independent of the Melnikov function. Therefore, the analytical criteria for chaotic motions in nonlinear dynamical systems need to be further developed. The methodology presented in this paper is applicable to nonlinear dynamical systems without any separatrix.

Switching Mechanism and Complex Motions in an Extended Fermi-Acceleration Oscillator

2010 ◽  
Vol 5 (4) ◽  
Author(s):  
Albert C. J. Luo ◽  
Yu Guo
Keyword(s):  
Dynamical Systems ◽  
Extended Model ◽  
Time Varying ◽  
Fermi Acceleration ◽  
Periodic Motions ◽  
Chaotic Motions ◽  
Bouncing Ball ◽  
Mapping Structures ◽  
The Masses ◽  
First Time

In this paper, an extended model of the Fermi-acceleration oscillator is presented to describe impacting chatters, grazing, and sticking between the particle (or bouncing ball) and piston. The sticking phenomenon in such a system is investigated for the first time. Even in the traditional Fermi-oscillator, such a sticking phenomenon still exists but one often ignored it. In this paper, the analytical conditions for the grazing and sticking phenomena between the particle and piston in the Fermi-acceleration oscillator are developed from the theory of discontinuous dynamical systems. Compared with existing studies, the four exact mappings are used to analyze the motion behaviors of the Fermi-oscillator instead of one or two mappings. Mapping structures formed by generic mappings are adopted for the analytical predictions of periodic motions in the Fermi-acceleration oscillator. Periodic and chaotic motions in such an oscillator are illustrated to show motion complexity and grazing and sticking mechanism. Once the masses of the ball and primary mass are in the same quantity level, the model presented in this paper will be very useful and significant. This idea can apply to a system possessing two independent oscillators with impact, such as gear transmission systems, bearing systems, and time-varying billiard systems.

Complex Dynamics of Projective Synchronization of Chua Circuits with Different Scrolls

2015 ◽  
Vol 25 (05) ◽  
pp. 1530016 ◽  
Author(s):  
Fuhong Min ◽  
Albert C. J. Luo
Keyword(s):  
Dynamical Systems ◽  
Complex Dynamics ◽  
Experimental Results ◽  
Projective Synchronization ◽  
Chaotic Motions ◽  
Invariant Domain

In this paper, the dynamics mechanism of the projective synchronization of Chua circuits with different scrolls is investigated analytically through the theory of discontinuous dynamical systems. The analytical conditions for the projective synchronization of Chua circuits with chaotic motions are developed. From these conditions, the parameter characteristics of the projective synchronization of Chua circuits with different scrolls are discussed, and the corresponding parameter maps and the invariant domain for such projective synchronization of Chua circuits are presented. Illustrations for partial and full projective synchronizations of the Chua circuits are given. The projective synchronization of Chua circuits is implemented experimentally, and numerical and experimental results are compared.

PARAMETRIC ANALYSIS OF BIFURCATION AND CHAOS IN A PERIODICALLY DRIVEN HORIZONTAL IMPACT PAIR

2012 ◽  
Vol 22 (11) ◽  
pp. 1250268 ◽  
Author(s):  
YU GUO ◽  
ALBERT C. J. LUO
Keyword(s):  
Dynamical Systems ◽  
Parametric Analysis ◽  
Periodic Motions ◽  
Bifurcation And Chaos ◽  
Chaotic Motions ◽  
Periodically Driven ◽  
Mapping Structures ◽  
Stability And Bifurcation Analysis ◽  
Different Types ◽  
The Impact

In this paper, complex motions and chaos in the periodically driven horizontal impact pair are investigated using the theory of switchability for discontinuous dynamical systems. Domains and boundaries are defined due to the discontinuity caused by impacts. Analytical conditions for switching of stick and grazing motions are derived in detail. Generic mappings are introduced to describe different periodic motions and to identify the mapping structures of chaos. The periodic motions in such impact pair are analytically predicted, and the corresponding stability and bifurcation analysis of periodic motions are carried out. Parameter maps with different types of motions are developed. Periodic and chaotic motions with different mapping structures are illustrated numerically for a better understanding of physics of ball motions in the impact pair.

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