scholarly journals Confusion threshold study of the Duffing oscillator with a nonlinear fractional damping term

2020 ◽  
pp. 146134842092268
Author(s):  
Wang Mei-Qi ◽  
Ma Wen-Li ◽  
Chen En-Li ◽  
Yang Shao-Pu ◽  
Chang Yu-Jian ◽  
...  
Keyword(s):  
Chaotic Motion ◽  
Numerical Solutions ◽  
Duffing Oscillator ◽  
Time History ◽  
Nonlinear Damping ◽  
Critical Conditions ◽  
Fractional Damping ◽  
Nonlinear Parameters ◽  
Smale Horseshoe ◽  
Melnikov Theory

In this study, the critical conditions for generating chaos in a Duffing oscillator with nonlinear damping and fractional derivative are investigated. The Melnikov function of the Duffing oscillator is established based on Melnikov theory. The necessary analytical conditions and critical value curves of chaotic motion in the sense of Smale horseshoe are obtained. The numerical solutions of chaotic motion, including time history diagram, frequency spectrum diagram, phase diagram, and Poincare map, are studied. The correctness of the analytical solution is verified through a comparison of numerical and analytical calculations. The effects of linear and nonlinear parameters on chaotic motion are also analyzed. These results are relevant to the study of system dynamics.

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>

The Chaotic Behavior of Symmetric Laminated Composite Arch

2010 ◽  
Vol 29-32 ◽  
pp. 287-292
Author(s):  
Jian Jun Wang ◽  
Zhi Jun Han ◽  
Chao Kang ◽  
Guo Yun Lu ◽  
Shan Yuan Zhang
Keyword(s):  
Chaotic Motion ◽  
Chaotic Behavior ◽  
Steady Motion ◽  
Time History ◽  
Laminated Composite ◽  
Critical Conditions ◽  
Loading Frequency ◽  
Periodic Excitation ◽  
Chaotic Region ◽  
Nonlinear Dynamic Equations

Chaotic motion of symmetric laminated composite arch with two hinge supports under transverse periodic excitation was investigated. The nonlinear dynamic equations of the arch are changed into the square-order and cubic nonlinear differential dynamic system by Galerkin method, and its homoclinic orbit parameter equations are also acquired. The critical conditions of horseshoe-type chaos are obtained by using Melnikov function. The influence of loading frequency on chaotic region are analysed by numerical calculation. The motion behaviors of system are described through the bifurcation diagrams, the time-history curve, phase portrait and Poincaré map. The results are given as follows. The influence of loading frequency on chaotic region are significant. When the height of arch reach some value, the system can occur horseshoe-type chaos. The system of symmetric laminated composite arch under transverse periodic excitation may occur steady motion and chaotic motion.

scholarly journals Chaotic Motion in Forced Duffing System Subject to Linear and Nonlinear Damping

2017 ◽  
Vol 2017 ◽  
pp. 1-8 ◽  
Author(s):  
Tai-Ping Chang
Keyword(s):  
Lyapunov Exponent ◽  
Nonlinear System ◽  
Chaotic Motion ◽  
Chaotic Behavior ◽  
Duffing Oscillator ◽  
Nonlinear Damping ◽  
Critical Value ◽  
Duffing System ◽  
Linear And Nonlinear

This paper investigates the chaotic motion in forced Duffing oscillator due to linear and nonlinear damping by using Melnikov technique. In particular, the critical value of the forcing amplitude of the nonlinear system is calculated by Melnikov technique. Further, the top Lyapunov exponent of the nonlinear system is evaluated by Wolf’s algorithm to determine whether the chaotic phenomenon of the nonlinear system actually occurs. It is concluded that the chaotic motion of the nonlinear system occurs when the forcing amplitude exceeds the critical value, and the linear and nonlinear damping can generate pronounced effects on the chaotic behavior of the forced Duffing oscillator.

ANALYTICAL ESTIMATES OF THE EFFECT OF NONLINEAR DAMPING IN SOME NONLINEAR OSCILLATORS

2000 ◽  
Vol 10 (09) ◽  
pp. 2257-2267 ◽  
Author(s):  
JOSÉ L. TRUEBA ◽  
JOAQUÍN RAMS ◽  
MIGUEL A. F. SANJUÁN
Keyword(s):  
Nonlinear Oscillator ◽  
Duffing Oscillator ◽  
Nonlinear Oscillators ◽  
Nonlinear Damping ◽  
Damping Coefficient ◽  
Critical Parameters ◽  
Nonlinear Dissipation ◽  
Simple Pendulum ◽  
Melnikov Theory ◽  
Damped Systems

This paper reports on the effect of nonlinear damping on certain nonlinear oscillators, where analytical estimates provided by the Melnikov theory are obtained. We assume general nonlinear damping terms proportional to the power of velocity. General and useful expressions for the nonlinearly damped Duffing oscillator and for the nonlinearly damped simple pendulum are computed. They provide the critical parameters in terms of the damping coefficient and damping exponent, that is, the power of the velocity, for which complicated behavior is expected. We also consider generalized nonlinear damped systems, which may contain several nonlinear damping terms. Using the idea of Melnikov equivalence, we show that the effect of nonlinear dissipation can be equivalent to a linearly damped nonlinear oscillator with a modified damping coefficient.

scholarly journals Effect of Nonlinear Dissipation on the Basin Boundaries of a Driven Two-Well Modified Rayleigh–Duffing Oscillator

2015 ◽  
Vol 25 (02) ◽  
pp. 1550024 ◽  
Author(s):  
C. H. Miwadinou ◽  
A. V. Monwanou ◽  
J. B. Chabi Orou
Keyword(s):  
Initial Conditions ◽  
Duffing Oscillator ◽  
Excitation Amplitude ◽  
Nonlinear Damping ◽  
Homoclinic Bifurcation ◽  
Nonlinear Dissipation ◽  
Transition To Chaos ◽  
Melnikov Criterion ◽  
Quadratic Nonlinearities ◽  
Basin Boundaries

This paper considers the effect of nonlinear dissipation on the basin boundaries of a driven two-well modified Rayleigh–Duffing oscillator where pure cubic, unpure cubic, pure quadratic and unpure quadratic nonlinearities are considered. By analyzing the potential, an analytic expression is found for the homoclinic orbit. The Melnikov criterion is used to examine a global homoclinic bifurcation and transition to chaos. Unpure quadratic parameter and parametric excitation amplitude effects are found on the critical Melnikov amplitude μ cr . Finally, the phase space of initial conditions is carefully examined in order to analyze the effect of the nonlinear damping, and particularly how the basin boundaries become fractalized.

Stability and Bifurcation Analysis of Periodic Motions in a Periodically Forced Duffing Oscillator

2011 ◽  
Author(s):  
Albert C. J. Luo ◽  
Jianzhe Huang
Keyword(s):  
Approximate Solution ◽  
Bifurcation Analysis ◽  
Duffing Oscillator ◽  
Approximate Solutions ◽  
Nonlinear Damping ◽  
Periodic Motions ◽  
Stability And Bifurcation ◽  
Stability And Bifurcation Analysis

In this paper, the analytical, approximate solutions of period-1 motions in the nonlinear damping, periodically forced, Duffing oscillator is obtained. The corresponding stability and bifurcation analysis of the HB2 approximate solution of period-1 motions in the forced Duffing oscillator is carried out. Numerical illustrations of period-1 motions are presented.

Chaotic Motion in the Vicinity of Separatrix of a Parametrically Driven Duffing Oscillator With a Twin-Well Potential

2001 ◽  
Author(s):  
Albert C. J. Luo
Keyword(s):  
Chaotic Motion ◽  
Duffing Oscillator

Abstract The conditions for the (M:1) and (2M:1) resonances inside and outside of the separatrix of the parametrically driven Duffing oscillator are determined. The onset of such resonance in the vicinity of separatrix is investigated analytically and numerically. The results presented in this article can be applied to the post-buckled structures under parametric excitations.

scholarly journals Nonlinear Dynamic Analysis of Rotor Rub-Impact System

2019 ◽  
Vol 2019 ◽  
pp. 1-20
Author(s):  
Youfeng Zhu ◽  
Zibo Wang ◽  
Qiang Wang ◽  
Xinhua Liu ◽  
Hongyu Zang ◽  
...  
Keyword(s):  
Periodic Motion ◽  
Chaotic Motion ◽  
Nonlinear Dynamic Analysis ◽  
Period Doubling ◽  
Time History ◽  
Rotor System ◽  
Quasiperiodic Motion ◽  
Rotor Bearing ◽  
Motion Period ◽  
Bearing System

A dynamic model of a double-disk rub-impact rotor-bearing system with rubbing fault is established. The dynamic differential equation of the system is solved by combining the numerical integration method with MATLAB. And the influence of rotor speed, disc eccentricity, and stator stiffness on the response of the rotor-bearing system is analyzed. In the rotor system, the time history diagram, the axis locus diagram, the phase diagram, and the Poincaré section diagram in different rotational speeds are drawn. The characteristics of the periodic motion, quasiperiodic motion, and chaotic motion of the system in a given speed range are described in detail. The ways of the system entering and leaving chaos are revealed. The transformation and evolution process of the periodic motion, quasiperiodic motion, and chaotic motion are also analyzed. It shows that the rotor system enters chaos by the way of the period-doubling bifurcation. With the increase of the eccentricity, the quasi-periodicity evolution is chaotic. The quasiperiodic motion evolves into the periodic three motion phenomenon. And the increase of the stator stiffness will reduce the chaotic motion period.

scholarly journals The direct method of Lyapunov for nonlinear dynamical systems with fractional damping

2020 ◽  
Vol 102 (4) ◽  
pp. 2017-2037
Author(s):  
Matthias Hinze ◽  
André Schmidt ◽  
Remco I. Leine
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Dynamical Systems ◽  
Direct Method ◽  
Infinite Delay ◽  
Functional Differential Equations ◽  
Nonlinear Damping ◽  
Degree Of Freedom ◽  
Lyapunov Functionals ◽  
State Variables ◽  
Fractional Damping

AbstractIn this paper, we introduce a generalization of Lyapunov’s direct method for dynamical systems with fractional damping. Hereto, we embed such systems within the fundamental theory of functional differential equations with infinite delay and use the associated stability concept and known theorems regarding Lyapunov functionals including a generalized invariance principle. The formulation of Lyapunov functionals in the case of fractional damping is derived from a mechanical interpretation of the fractional derivative in infinite state representation. The method is applied on a single degree-of-freedom oscillator first, and the developed Lyapunov functionals are subsequently generalized for the finite-dimensional case. This opens the way to a stability analysis of nonlinear (controlled) systems with fractional damping. An important result of the paper is the solution of a tracking control problem with fractional and nonlinear damping. For this problem, the classical concepts of convergence and incremental stability are generalized to systems with fractional-order derivatives of state variables. The application of the related method is illustrated on a fractionally damped two degree-of-freedom oscillator with regularized Coulomb friction and non-collocated control.

Topological vortex dynamics in axisymmetric viscous flows

1994 ◽  
Vol 260 ◽  
pp. 57-80 ◽  
Author(s):  
Mogens V. Melander ◽  
Fazle Hussain
Keyword(s):  
Phase Portrait ◽  
Critical Points ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Saddle Points ◽  
Time History ◽  
Vorticity Field ◽  
Vortex Lines ◽  
Short Period ◽  
Virtual Velocity

The topology of vortex lines and surfaces is examined in incompressible viscous axisymmetric flows with swirl. We argue that the evolving topology of the vorticity field must be examined in terms of axisymmetric vortex surfaces rather than lines, because only the surfaces enjoy structural stability. The meridional cross-sections of these surfaces are the orbits of a dynamical system with the azimuthal circulation being a Hamiltonian H and with time as a bifurcation parameter μ. The dependence of H on μ is governed by the Navier–Stokes equations; their numerical solutions provide H. The level curves of H establish a time history for the motion of vortex surfaces, so that the circulation they contain remains constant. Equivalently, there exists a virtual velocity field in which the motion of the vortex surfaces is frozen almost everywhere; the exceptions occur at critical points in the phase portrait where the virtual velocity is singular. The separatrices emerging from saddle points partition the phase portrait into islands; each island corresponds to a structurally stable vortex structure. By using the flux of the meridional vorticity field, we obtain a precise definition of reconnection: the transfer of flux between islands. Local analysis near critical points shows that the virtual velocity (because of its singular behaviour) performs ‘cut-and-connect’ of vortex surfaces with the correct rate of circulation transfer - thereby validating the long-standing viscous ‘cut-and-connect’ scenario which implicitly assumes that vortex surfaces (and vortex lines) can be followed over a short period of time in a viscous fluid. Bifurcations in the phase portrait represent (contrary to reconnection) changes in the topology of the vorticity field, where islands spontaneously appear or disappear. Often such topology changes are catastrophic, because islands emerge or perish with finite circulation. These and other phenomena are illustrated by direct numerical simulations of vortex rings at a Reynolds number of 800.

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