Stability analysis for periodic solutions of the Van der Pol–Duffing forced oscillator
TL;DR: In this article, the stable/unstable periodic solutions of the Van der Pol-Duffing forced oscillator with the variation of the forced frequency are analyzed by using Floquet theory.
Abstract: Based on the homotopy analysis method (HAM), the high accuracy frequency response curve and the stable/unstable periodic solutions of the Van der Pol-Duffing forced oscillator with the variation of the forced frequency are obtained and studied. The stability of the periodic solutions obtained is analyzed by use of Floquet theory. Furthermore, the results are validated in the light of spectral analysis and bifurcation theory.
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TL;DR: In this paper , a fractal-differential model for nonlinear vibration system in fractal space is presented, and the stability criterion for the equation under consideration is obtained by using the linearized stability theory in the autonomous arrangement.
Abstract: A critical hurdle of a nonlinear vibration system in a fractal space is the inefficiency in modelling the system. Specifically, the differential equation models cannot elucidate the effect of porosity size and distribution of the periodic property. This paper establishes a fractal-differential model for this purpose, and a fractal Duffing-Van der Pol oscillator (DVdP) with two-scale fractal derivatives and a forced term is considered as an example to reveal the basic properties of the fractal oscillator. Utilizing the two-scale transforms and He-Laplace method, an analytic approximate solution may be attained. Unfortunately, this solution is not physically preferred. It has to be modified along with the nonlinear frequency analysis, and the stability criterion for the equation under consideration is obtained. On the other hand, the linearized stability theory is employed in the autonomous arrangement. Consequently, the phase portraits around the equilibrium points are sketched. For the non-autonomous organization, the stability criteria are analyzed via the multiple time scales technique. Numerical estimations are designed to confirm graphically the analytical approximate solutions as well as the stability configuration. It is revealed that the exciting external force parameter plays a destabilizing role. Furthermore, both of the frequency of the excited force and the stiffness parameter, execute a dual role in the stability picture.
48 citations
TL;DR: In this article, the authors investigate the dynamics of a Van der Pol-Duffing forced oscillator, which is modelled by a five-parameter second order nonautonomous nonlinear ordinary differential equation.
Abstract: We investigate the dynamics of a Van der Pol–Duffing forced oscillator, which is modelled by a five-parameter second order nonautonomous nonlinear ordinary differential equation. Firstly we fix three of these parameters, and investigate the dynamics of this system by varying the other two, namely the amplitude and the angular frequency of the external forcing. We also investigate the existence of different attractors, periodic, quasiperiodic, and chaotic. Finally, we investigate the occurrence of multistability in the considered Van der Pol–Duffing forced oscillator, for some fixed sets of parameters.
19 citations
TL;DR: In this paper, a Van der Pol-Duffing (VdPD) jerk oscillator is designed and analyzed analytically, numerically and analogically, and numerical results indicate that the proposed VdPD jerk oscillators displays chaotic oscillations, symmetrical bifurcations and coexisting attractors.
Abstract: In this paper, a Van der Pol–Duffing (VdPD) jerk oscillator is designed. The proposed VdPD jerk oscillator is built by converting the autonomous two-dimensional VdPD oscillator to a jerk oscillator. Dynamical behaviours of the proposed VdPD jerk oscillator are investigated analytically, numerically and analogically. The numerical results indicate that the proposed VdPD jerk oscillator displays chaotic oscillations, symmetrical bifurcations and coexisting attractors. The physical existence of the chaotic behaviour found in the proposed VdPD jerk oscillator is verified by using Orcad-PSpice software. A good qualitative agreement is shown between the numerical simulations and the PSpice results. Moreover, the fractional-order form of the proposed VdPD jerk oscillator is studied using stability theorem of fractional-order systems and numerical simulations. It is found that chaos, periodic oscillations and coexistence of attractors exist in the fractional-order form of the proposed jerk oscillator with order less than three. The effect of fractional-order derivative on controlling chaos is illustrated. It is shown that chaos control is achieved in fractional-order form of the proposed VdPD jerk oscillator only for the values of linear controller used. Finally, the problem of drive–response synchronisation of the fractional-order form of the chaotic proposed VdPD jerk oscillators is considered using active control technique.
19 citations
TL;DR: In this paper, the authors investigated the dynamics of both symmetric and asymmetric Van der Pol-Duffing oscillators driven by a periodic force F(t) = f cos ωt.
Abstract: We investigate numerically the dynamics of both symmetric and asymmetric Van der Pol-Duffing oscillators driven by a periodic force F(t) = f cosωt. Each system is modeled by a different second order nonautonomous nonlinear ordinary differential equation controlled by five parameters. Our investigation takes into account the (ω, f) parameter-space in the two systems, keeping the other three parameters fixed. We verify the existence of parameter regions for which the corresponding trajectories in the phase-space are periodic, quasiperiodic, and chaotic, for the symmetric case. In the asymmetric case we verify the existence only of periodic and chaotic regions in the (ω, f) parameter-space. Finally, we also investigate the organization of the dynamics in the two systems, identifying Fibonacci and period-adding sequences of periodic structures.
12 citations
13 Aug 2019
TL;DR: An encryption algorithm is designed by the pseudo-random sequences generated from the VdPVP, which consists of chaos scrambling and chaos XOR (exclusive-or) operation, and the statistical analyses show that it has good security and encryption effectiveness.
Abstract: The Van der Pol oscillator is investigated by the parameter control method. This method only needs to control one parameter of the Van der Pol oscillator by a simple periodic function; then, the Van der Pol oscillator can behave chaotically from the stable limit cycle. Based on the new Van der Pol oscillator with variable parameter (VdPVP), some dynamical characteristics are discussed by numerical simulations, such as the Lyapunov exponents and bifurcation diagrams. The numerical results show that there exists a positive Lyapunov exponent in the VdPVP. Therefore, an encryption algorithm is designed by the pseudo-random sequences generated from the VdPVP. This simple algorithm consists of chaos scrambling and chaos XOR (exclusive-or) operation, and the statistical analyses show that it has good security and encryption effectiveness. Finally, the feasibility and validity are verified by simulation experiments of image encryption.
10 citations
References
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TL;DR: Van der Pol's equation for a relaxation oscillator is generalized by the addition of terms to produce a pair of non-linear differential equations with either a stable singular point or a limit cycle, which qualitatively resembles Bonhoeffer's theoretical model for the iron wire model of nerve.
5,430 citations
Book•
10 Jul 2012
TL;DR: In this paper, a convergence series for Divergent Taylor Series is proposed to solve nonlinear initial value problems and nonlinear Eigenvalue problems with free or moving boundary in heat transfer.
Abstract: Basic Ideas.- Systematic Descriptions.- Advanced Approaches.- Convergent Series For Divergent Taylor Series.- Nonlinear Initial Value Problems.- Nonlinear Eigenvalue Problems.- Nonlinear Problems In Heat Transfer.- Nonlinear Problems With Free Or Moving Boundary.- Steady-State Similarity Boundary-Layer Flows.- Unsteady Similarity Boundary-Layer Flows.- Non-Similarity Boundary-Layer Flows.- Applications In Numerical Methods.
852 citations
Book•
04 Apr 2011
TL;DR: In this article, the authors present a survey of the literature on nonlinear dynamics of pendulum and nonlinear oscillators, including a brief biography of Georg Duffing, and some of the most relevant works.
Abstract: List of Contributors. Preface. 1 Background: On Georg Duffing and the Duffing Equation (Ivana Kovacic and Michael J. Brennan). 1.1 Introduction. 1.2 Historical perspective. 1.3 A brief biography of Georg Duffing. 1.4 The work of Georg Duffing. 1.5 Contents of Duffing's book. 1.6 Research inspired by Duffing s work. 1.7 Some other books on nonlinear dynamics. 1.8 Overview of this book. References. 2 Examples of Physical Systems Described by the Duffing Equation (Michael J. Brennan and Ivana Kovacic). 2.1 Introduction. 2.2 Nonlinear stiffness. 2.3 The pendulum. 2.4 Example of geometrical nonlinearity. 2.5 A system consisting of the pendulum and nonlinear stiffness. 2.6 Snap-through mechanism. 2.7 Nonlinear isolator. 2.8 Large deflection of a beam with nonlinear stiffness. 2.9 Beam with nonlinear stiffness due to inplane tension. 2.10 Nonlinear cable vibrations. 2.11 Nonlinear electrical circuit. 2.12 Summary. References. 3 Free Vibration of a Duffing Oscillator with Viscous Damping (Hiroshi Yabuno). 3.1 Introduction. 3.2 Fixed points and their stability. 3.3 Local bifurcation analysis. 3.4 Global analysis for softening nonlinear stiffness ( < 0). 3.5 Global analysis for hardening nonlinear stiffness ( < 0). 3.6 Summary. Acknowledgments. References. 4 Analysis Techniques for the Various Forms of the Duffing Equation (Livija Cveticanin). 4.1 Introduction. 4.2 Exact solution for free oscillations of the Duffing equation with cubic nonlinearity. 4.3 The elliptic harmonic balance method. 4.4 The elliptic Galerkin method. 4.5 The straightforward expansion method. 4.6 The elliptic Lindstedt Poincare method. 4.7 Averaging methods. 4.8 Elliptic homotopy methods. 4.9 Summary. References. Appendix AI: Jacob elliptic function and elliptic integrals. Appendix 4AII: The best L2 norm approximation. 5 Forced Harmonic Vibration of a Duffing Oscillator with Linear Viscous Damping (Tamas Kalmar-Nagy and Balakumar Balachandran). 5.1 Introduction. 5.2 Free and forced responses of the linear oscillator. 5.3 Amplitude and phase responses of the Duffing oscillator. 5.4 Periodic solutions, Poincare sections, and bifurcations. 5.5 Global dynamics. 5.6 Summary. References. 6 Forced Harmonic Vibration of a Duffing Oscillator with Different Damping Mechanisms (Asok Kumar Mallik). 6.1 Introduction. 6.2 Classification of nonlinear characteristics. 6.3 Harmonically excited Duffing oscillator with generalised damping. 6.4 Viscous damping. 6.5 Nonlinear damping in a hardening system. 6.6 Nonlinear damping in a softening system. 6.7 Nonlinear damping in a double-well potential oscillator. 6.8 Summary. Acknowledgments. References. 7 Forced Harmonic Vibration in a Duffing Oscillator with Negative Linear Stiffness and Linear Viscous Damping (Stefano Lenci and Giuseppe Rega). 7.1 Introduction. 7.2 Literature survey. 7.3 Dynamics of conservative and nonconservative systems. 7.4 Nonlinear periodic oscillations. 7.5 Transition to complex response. 7.6 Nonclassical analyses. 7.7 Summary. References. 8 Forced Harmonic Vibration of an Asymmetric Duffing Oscillator (Ivana Kovacic and Michael J. Brennan). 8.1 Introduction. 8.2 Models of the systems under consideration. 8.3 Regular response of the pure cubic oscillator. 8.4 Regular response of the single-well Helmholtz Duffing oscillator. 8.5 Chaotic response of the pure cubic oscillator. 8.6 Chaotic response of the single-well Helmholtz Duffing oscillator. 8.7 Summary. References. Appendix Translation of Sections from Duffing's Original Book (Keith Worden and Heather Worden). Glossary. Index.
650 citations
TL;DR: In this paper, a totally analytic solution for thin film flow of a fourth grade fluid down a vertical cylinder is obtained using homotopy analysis method (HAM), and the series solution is developed and the recurrence relations are given explicitly.
254 citations
TL;DR: In this paper, Chaotically transitional processes in the forced negative-resistance oscillator were investigated using analog and digital computers and the difference between the almost periodic oscillations and the Chaotic Transition Process (CTP) was clarified.
Abstract: This paper deals with chaotically transitional phenomena which occur In the forced negative-resistance oscillator. Experimental studies using analog and digital computers have been carried out. The difference between the almost periodic oscillations and the chaotically transitional processes is clarified. Various strange attractors representing chaotically transitional processes and their average power spectra are given. They are discussed in detail and compared with the results obtained in the forced oscillatory systems.
144 citations