Discrete Nonlinear Oscillator Dynamical System and the Inertial Proximal Algorithm

2019 ◽  
pp. 207-218
Author(s):  
Behzad Djafari-Rouhani ◽  
Hadi Khatibzadeh
Keyword(s):  
Dynamical System ◽  
Nonlinear Oscillator ◽  
Proximal Algorithm ◽  
Inertial Proximal Algorithm

scholarly journals Inexact Inertial Proximal Algorithm for Maximal Monotone Operators

2015 ◽  
Vol 23 (2) ◽  
pp. 133-146
Author(s):  
Hadi Khatibzadeh ◽  
Sajad Ranjbar
Keyword(s):  
Nonlinear Oscillator ◽  
Proximal Point Algorithm ◽  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Proximal Algorithm ◽  
Weak And Strong Convergence ◽  
Proximal Point ◽  
Convergence Results ◽  
Inertial Proximal Algorithm

Abstract In this paper, convergence of the sequence generated by the inexact form of the inertial proximal algorithm is studied. This algorithm which is obtained by the discretization of a nonlinear oscillator with damping dynamical system, has been introduced by Alvarez and Attouch (2001) and Jules and Maingé (2002) for the approximation of a zero of a maximal monotone operator. We establish weak and strong convergence results for the inexact inertial proximal algorithm with and without the summability assumption on errors, under different conditions on parameters. Our theorems extend the results on the inertial proximal algorithm established by Alvarez and Attouch (2001) and rules and Maingé (2002) as well as the results on the standard proximal point algorithm established by Brézis and Lions (1978), Lions (1978), Djafari Rouhani and Khatibzadeh (2008) and Khatibzadeh (2012). We also answer questions of Alvarez and Attouch (2001).

Bifurcation Trees of Period-1 Motions to Chaos in a Two-Degree-of-Freedom, Nonlinear Oscillator

2015 ◽  
Vol 25 (13) ◽  
pp. 1550179 ◽  
Author(s):  
Albert C. J. Luo ◽  
Bo Yu
Keyword(s):  
Dynamical System ◽  
Fourier Series ◽  
Analytical Solutions ◽  
Nonlinear Oscillator ◽  
Degree Of Freedom ◽  
Harmonic Amplitude ◽  
Periodic Motions ◽  
Series Transformation ◽  
Two Degree Of Freedom ◽  
Analytical Bifurcation Trees

In this paper, analytical solutions for period-[Formula: see text] motions in a two-degree-of-freedom (2-DOF) nonlinear oscillator are developed through the finite Fourier series. From the finite Fourier series transformation, the dynamical system of coefficients of the finite Fourier series is developed. From such a dynamical system, the solutions of period-[Formula: see text] motions are obtained and the corresponding stability and bifurcation analyses of period-[Formula: see text] motions are carried out. Analytical bifurcation trees of period-1 motions to chaos are presented. Displacements, velocities and trajectories of periodic motions in the 2-DOF nonlinear oscillator are used to illustrate motion complexity, and harmonic amplitude spectrums give harmonic effects on periodic motions of the 2-DOF nonlinear oscillator.

Subharmonic Orbits of a Strongly Nonlinear Oscillator Forced by Closely Spaced Harmonics

2010 ◽  
Vol 6 (1) ◽  
Author(s):  
Themistoklis P. Sapsis ◽  
Alexander F. Vakakis
Keyword(s):  
Dynamical System ◽  
Nonlinear Oscillator ◽  
Resonance Condition ◽  
Asymptotic Results ◽  
Strongly Nonlinear ◽  
Singular Perturbation Analysis ◽  
Local Bifurcation Analysis ◽  
The Family ◽  
Subharmonic Orbits ◽  
Strongly Nonlinear Oscillator

We study asymptotically the family of subharmonic responses of an essentially nonlinear oscillator forced by two closely spaced harmonics. By expressing the original oscillator in action-angle form, we reduce it to a dynamical system with three frequencies (two fast and one slow), which is amenable to a singular perturbation analysis. We then restrict the dynamics in neighborhoods of resonance manifolds and perform local bifurcation analysis of the forced subharmonic orbits. We find increased complexity in the dynamics as the frequency detuning between the forcing harmonics decreases or as the order of a secondary resonance condition increases. Moreover, we validate our asymptotic results by comparing them to direct numerical simulations of the original dynamical system. The method developed in this work can be applied to study the dynamics of strongly nonlinear (nonlinearizable) oscillators forced by multiple closely spaced harmonics; in addition, the formulation can be extended to the case of transient excitations.

A New Megastable Oscillator with Rational and Irrational Parameters

2019 ◽  
Vol 29 (13) ◽  
pp. 1950176 ◽  
Author(s):  
Zhen Wang ◽  
Ibrahim Ismael Hamarash ◽  
Payam Sadeghi Shabestari ◽  
Sajad Jafari
Keyword(s):  
Dynamical System ◽  
Infinite Number ◽  
Limit Cycles ◽  
Nonlinear Oscillator ◽  
Analog Circuit ◽  
Strange Attractors ◽  
Two Dimensional ◽  
System A ◽  
New System

In this paper, a new two-dimensional nonlinear oscillator with an unusual sequence of rational and irrational parameters is introduced. This oscillator has endless coexisting limit cycles, which make it a megastable dynamical system. By periodically forcing this system, a new system is designed which is capable of exhibiting an infinite number of coexisting asymmetric torus and strange attractors. This system is implemented by an analog circuit, and its Hamiltonian energy is calculated.

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