Nonlinear Dynamics of the Quadratic-Damping Helmholtz Oscillator

Abstract

In this paper, the Helmholtz equation with quadratic damping themes is used for modeling the dynamics of a simple prey-predator system also called a simple Lotka–Volterra system. From the Helmholtz equation with quadratic damping themes obtained after modeling, the equilibrium points have been found, and their stability has been analyzed. Subsequently, the harmonic oscillations have been studied by the harmonic balance method, and the phenomena of resonance and hysteresis are observed. The primary and secondary resonances have been researched by the multiple-scale method, and the conditions of stability of the amplitudes of oscillations are determined. Chaos is detected analytically by the Melnikov method and numerically using the basin of attraction, the bifurcation diagram, the Lyapunov exponent, the phase portrait, and the Poincaré section. The effects of all the parameters of the system are analyzed in detail, and special emphasis is placed on the new parameters. Through this analysis, the complex phenomena such as hysteresis, bistability, amplitude jump, resonances, and chaos have been obtained. The control of the parameters and the necessary conditions to control the aforementioned phenomena have been found.

Nonlinear Dynamics of a Nonlinear Damping Gyroscope and Its Passive Control

“…We also noticed the increase or decrease of these instability domains when the control parameters K; U of the control force of the system increase or decrease. )e effect of the control process on chaotic dynamic and on coexistence of attractors was effective with ε � −11 [32]. )e high amplitude of harmonic oscillations, chaotic states, and coexistence of attractors was successfully controlled by the passive control investigated in this work.…”

“…In the presence of the control force, we plotted for the same parameters as Figures5(a), 5(c), and 5(d) for ε � 1 and Figures5(e) and 5(f ) for ε � −1. It noticed through these figures that the chaos is accentuated for ε � 1 but totally reduced for ε � −1[32].…”

“…Our goal in this subsection is to reduce or remove this chaotic dynamics by using a passive control force [31]. To achieve this goal, we used the fourth-order Runge-Kutta algorithm [32] to numerically solve equation ( 3), and the resulting bifurcation diagrams and its corresponding Lyapunov exponents are plotted using the amplitude of the external excitation forces as the bifurcation parameter. For the simulations, the parameters used are α � 10; β � 1; Ω � 2; C 1 � 0.…”

Chaotic Dynamics of a Mixed Rayleigh–Liénard Oscillator Driven by Parametric Periodic Damping and External Excitations

“…In recent years, the nonlinear dynamics of this class of oscillators has been intensively studied, and interesting results such as perioddoubling leading to chaotic motion, strange attractors, reverse period-doubling bifurcation, symmetry breaking, antimonotonicity, existence of horseshoe chaos, and so on have been obtained [17][18][19][20][21][22][23][24][25]. In most of these studies, the Melnikov perturbation method [18,24,26,27] has been widely used to detect chaotic dynamics and to analyze nearhomoclinic motion with deterministic or random perturbation. is method is today considered as a powerful analytical tool to provide an approximate criterion for the occurrence of hetero/homoclinic chaos in a wide class of dynamical systems.…”

Dynamics and active control of chemical oscillations modeled as a forced generalized Rayleigh oscillator with asymmetric potential

“…We made this choice because it has already been demonstrated that a modified Van der Pol-Duffing oscillator can be used to model the nonlinear chemical oscillations like BZ reactions [21][22][23][24][25][26]. Since the control of regular and irregular motions is an interesting issue in several areas [27], the dynamical behavior of a forced generalized Rayleigh oscillator, which constitutes a new model for describing the nonlinear chemical oscillations, may be investigated. For this, we use the Melnikov method to analyze the chaotic behavior of this new chemical oscillator.…”