scholarly journals Chaos in a Magnetic Pendulum Subjected to Tilted Excitation and Parametric Damping

2012 ◽  
Vol 2012 ◽  
pp. 1-18 ◽  
Author(s):  
C. A. Kitio Kwuimy ◽  
C. Nataraj ◽  
M. Belhaq
Keyword(s):  
Lower Bound ◽  
Tilt Angle ◽  
Chaotic Dynamics ◽  
Basin Of Attraction ◽  
Melnikov Method ◽  
Harmonic Excitation ◽  
Gauss Hypergeometric Function ◽  
Vertical Position ◽  
Periodic Fluctuation ◽  
The Basin Of Attraction

The effect of tilted harmonic excitation and parametric damping on the chaotic dynamics in an asymmetric magnetic pendulum is investigated in this paper. The Melnikov method is used to derive a criterion for transition to nonperiodic motion in terms of the Gauss hypergeometric function. The regular and fractal shapes of the basin of attraction are used to validate the Melnikov predictions. In the absence of parametric damping, the results show that an increase of the tilt angle of the excitation causes the lower bound for chaotic domain to increase and produces a singularity at the vertical position of the excitation. It is also shown that the presence of parametric damping without a periodic fluctuation can enhance or suppress chaos while a parametric damping with a periodic fluctuation can increase the region of regular motions significantly.

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

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-18
Author(s):  
Yélomè Judicaël Fernando Kpomahou ◽  
Laurent Amoussou Hinvi ◽  
Joseph Adébiyi Adéchinan ◽  
Clément Hodévèwan Miwadinou
Keyword(s):  
Chaotic Dynamics ◽  
Initial Conditions ◽  
Equilibrium Points ◽  
Basin Of Attraction ◽  
Melnikov Method ◽  
Constant Term ◽  
Geometrical Shape ◽  
Analytical Prediction ◽  
Coexistence Of Attractors ◽  
The Basin Of Attraction

In this paper, chaotic dynamics of a mixed Rayleigh–Liénard oscillator driven by parametric periodic damping and external excitations is investigated analytically and numerically. The equilibrium points and their stability evolutions are analytically analyzed, and the transitions of dynamical behaviors are explored in detail. Furthermore, from the Melnikov method, the analytical criterion for the appearance of the homoclinic chaos is derived. Analytical prediction is tested against numerical simulations based on the basin of attraction of initial conditions. As a result, it is found that for ω = ν , the chaotic region decreases and disappears when the amplitude of the parametric periodic damping excitation increases. Moreover, increasing of F 1 and F 0 provokes an erosion of the basin of attraction and a modification of the geometrical shape of the chaotic attractors. For ω ≠ ν and η = 0.8 , the fractality of the basin of attraction increases as the amplitude of the external periodic excitation and constant term increase. Bifurcation structures of our system are performed through the fourth-order Runge–Kutta ode 45 algorithm. It is found that the system displays a remarkable route to chaos. It is also found that the system exhibits monostable and bistable oscillations as well as the phenomenon of coexistence of attractors.

scholarly journals New Developments in Dynamics: Hyperbolicity and Chaotic Dynamics

1992 ◽  
Vol 152 ◽  
pp. 363-368
Author(s):  
J. Palis
Keyword(s):  
Chaotic Dynamics ◽  
Basin Of Attraction ◽  
Dissipative Systems ◽  
New Developments ◽  
General Systems ◽  
Conservative Systems ◽  
Area Preserving Maps ◽  
The Sixties ◽  
Area Preserving ◽  
The Basin Of Attraction

Two important theories in Dynamical Systems were constructed in the sixties: the hyperbolic theory for general systems and the KAM (after Kolmogorov, Arnold and Moser) theory for conservative systems as the ones that appear in Celestial Mechanics. Most of our discussions here concern dissipative (or locally dissipative) systems, although most questions are now being posed for area preserving maps (symplectic maps in higher dimensions). Moreover, one can argue that understanding dynamically small dissipative perturbations of conservative systems is of much importance: indeed it has been recently shown that a KAM curve (tori in higher dimension) can be destroyed and in fact engulfed in the basin of attraction of a Hénon-like strange attractor as defined below.

scholarly journals Melnikov Chaos in a Modified Rayleigh–Duffing Oscillator with ϕ6 Potential

2016 ◽  
Vol 26 (05) ◽  
pp. 1650085 ◽  
Author(s):  
C. H. Miwadinou ◽  
A. V. Monwanou ◽  
L. A. Hinvi ◽  
A. A. Koukpemedji ◽  
C. Ainamon ◽  
...  
Keyword(s):  
Chaotic Behavior ◽  
Duffing Oscillator ◽  
Basin Of Attraction ◽  
Melnikov Method ◽  
Nonlinear Damping ◽  
External Excitation ◽  
Melnikov Criterion ◽  
Single Well ◽  
Quadratic Damping ◽  
The Basin Of Attraction

The chaotic behavior of the modified Rayleigh–Duffing oscillator with [Formula: see text] potential and external excitation is investigated both analytically and numerically. The so-called oscillator models, for example, ship rolling motions. The single well and triple well potential cases are considered. Melnikov method is applied and the conditions for the existence of homoclinic and heteroclinic chaos are obtained. The effects of nonlinear damping on roll motion of ships are analyzed in detail. As it is known, nonlinear roll damping is a very important parameter in estimating ship responses. It is noted that the pure and unpure quadratic damping parameters affect the Melnikov criterion in the heteroclinic and homoclinic cases respectively while the pure cubic parameter affects the amplitude in both cases. The predictions have been tested with numerical simulations based on the basin of attraction. It is pointed out that certain quadratic damping effects are contrary to cubic damping effect.

Multi-Pulse Chaotic Dynamics of Four-Dimensional Non-Autonomous Nonlinear System for a Truss Core Sandwich Plate

2014 ◽  
Author(s):  
Wei Zhang ◽  
Qi-liang Wu
Keyword(s):  
Nonlinear System ◽  
Chaotic Dynamics ◽  
Sandwich Plate ◽  
Melnikov Method ◽  
Order Mode ◽  
Simply Supported ◽  
Truss Core ◽  
Extended Melnikov Method ◽  
Kagome Truss ◽  
Truss Core Sandwich Plate

In this paper, an extended high-dimensional Melnikov method is used to investigate global and chaotic dynamics of a simply supported 3D-kagome truss core sandwich plate subjected to the transverse and the in-plane excitations. Based on the motion equation derived by Zhang and the method of multiple scales, the averaged equation is obtained for the case of principal parametric resonance and 1:2 sub-harmonic resonance for the first-order mode and primary resonance for the second-order mode. From the averaged equation obtained, the system is simplified to a three order standard form with a double zero and a pair of pure imaginary eigenvalues by using the theory of normal form. Then, the extended Melnikov method is utilized to investigate the Shilnikov-type multi-pulse heteroclinic bifurcations and existence of chaos. The analysis of the extended Melnikov method demonstrates that there exist the Shilnikov-type multi-pulse heteroclinic bifurcations and chaos in the four-dimensional non-autonomous nonlinear system. Finally, the results of numerical simulations also show that for the nonlinear system of simply supported 3D-kagome truss core sandwich plate with the transverse and the in-plane excitations, the Shilnikov-type multi-pulse motion of chaos can happen and further verify the result of theoretical analysis.

scholarly journals Optimization and Stability of Heat Engines: The Role of Entropy Evolution

Entropy ◽  
2018 ◽  
Vol 20 (11) ◽  
pp. 865 ◽  
Author(s):  
Julian Gonzalez-Ayala ◽  
Moises Santillán ◽  
Maria Santos ◽  
Antonio Calvo Hernández ◽  
José Mateos Roco
Keyword(s):  
Local Stability ◽  
Production Efficiency ◽  
Basin Of Attraction ◽  
Heat Engine ◽  
Time Constraints ◽  
Thermal Gradients ◽  
Heat Engines ◽  
Low Dissipation ◽  
The Basin Of Attraction

Local stability of maximum power and maximum compromise (Omega) operation regimes dynamic evolution for a low-dissipation heat engine is analyzed. The thermodynamic behavior of trajectories to the stationary state, after perturbing the operation regime, display a trade-off between stability, entropy production, efficiency and power output. This allows considering stability and optimization as connected pieces of a single phenomenon. Trajectories inside the basin of attraction display the smallest entropy drops. Additionally, it was found that time constraints, related with irreversible and endoreversible behaviors, influence the thermodynamic evolution of relaxation trajectories. The behavior of the evolution in terms of the symmetries of the model and the applied thermal gradients was analyzed.

scholarly journals Time Delay and Feedback Control of an Inverted Pendulum With Stick Slip Friction

2007 ◽  
Author(s):  
Sue Ann Campbell ◽  
Stephanie Crawford ◽  
Kirsten Morris
Keyword(s):  
Time Delay ◽  
Basin Of Attraction ◽  
Critical Time ◽  
Viscous Friction ◽  
Experimental System ◽  
Friction Model ◽  
Stable Equilibrium Point ◽  
Feedback Controllers ◽  
Stick Slip ◽  
The Basin Of Attraction

We consider an experimental system consisting of a pendulum, which is free to rotate 360 degrees, attached to a cart which can move in one dimension. There is stick slip friction between the cart and the track on which it moves. Using two different models for this friction we design feedback controllers to stabilize the pendulum in the upright position. We show that controllers based on either friction model give better performance than one based on a simple viscous friction model. We then study the effect of time delay in this controller, by calculating the critical time delay where the system loses stability and comparing the calculated value with experimental data. Both models lead to controllers with similar robustness with respect to delay. Using numerical simulations, we show that the effective critical time delay of the experiment is much less than the calculated theoretical value because the basin of attraction of the stable equilibrium point is very small.

Stability of Subharmonic Oscillations in a Pipe Conveying Fluid under Harmonic Excitation

2013 ◽  
Vol 444-445 ◽  
pp. 796-800
Author(s):  
Yi Xiang Geng ◽  
Han Ze Liu
Keyword(s):  
Averaging Method ◽  
Melnikov Method ◽  
Equation Of Motion ◽  
Harmonic Excitation ◽  
Angle Variable ◽  
Galerkin Approach ◽  
Existence And Stability ◽  
The Stability ◽  
Action Angle Variable ◽  
Conveying Fluid

The existence and stability of subharmonic oscillations in a two end-fixed fluid conveying pipe whose base is subjected to a harmonic excitation are investigated. A Galerkin approach is utilized to reduce the equation of motion to a second order nonlinear differential equation. The conditions for the existence of subharmonic oscillations are given by using Melnikov method. The stability of subharmonic oscillations is discussed in detail by using action-angle variable and averaging method. It is shown that the velocity of fluid plays an important role in the stability of subharmonic oscillations.

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