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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.

Various nonlinear phenomena have been found in physical systems and chaotic behavior has been reported in various engineering systems with applications in microelectromechanical [

Current literature examines various nontrivial phenomena caused by a high-frequency excitation in physical systems. Thomsen [

The effect of a fast parametric excitation on self-excited vibrations in a delayed van der Pol oscillator was reported in [

It thus appears from this paper that, parametrically excited magnetic pendulum is an interesting system from both the mathematical and physical points of view. The present work was motivated by the experimental work carried out by Mann [

To identify the conditions leading to nonperiodic response, Melnikov method is applied using the Gauss hypergeometric function. The use of this function can be considered as an extension of the work by Litak et al. [

The paper is organized as follows. In Section

A schematic representation of the pendulum is shown in Figure

Schematic representation of the pendulum.

This potential has two stable and one unstable equilibrium points given by

Figure

Asymmetric potential and separatrix showing the depth and shape difference between the left and right side. (a) Asymmetric potential. (b) Asymmetric separatrix.

In order to perform the Melnikov analysis, the perturbed Hamiltonian equation (

The integrals

The Gauss hypergeometric function, for complex or real argument, can be evaluated using the Gnu Scientific Library (GSL) via a PYGSL code [

Thus, one has, for the integral

For the integral

For the integrals

Finally, the integral

Using the Melnikov criterion [

This criterion defines the threshold value for the appearance of transverse intersection between the perturbed and unperturbed manifolds. This threshold condition is plotted in Figure

Critical amplitude

Figure

To test the validity of the Melnikov predictions, we investigate the regular or irregular (fractal) shape of the basins of attraction [

Evolution of the shape of the basin of attraction of the right well as

The idea of controlling a pendulum via feedback parametric damping was recently investigated experimentally by Kraftmakher [

From (

In the absence of the periodic feedback fluctuation (

Figure

Effect of the parametric damping on the threshold condition for chaos with

In the presence of the periodic fluctuation in the parametric damping (

Hence, the periodic motion is guaranteed if

Figure

Effect of sinusoidal fluctuation on the control strategy for

Figure

Effect of sinusoidal fluctuation on the control strategy for

The effect of a tilted parametric excitation and of parametric damping with periodic fluctuation on the appearance of chaos in an asymmetric magnetic pendulum was examined. The analysis was carried out using the Melnikov method to derive the analytical condition for chaotic motions. These analytical predictions were tested and validated by exploring the fractal and regular shapes of the basins of attraction. It was shown that in the absence of the parametric damping components (

Furthermore, in the presence of the incline of the excitation and of the parametric damping without periodic fluctuation (

The results of this work show that chaotic dynamics can be controlled in an asymmetric magnetic pendulum by acting either on the incline of parametric excitation or on feedback gains of a parametric damping, or on both. This provides some interesting possibilities for controlling the dynamics in asymmetric magnetic pendulums.

Consider the following:

Consider

Consider

This work is supported by the US Office of Naval Research under the Grant ONR N00014-08-1-0435. Thanks are due to Mr. Anthony Seman III of ONR and Dr. Stephen Mastro of NAVSEA, Philadelphia PA, USA.