Research Article | Open Access
C. A. Kitio Kwuimy, C. Nataraj, M. Belhaq, "Chaos in a Magnetic Pendulum Subjected to Tilted Excitation and Parametric Damping", Mathematical Problems in Engineering, vol. 2012, Article ID 546364, 18 pages, 2012. https://doi.org/10.1155/2012/546364
Chaos in a Magnetic Pendulum Subjected to Tilted Excitation and Parametric Damping
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 [1–3], electromechanical [4–6], mechanical [7–10], electronic [11–13], and others. Usually, numerical indicators such as the Lyapunov exponent and bifurcation diagram are used to determine and study the occurrence of chaos. The Melnikov method , on the other hand, predicts analytically the lower bound in parameter space separating regular and chaotic dynamics. The Melnikov method has been recently applied in experimental and theoretical research in various fields of science, including epidemiology , biology , and engineering systems [2, 7, 8]. Along these lines, Cicogna and Papoff  considered a Duffing type potential with an additional linear term and estimated the threshold condition for the appearance of chaos by using a Taylor expansion with respect to the asymmetric parameter. The optimal control of chaos was studied by Lenci and Rega  for the Helmholtz-Duffing oscillator. Litak et al.  revisited the Melnikov criteria for a driven system under a single and double well asymmetric potential and expressed the integrals to be evaluated for the appearance of chaos in terms of logarithm function. Cao et al.  applied the Melnikov theory to a driven Helmholtz-Duffing oscillator and derived the condition for appearance of fractal basin boundaries. Recently, a magnetic pendulum driven by a high-frequency excitation under a magnetic potential was considered . This paper aims to apply the well-known Melnikov theory to a fundamental physical device used in several engineering systems, namely, a magnetic pendulum, and discuss the possibility of chaos suppression in the system.
Current literature examines various nontrivial phenomena caused by a high-frequency excitation in physical systems. Thomsen  considered the stiffening, biasing, and smoothening in such systems, Bartuccelli et al.  and Schmitt and Bayly  showed that a high-frequency excitation of a horizontally or vertically shaken pendulum results in oscillations about a nonzero mean angle. Yabuno et al.  considered an inverted pendulum and showed that a tilt angle of the excitation produces stable equilibrium states different from the direction of the gravity and the excitation. The symmetry breaking bifurcation due to the tilt angle was also investigated qualitatively and through experiments by Mann and Koplow . In a related experimental work, Mann investigated the energy criterion for snap-through instability and nonperiodic motion.
The effect of a fast parametric excitation on self-excited vibrations in a delayed van der Pol oscillator was reported in [27, 28]. Fidlen and Juel Thomsen  analyzed this effect on the equilibrium of a strongly damped system comparing to the case of a slightly damped one. Mann and Koplow  showed that a small deviation from either a perfectly vertical or horizontal excitation will result in symmetry breaking bifurcations opposed to pitchfork bifurcations obtained for vertical or horizontal excitation. Also, the condition for well escapes in a bistable configuration of the potential energy has been studied . An earlier work on magnetic pendulum was done by Moon et al.  who showed evidence of homoclinic orbit and horseshoe chaos in a magnetic pendulum. Kraftmakher  reported that parametric damping has some strong effects on the stabilization dynamics of a pendulum. The effects of the sinusoidal fluctuation arising from a control strategy was also considered. Recently, Sah and Belhaq  investigated the tilting effect of a fast excitation on self-excited vibrations in a delayed van der Pol pendulum.
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  in which the basins of attraction have been computed and the influence of parametric excitation on escape phenomena have been studied. Specifically, we consider the experimental system as in  and we investigate analytically and numerically the tilting effect of the parametric excitation as well as the time-dependent parametric damping on the basins of attraction and on chaos domain in the parameter space of the system. In other words, the present paper can be considered as an extension of the experimental work  in adding other effects to the original system.
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. . To the best of our knowledge, the effect of tilted excitation and parametric damping on the appearance of chaos in magnetic pendulum has not been addressed. In addition to being of fundamental interest, this paper expects to impact practical design and control of systems involving a magnetic pendulum.
The paper is organized as follows. In Section 2, the mathematical model and the corresponding homoclinic orbit are given. In Section 3, the Melnikov theorem is used to obtain the threshold condition for the appearance of chaos and the basins of attraction are explored to support the Melnikov results. Attention is focused on the effects of the titling of the excitation on the chaos occurrence, in the absence of parametric damping. In Section 4, parametric damping is introduced and its effect is analyzed. Section 5 concludes the work.
2. The Model and Hamiltonian System
A schematic representation of the pendulum is shown in Figure 1 where is the angular deviation, and are the mass and the effective length of the pendulum, respectively, and are the amplitude and the frequency of the excitation, respectively, and is the tilt angle of the excitation with respect to the gravity direction. Using Newton's laws it is shown that the mathematical model describing the angular oscillations of the system can be written as [21, 26] where is the angular deviation, is the time, is the damping coefficient, and are the mass and the effective length of the pendulum, respectively, is the acceleration due to gravity, and are the amplitude and the frequency of excitation, respectively, is the tilt angle of the excitation with respect to the horizontal direction, and is a nonsymmetric potential defining the nonlinear restoring force of the magnet given as  where , , and are physical constants. By expanding and up to the third order, (2.1) becomes in dimensionless form with where and are, respectively, the characteristic angle and the time used to rescale the modeling equation. It is important to note that the case corresponds to the horizontal excitation (), while the case corresponds to the vertical excitation (). The signs of , , and change with the characteristic of the magnetic potential; see for instance the experiment carried out in  in which , , and . In order to perform a general analysis, the dimensionless parameters are used and their values are meaningfully selected according to the available literature. Without loss of generality, we choose in the rest of the paper , , and . The following Hamiltonian system, obtained from (2.3) leads to the Hamiltonian function where is given by following asymmetric potential
This potential has two stable and one unstable equilibrium points given by and , where the letters and stand for the left and right hand side, respectively. It is obvious that and by comparing and , one concludes that the left well is always deeper if and it is deeper for if . The Hamiltonian defined in (2.6) has an hyperbolic fixed point at and two elliptic fixed points at leading to the homoclinic orbit given by with .
Figure 2(a) shows asymmetric potentials and orbits with different depths for the left and right wells. The shape of the orbits on the left and right sides of the saddle point is different as shown in Figure 2(b). As the asymmetric term increases, the deepness of the left well becomes large compared to the right one. In order to analyze the condition for the appearance of chaos in the magnetic pendulum with a tilted excitation, we apply the Melnikov method.
3. Melnikov Analysis and Basin of Attraction
In order to perform the Melnikov analysis, the perturbed Hamiltonian equation (2.5) is rewritten as where where being a phase angle, is a perturbation parameter, , and . In the rest of the paper, the stars are removed for simplicity. For , the homoclinic orbit connects the unstable point and the eigenvalues of the linearized problem around are real and of opposite sign. For , the Melnikov theorem can be used to detect transverse intersections between perturbed stable and unstable manifolds in the system. According to , the distance between the perturbed and unperturbed manifolds is given by the Melnikov function: which can be written as
The integrals and , are evaluated hereafter. For the integral we have where is defined for the left and the right side such that , is the Gamma function and is the associated Legendre function of the second kind. Using the definition, the functional relation of the Gamma function and the associated Legendre function of the second kind (listed in Appendices A and B), the integral becomes where , , and is the hypergeometric function. Unfortunately, further simplification of the hypergeometric function can be made only for some specific values of the argument, see  for details. Note that in the absence of the magnetic potential (, ), the device is under a symmetric Duffing potential and then the integral becomes [2, 8, 19]
The Gauss hypergeometric function, for complex or real argument, can be evaluated using the Gnu Scientific Library (GSL) via a PYGSL code . In order to evaluate the other integrals, we define the following functions: with , , and . It can be shown (see Appendix C) that is always a positive real number. is considered as a product of odd and even function.
For the integral and one obtains which takes the expression corresponding to the parametrically driven Duffing equation 
For the integrals , we obtain, respectively
Finally, the integral , are given by
Using the Melnikov criterion , one concludes that nonperiodic motions appear when
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 3 as a function of the excitation frequency for different values of the incline and as a function of for with , , , , and for the plus sign in (3.2).
Figure 3(a) shows a classical result, that is, in the chosen interval , the threshold value of decreases for small values of the frequency and increases for large values of the frequency. A singularity is obtained for (red lines) indicating that the bifurcation curve splits into two lobes creating a domain of a regular motion near a certain value of the frequency . Figure 3(b) illustrates the effect of the tilted angle of the excitation on the bifurcation curves for . It appears from this figure that the threshold increases with until a critical value and then decreases. The domain of periodic motion is larger near the vertical excitation () than near the horizontal one ().
To test the validity of the Melnikov predictions, we investigate the regular or irregular (fractal) shape of the basins of attraction . A basin with smooth shape will suggest regular dynamics of the system while a fractal basin will indicate nonregular dynamic. The basins are plotted using a simplified criterion which distinguish solutions of mean value displacement. This is done by scanning the initial values and of and in domain, solving numerically (2.3) and collecting the initial conditions which attract the dynamics in the right (for instance) well of the potential (marked region of the basin). The unmarked region represents the domain for which the system dynamics is attracted to the other well. In a more detailed approach, a larger number of basins would illustrate other signatures of the system response hidden in the figure. The effect of on the shape of the basins of attraction of Figure 4 are plotted for the values of Figure 3(a). The graph of Figure 4(a) is plotted for , , and showing a regular behavior of the system which is consistent with the Melnikov prediction. Figure 4(b) shows how the shape of the basin becomes fractal for . The fractal shape is more pronounced as increases as illustrated in Figure 4(c) () and Figure 4(d) (). Choosing close to the critical value may not lead to the same conclusion as illustrated in Figure 4(b) since the Melnikov function is a first-order approximation . These results show that the final state of the system highly depends upon initial conditions and the tilt angle is the key parameter. This strong dependency of the system upon initial conditions was obtained experimentally by Mann  for a perfectly horizontal excitation based on energy criterion analysis.
4. Effect of Parametric Damping
The idea of controlling a pendulum via feedback parametric damping was recently investigated experimentally by Kraftmakher . In his experiment, the voltage was collected through a coil, amplified and reinjected in the system for the purpose of control. In such a process the system may experience fluctuations arising as noise, lost signal, and additional resonant excitation. These fluctuations can be modeled as a harmonic function of time. In such a situation, (2.3) under parametric damping can be written as where and are the feedback gains of parametric damping. The plus sign is taken for negative feedback and the minus is for positive feedback . By applying the Melnikov theorem, one obtains with and .
From (4.2), the Melnikov function reads in which with , , , and as a product of even and odd function. Thus the Melnikov function becomes
In the absence of the periodic feedback fluctuation (), the threshold condition for the appearance of chaos of the asymmetric pendulum with parametric damping and tilted excitation is given by
Figure 5 illustrates the effect of the parametric damping component () on the threshold of Figure 3 in the absence of the harmonic fluctuation (). In this figure, the red line is plotted for a positive feedback (), the green line is for a negative feedback (), and the blue line corresponds to the case without feedback gain (). The graphs show that the negative feedback is effictive for enhancement of chaos by increasing the bound value delimiting chaotic domain, while the positive feedback is effictive in suppressing chaos by decreasing the threshold curve for chaos. A rapid analysis of (4.8) shows clearly that increases with (negative feedback) and linearly decreases with (positive feedback).
In the presence of the periodic fluctuation in the parametric damping (), one obtains from (4.7) the following under resonance conditions
Hence, the periodic motion is guaranteed if
Figure 5 shows the effect of the feedback gain on the domain of chaotic dynamics for and . It can be seen from the plots that by decreasing the feedback gain from (Figure 6(a)) to (Figure 6(c)), the area of regular motions (marked domain) increases.
Figure 7 illustrates the influence of the incline of the excitation on the chaotic domain for the given value . The plots in this figure indicate that as increases from horizontal (, Figure 7(a)) to vertical (, Figure 7(c)), the area of regular motions decreases with a singularity at as shown in Figures 3(b) and 7(c).
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 (, ), the incline angle of the excitation can influence the chaotic dynamic of the magnetic pendulum. In other words, as the tilt angle of the excitation increases from horizontal () to the vertical () position, the domain of regular motion increases (Figure 2). This suggests that stability can be gained when the tilting excitation approaches the vertical position, which is consistent with the results given in .
Furthermore, in the presence of the incline of the excitation and of the parametric damping without periodic fluctuation (, ), the results show that a positive feedback gain () enlarges the domain of periodicity while a negative feedback gain () extends the region of chaotic regime (Figure 5). On the other hand, in the case where the parametric damping with harmonic fluctuation is introduced, the area of regular motions increases with decreasing the feedback gain (Figure 6) and this area increases by increasing the incline from a horizontal to a vertical direction, as expected (Figure 7).
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.
A. Functional Relation of the Gamma Function
Consider the following:
B. Functional Relation between the Associated Lengendre Function of the Second Kind and the Gauss Hypergeometric Function
C. Properties of the Gauss Hypergeometric Function
Consider and the Pochhammer symbol is defined as then, if , one has this last expression of is always a positive real number.
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.
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