Department of Mechanical and Industrial Engineering, Southern Illinois University at Edwardsville,. Edwardsville, IL 62026-1805, U.S.A.. RAY P. S. HAN.
Nonlinear Dynamics 19: 37–48, 1999. © 1999 Kluwer Academic Publishers. Printed in the Netherlands.
Resonant-Separatrix Webs in Stochastic Layers of the Twin-Well Duffing Oscillator ∗ ALBERT C. J. LUO and KEQIN GU Department of Mechanical and Industrial Engineering, Southern Illinois University at Edwardsville, Edwardsville, IL 62026-1805, U.S.A.
RAY P. S. HAN Department of Mechanical Engineering, University of Iowa, Iowa City, IA 52242-1527, U.S.A. (Received: 21 October 1998; accepted: 20 January 1999) Abstract. The excitation strength for the onset of a new resonant-separatrix in the stochastic layer of the Duffing oscillator is predicted through the energy change in minimum and maximum energy spectra. The widths of stochastic layers are estimated through the use of the maximum and minimum energy which can be measured experimentally. The energy spectrum approach, rather than the Poincaré mapping section method, is applied to detect the resonant-separatrix web in the stochastic layer, and it is applicable for the onset of resonant layers in nonlinear dynamic systems. The analytical condition for the onset of a new resonant-separatrix in the stochastic layer is also presented. The analytical and numerical predictions are in good agreement. Keywords: Resonant-separatrix web, stochastic layer, energy spectrum, Duffing oscillator.
1. Introduction The resonant-separatrix webs in the stochastic layers in the twin-well Duffing oscillator are first observed through the Poincaré mapping section before a detailed investigation of the resonant-separatrix webs in the stochastic layer is undertaken. Consider an undampened, twinwell Duffing oscillator x¨ − α1 x + α2 x 3 = Q0 cos(t),
(1)
where α1 > 0 and α2 > 0 are system parameters, Q0 and are the excitation strength and frequency, respectively. For numerical simulations of the resonant-separatrix webs in stochastic layers, we use a 2nd-order symplectic scheme [1, 2] with time step 1t = 10−5 ∼ 10−7 T , where T = 2π/ , and a precision of 10−6 . When α1 = α2 = 1.0 and = 4.0, the resonant-separatrix webs in the stochastic layers generated by 20,000 Poincaré mappings of Equation (1) are illustrated in Figure 1 for Q0 = 0.98 and Q0 = 0.2. As discussed in [3], the stochastic layer of the Duffing oscillator is separated into inner and outer stochastic layers by the homoclinic orbit, as shown in Figure 1, and the dynamics behavior of the two stochastic layers are very distinguishing because of the different resonances. In the upper plot of Figure 1, the 3rd and 5th-order resonant-separatrix webs are in the outer stochastic layer at Q0 = 0.98, and the sub-resonant separatrix in the vicinity of the 5th-order resonant-separatrix is also clearly observed. When the excitation strength decreases ∗ Contributed by Professor D. T. Mook.
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Figure 1. Poincar´e mapping sections of resonant-separatrix webs in the stochastic layer at = 4.0. The outer resonance of the 3rd and 5th-order for Q0 = 0.98 (upper), and the outer resonance of the 7th-order and the inner resonance of the 4th-order for Q0 = 0.2 (lower).
to Q0 = 0.2, the 7th-order resonant-separatrix appears in the outer stochastic layer, and the 4th order resonant-separatrix in the inner-stochastic layer is observed in the lower plot of Figure 1. From the above observations, the appearance of a new resonant-separatrix in the stochastic layer depends on excitation strength, and the width of the stochastic layer increases with excitation strength. As discussed in [3], the appearance of a resonant separatrix in the stochastic layers is also related to the excitation frequency. The dynamics of stochastic layers in Hamiltonian systems is very complicated. The earliest study appeared in [4] in the end of the 19th century, where Poincaré qualitatively described a stochastic layer formed by separatrix splitting. In 1962, Melnikov [5] gave a quantitative investigation of the separatrix splitting (see also [6]). The stochastic layer in an undampened Duffing oscillator possessing two potential-wells is of great interest due to extensive applications in physics [7–12]. In 1968, Zaslavsky and Filonenko [13] investigated the one-
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dimensional motion of a charged particle in the field of a traveling wave under the influence of a perturbed traveling wave using a separatrix map (or whisker map in [14]). Since then, this map has been used to investigate stochastic layers in nonlinear dynamics [15–21]. However, three approximations have been used in the derivation of the whisker map: √ (i) The first complete elliptic integral is approximated by K(k) ≈ log(4/ 1 − k 2 ) for periodic orbits of the unperturbed system close to the separatrix [8]. (ii) Only the period of libration (e.g., in pendulum) is considered in the computation of phase change without any consideration of rotation. (iii) The approximate computation of energy increments is based only on the homoclinic orbit. For the stochastic instability of trapped particles, Zaslavsky and Filonenko [13] obtained a standard map by linearization of the whisker map in the neighborhood of an arbitrarilyassigned primary resonance (see also [15, 16]). In addition to the three approximations in the whisker map, the linearization results in two more approximations for the standard map: (iv) The linearization based on a primary resonance is employed. (v) The primary resonance is chosen arbitrarily without any due regard for libration and rotation. Because of the above approximations (iv) and (v), the standard map cannot give the satisfactory prediction of the stochastic layer. In recent years, the whisker map has been favored over the standard map in investigations of stochastic layers (e.g., [10–12, 17–21]). In 1990, Rom-Kader [17] derived the energy relationship in the whisker map using the Melnikov function. In 1994, she investigated transport rates in the stochastic layer and escape rates in the vicinity of homoclinic tangles statistically and numerically. However, the analytic condition does not correlate well with resonance arising from the original system [18]. In the following year, Rom-Kader [19] considered the secondary homoclinic bifurcation in the stochastic layer, and presented the secondary homoclinic bifurcation theorems for the secondary intersection and tangency of the stable and unstable manifolds of the hyperbolic periodic orbits. However, in the stochastic layer, there is an infinite number of primary resonances clustered in the neighborhood of the separatrix of the unperturbed system [15]. The secondary homoclinic bifurcation theorems cannot apply to stochastic layers characterized by resonantseparatrix webs. In 1995, Zaslavsky and Abdullaev [11] introduced a shifted separatrix map to study the scaling properties and anomalous transport of particles inside the stochastic layer. Both the whisker and standard map mentioned above have inherent limitations caused by derivation. Therefore, the two maps cannot provide consistently satisfactory predictions of the stochastic layer except for lower-order resonance and very weak excitations. In 1995, Luo [22] developed the incremental energy approach to avoid those limitations and derived the accurate whisker map to investigate the mechanism of chaotic motion in stochastic layers (see also [3]). In this paper, resonant-separatrix webs in stochastic layers are investigated numerically, and the excitation strength for the onset of a new resonant separatrix in the stochastic layer is determined by the minimum and maximum energy spectra. The width of the stochastic layer is computed through the minimum and maximum energy. For comparison, the analytical condition for the onset of a new resonant-separatrix is also presented.
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2. Incremental Energy Approach Consider a two-dimensional time-periodic system defined by x x˙ = f(x) + g(x, t); x = ∈
