Technical Physics, Vol. 49, No. 5, 2004, pp. 521–525. Translated from Zhurnal TekhnicheskoÏ Fiziki, Vol. 74, No. 5, 2004, pp. 1–5. Original Russian Text Copyright © 2004 by Vecheslavov. THEORETICAL AND MATHEMATICAL PHYSICS Chaotic Layer of a Pendulum under Low- and Medium-Frequency Perturbations V. V. Vecheslavov Budker Institute of Nuclear Physics, Siberian Division, Russian Academy of Sciences, pr. Akademika Lavrent’eva 11, Novosibirsk, 630090 Russia e-mail: [email protected] Received April 22, 2003 Abstract—The amplitude of the separatrix map and the size of a pendulum chaotic layer are studied numeri- cally and analytically as functions of the adiabaticity parameter at low and medium perturbation frequencies. Good agreement between the theory and numerical experiment is found at low frequencies. In the medium-fre- quency range, the efficiency of using resonance invariants of separatrix mapping is high. Taken together with the known high-frequency asymptotics, the results obtained in this work reconstruct the chaotic layer pattern throughout the perturbation frequency range. © 2004 MAIK “Nauka/Interperiodica”. INTRODUCTION important that actually both unperturbed separatrices Interaction between nonlinear resonances with the consist of two spatially coincident trajectories for for- formation of dynamic chaos in Hamiltonian systems is ward and backward time, respectively. a complex problem, which is far from being solved. In In the case of an analytical potential (as in (1)), the a number of cases, this problem may be reduced to presence of at least one (!) perturbing resonance always studying a pendulum (a fundamental resonance near results in splitting either of the separatrices into two which initial conditions are chosen) subjected to a branches (“whiskers” after Arnold). These branches do quasi-periodic perturbation not return to the saddle point and are no longer coinci- 2 dent with each other. They intersect at so-called (),, p ω2 () (), 1 Hxpt = ----- ++0 cos x Vxt, (1) homoclinic points. The free ends of these branches 2 form an infinite number of loops with an infinitely Vxt(), = εcos()a x – Ω t + ε cos()a x – Ω t , (2) increasing length that fill a narrow region near the 1 1 1 2 2 2 unperturbed separatrices and create a chaotic layer. ε ε where harmonic amplitudes 1 and 2 are assumed to be This layer can be subdivided into three parts: the top ε ε Ӷ small ( 1, 2 1). Note that either of the harmonics is part (the phase x rotates at the top, p > 0), the middle also of resonant nature and, hence, may be a fundamen- part (the phase oscillates), and the bottom part (the tal resonance in a related region of the phase space. phase x rotates at the bottom, p < 0). Determination of The set of equations (1) and (2) has been the subject the sizes of these parts, which may be substantially dif- of intensive research (see, e.g., [1–3]). The situation in ferent for an asymmetrical perturbation like (2), is of the vicinity of the separatrices of the fundamental reso- great importance from the practical standpoint [8–10]. nance and the formation of a chaotic layer have been The formation of a chaotic layer in the case of a studied most frequently. Our goal is to investigate the symmetrical high-frequency perturbation of the type fundamental chaotic layer throughout the range of per- turbation frequencies (it is appropriate here to recall the m m Vxt(), = εcos ----x – Ωt + εcos ----x + Ωt , (3) mechanism of its formation [1]). We will start with 2 2 describing unperturbed separatrices. First of all, they always have a saddle, a fixed point, Ω ӷ ω where 0 and m is an integer, was studied in detail which should be treated as an independent trajectory by Chirikov [1]. (an unperturbed pendulum may follow this trajectory for an infinite time). Two trajectories (separatrices) Using the properties of standard mapping and his originate from the saddle point and then asymptotically criterion for resonance overlapping, Chirikov showed approach it. Either is the boundary between the off-res- onance phase rotation and resonant phase oscillation. In 1 Such a splitting does not always happen in systems with a smooth potential. Striking situations where the separatrices of both frac- the neighborhood of the saddle point in the phase plane, tional and integer resonances remain intact in piecewise linear there appears a characteristic cross with two outgoing systems despite the presence of a perturbation and strong local and two incoming trajectories (Fig. 2.1 in [1]). It is chaos are discussed in [4–6]. 1063-7842/04/4905-0521 $26.00 © 2004 MAIK “Nauka/Interperiodica” 522 VECHESLAVOV that, in the high-frequency limit, all three parts of the frequencies of the perturbation, ε = 0.01, and weak ω layer are of equal size: oscillations, 0 = 1.0 (unless otherwise specified). The high-frequency range considered above is comple- w ===w w λW, λ∞, (4) tp md bt mented by the ranges of medium and low frequencies, where λ = Ω/ω is the so-called adiabaticity parameter which makes it possible to characterize the layer 0 λ ∞ ω2 throughout the range 0 < < . and w = H(x, p, t)/0 – 1 is the relative deviation from the unperturbed separatrix in terms of energy. The variable W appearing in (4) is the amplitude of 1. SEPARATRIX MAP AMPLITUDE the harmonic (with a frequency Ω) of the separatrix map for the set of equations (1) and (3). This map, first Two basic methods for calculating the sizes of the introduced in [7], approximately describes the dynam- chaotic layer are known. In the first method, the mini- ics of a system near the separatrix when it passes mum period T0, min of motion in a given region of the through the states of stable equilibrium (see also [1]). layer is found (T0 is the time interval between two suc- cessive intersections of the equilibrium phase x = π). The theoretical value W = WT is related to the λ Then, the energy size of this region is determined by the Mel’nikov–Arnold integrals Am( ) [1] by the relationship formula [1] ()λ ελ ()λ WT = Am . (5) ()ω w = 32exp – 0T , (9) In this paper, these integrals are also used. Accord- ing to [1], they are defined as where T means T0, min. 2π exp()πλ/2 m – 1 The second method implies the construction of the A (λ > 0 ) = ---------------------------------------------- ()2λ []1 + f ()λ , m ()m – 1 ! sinh()πλ m separatrix map of the system, (6) ()λ < ()m ()λ () πλ ψ ψ ψλ32 Am 0 = –1 Am exp– , (7) wwW==+ sin , + ln------ , w (10) () ()mm– 1 ψΩ ()π f 1 ==f 2 0, f m + 1 =f m – 1 + f m – 1 ---------------------- , = T0mod 2 4λ2 (8) and the application of the iteration procedure. m ≥ 3. We will briefly recall the numerical algorithm for It should be stressed that expressions (6)–(8) were constructing such a map (for details, see [8]). First, a derived in [1] for the general case, i.e., without any central homoclinic point p as a boundary between the assumptions or simplifications. Therefore, they are pb valid for any λ from the range 0 < λ < ∞. oscillation and rotation of the phase is found with a high accuracy at the line of symmetry x = π. A narrow The pattern drastically and qualitatively changes δ interval ppb + p is chosen in the vicinity of this point at when a high-frequency perturbation becomes asym- π Ω ≠ Ω x = , and a random path is emitted from the interval. metric, 1 2. Even early numerical simulations This path either executes a prescribed number of cycles [8, 9] showed that the separatrix map spectrum of such Ω of motion or is interrupted because of the transition to a system involves (along with the frequencies 1 and another region of the layer. In both cases, a new random Ω 2, appearing in perturbation (2) in explicit form) path is emitted from the same region until a desired ε ε ∆Ω mixed harmonics (~ 1 2) with the frequencies + = number of cycles N is achieved. The mean energy w is Ω Ω ∆Ω Ω Ω 1 + 2 and – = 2 – 1. Still more surprising is the calculated for each of the cycles by formula (9). By fact that these harmonics completely determine the size determining the energy variation δw = w – w for each of the chaotic layer under certain conditions. The case pair of adjacent cycles and assigning it to a time T that when the contribution of the mixed harmonic with 0 ∆Ω is common for a given pair, one can construct separatrix + = 3 to the separatrix map amplitude is several hun- δ map (10) ( w)k with T0, k, where k = 1, 2, …, N – 1. To dred times greater than the contributions from the pri- be definite, we will investigate the top part of the layer, mary harmonics is described in [9]. The impression that since its size is equal to that of the bottom part, wtp = weak initial harmonics give rise to an intense secondary w , in the case of symmetric perturbation. These outer harmonic and then do not contribute to the formation of bp chaos has been conclusively supported by numerical parts are of primary interest, since they are responsible simulation performed in [9]. for overlapping adjacent resonances and generating global chaos. As we shall see, map (10) may involve not In Sect. 2, we will show that the specific role of only one but also several harmonics (see Sect. 3). mixed harmonics disappears at low frequencies and their influence becomes insignificant. For this reason, Numerical (estimated) results for the separatrix map we consider only the fundamental chaotic layer of sys- amplitude WE should be compared with the theoretical tem (1) subjected to symmetric perturbation (3) at fixed values WT calculated via the Mel’nikov–Arnold inte- TECHNICAL PHYSICS Vol.