Summary

Proceedings of the 2013 International Symposium on Nonlinear Theory and its Applications

2013

Session Number:C3L-D

Session:

Number:457

Lyapunov Exponents of Saddle Quasi-Periodic Solutions

Kyohei Kamiyama,  Motomasa Komuro,  Tetsuro Endo,  

pp.457-458

Publication Date:

Online ISSN:2188-5079

DOI:10.15248/proc.2.457

PDF download (513KB)

Summary:
We developed the algorithm for obtaining saddle quasi-periodic solutions and demonstrated them in a ring of coupled bistable oscillators [1]. This algorithm worked when a saddle quasi-periodic solution existed in the basin boundary of two attractors. The reason why we derived the saddle quasi-periodic solution was that it was indispensable to elucidate the bifurcation mechanism of quasi-periodic solutions such as saddle-node, Neimark-Sackker, and pitchfork, etc.
In this paper, we develop the algorithm for calculating Lyapunov exponents (LEs) of the saddle quasi-periodic solution. Usually, LEs are calculated for “attractors,” therefore the algorithm using variational equation along a stable flow works well. However, when the flow is unstable as is the case of saddle quasi-periodic solution, we have to calculate the variational equation by correcting the unstable flow for every short time span. We demonstrate the calculation of LEs of pitchfork and saddle-node bifurcations of saddle quasi-periodic solutions in a ring of several number of coupled hard-type oscillators.
As an example, we will introduce LEs for the saddle quasi-periodic solution obtained for the two coupled hard-type oscillator system shown in Eq. (1).
From Eq. (1) we can obtain SICC (a stable (nodal) quasi-periodic solution) and UICC1 (an unstable (saddle) quasi-periodic solution) as shown in Fig. 1 [2]. To obtain UICC1 we use the algorithm to calculate a saddle quasi-periodic orbit shown in NOLTA2012 [1]. Fig. 2 shows the variation of Lyapunov exponents for UICC1 and SICC in terms of β. The upper trace of Fig. 2 shows the variation of three LEs of UICC1 in descending order. They show one positive, two almost zero LEs which are one of the evidences of saddle quasi-periodic solution. The lower trace of Fig. 2 shows that of three LEs in the same order. They show two almost zero and one negative LEs which are one of the evidences of stable (nodal) quasi-periodic solution. From this diagram we notice that a saddle-node bifurcation occurs clearly at β = β*=3.909.
For calculating LEs of SICC, we can use ordinary algorithm. In contrast, for calculating LEs of UICC, we must use our developed new algorithm. The algorithm is presented by flowchart shown in Fig. 3 [3]. In Fig. 3 an operator a´=Solve(ta, tb; a) is defined such that numerically solving Eq. (1) with initial condition a at t = ta gives a new value a´ at t = tb. Therefore, the operator Solve(t, t + T; at) gives the numerically calculated value at+T at t = t + T where T is chosen as T = 0.1 sec. Then, by using the obtained saddle solution as a core orbit, we can calculate LEs for the saddle quasi-periodic solution.

References:

[1] K. Kamiyama, M. Komuro, and T. Endo, “Improvement of the saddle unstable invariant closed curve (uicc) pursuit algorithm —the case of a saddle uicc between two stable iccs—,” Proceedings of the 2012 International Symposium on Nonlinear Theory and its Applications (NOLTA2012), p. 66, Oct. 2012.

[2] ——, “Bifurcation of quasi-periodic oscillations in mutually coupled hard-type oscillators— demonstration of unstable quasi-periodic orbits—,” Int. J. Bifurcation and Chaos, vol. 22, no. 06, p. 1230022 (13 pages), Jun. 2012.

[3] ——, “Algorithms for obtaining a saddle torus between two attractors,” Int. J. Bifurcation and Chaos, To be submitted.