Complex mixed-mode periodic and chaotic oscillations in a simple three-variable model of nonlinear system

Abstract

A detailed study of a generic model exhibiting new type of mixed-mode oscillations is presented. Period doubling and various period adding sequences of bifurcations are observed. New type of a family of 1D (one-dimensional) return maps is found. The maps are discontinuous at three points and consist of four branches. They are not invertible. The model describes in a qualitative way mixed-mode oscillations with two types of small amplitude oscillations at local maxima and local minima of large amplitude oscillations, which have been observed recently in the Belousov-Zhabotinsky system. (c) 2000 American Institute of Physics.

Period-Adding Bifurcations in Mixed-Mode Oscillations in the Belousov−Zhabotinsky Reaction at Various Residence Times in a CSTR

“…13,14 These studies may be helpful in experimental searches not only for a particular MMO with a given pattern but also for sequences of bifurcations. In this paper the initial sequence of period-adding bifurcations for the asymptotic regime has been found.…”

“…13, 14 This model was also helpful in the successful searches of new types of MMO with patterns LS n s m . 9 Detailed studies of this model reveal a rich variety of bifurcation sequences such as period doubling, period adding, broken Farey Table 1 may be obtained for the following values of the bifurcation parameter: LS 1 for r ) 0.07; LS 2 for r ) 0.067; LS 3 for r ) 0.066; LS 4 for r ) 0.065; LS 5 for r ) 0.06457.…”

“…Studies of this model show that the most important phenomenon which appears with changes of a bifurcation parameter is the period-adding sequence of bifurcations. 13, 14 These results have been prompted us to look for the period adding in the experiments.…”

Mixed Mode, Sequential and Relaxation Oscillations in the Belousov-Zhabotinsky System

“…The simple three-variable model [2,3] which qualitatively describes various asymptotic MMO observed in the BZ reaction, has been studied recently. This model was also helpful in the successful searches of new types of MMO with LS n s m [4] patterns:…”

“…The system attained an equilibrium in the manner of Sniper bifurcation, which was carefully discussed [1]. The chemical reactions, used by us in our previous article [1] to describe Hopf and Sniper bifurcations, were not able to reproduce the mixed mode region, but the simple theoretical model [2,3] used previously [4] has done it quite well and it is shown in the work.…”