Abstract
In this paper, the Hopf-zero bifurcation of the ring unidirectionally coupled Toda oscillators with delay was explored. First, the conditions of the occurrence of Hopf-zero bifurcation were obtained by analyzing the distribution of eigenvalues in correspondence to linearization. Second, the stability of Hopf-zero bifurcation periodic solutions was determined based on the discussion of the normal form of the system, and some numerical simulations were employed to illustrate the results of this study. Lastly, the normal form of the system on the center manifold was derived by using the center manifold theorem and normal form method.
Highlights
The bifurcation theory in dynamical system has become a research hotspot, which is widely used in the fields of physics, chemistry, medicine, finance, biology, engineering and so on [3,12,13,15,19,20,22,24,30,31,32,33,34,35]
The mathematical models to solve practical problems have been described based on nonlinear dynamical systems, in which, how to study the dynamic characteristics of the high-dimensional nonlinear system is very important
We have investigated the Hopf-zero bifurcation in the ring of unidirectionally coupled Toda oscillators with delay
Summary
The bifurcation theory in dynamical system has become a research hotspot, which is widely used in the fields of physics, chemistry, medicine, finance, biology, engineering and so on [3,12,13,15,19,20,22,24,30,31,32,33,34,35]. Under the guidance of the bifurcation theory, our study is majorally under the premise that the external force does not exist (considering only the coupled effect between oscillators). Hopf-zero bifurcation of the ring unidirectionally coupled Toda oscillators with delay. With the threshold of self-oscillation birth, the further increase of coupled coefficient γ will be followed by the new ring resonance phenomenon, besides when the threshold of self-oscillation birth is exceeded, the interaction between external oscillations and the inner oscillatory mode of system (1) will result in the synchronization region in the parameter plane [6, 7].
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