Abstract
In this paper, the non-linear dynamical behavior of a 3D autonomous dissipative system of Hopf–Langford type is investigated. Through the help of a mode transformation (as the system’s energy is included) it is shown that the 3D nonlinear system can be separated of two coupled subsystems in the master (drive)-slave (response) synchronization type. After that, based on the computing first and second Lyapunov values for master system, we have attempted to give a general framework (from bifurcation theory point of view) for understanding the structural stability and bifurcation behavior of original system. Moreover, a family of exact solutions of the master system is obtained and discussed. The effect of synchronization on the dynamic behavior of original system is also studied by numerical simulations.
Highlights
Non-linear chaotic behavior has been observed in systems of different nature as this motion is based on homoclinic structures which instability accompanied by local divergence and global contraction [1,2,3]
The effect of synchronization on the dynamic behavior of original system is studied by numerical simulations
The introduction of energy as a new variable in the original system leads to an equivalent four-dimensional system which can be separated into two coupled subsystems in the master ( x3, E)– slave ( x1, x2 ) synchronization type
Summary
Non-linear chaotic behavior has been observed in systems of different nature as this motion is based on homoclinic (heteroclinic) structures which instability accompanied by local divergence and global contraction [1,2,3] The investigation of dynamical processes in coupled nonlinear systems is an interesting problem from both theoretical (mathematical) and applied (engineering) points of view. Phenomena such as stability in interacting subsystems can be observed in nature and science. The structural stability (roughness) investigation of steady state and of limit cycles or other types of trajectories is a main problem in bifurcation theory.
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