Hopf Bifurcation, Multistability and its Control in a Satellite System

In this paper, we investigate the dynamical behavior of a chaotic satellite system established by Ayub and collaborators Khan (Pramana 90:1-9, 2018). The basic properties of the model, including dissipation, equilibrium point, and Kaplan-Yorke dimension, have been investigated by Ayub and collaborators. Using numerical tools such as diagrams of bifurcation, the graph of the maximum Lyapunov exponent, phase portraits, the basin of initial conditions, and two-parameter diagrams, we demonstrated that, as the control parameter of the satellite system exceeds a critical value, subcritical Hopf bifurcation occurs. Moreover, we found that for a range of fixed parameters and depending on the initial conditions, the satellite system may describe the coexistence of several orbits (three chaotic attractors with one periodic attractor). Considering that satellite parameters are not accessible and according to satellite application, it is well known that the choice of the orbit described by the satellite depends on the mission assigned to it. Then, in this work, we applied a control strategy based on the linear augmentation scheme to drive the satellite from its chaotic orbit to a periodic one (i.e. control of the multistability). As a result, one can easily observe that three of the four coexisting attractors were annihilated at a critical value of the coupling strength, with only a periodic one remaining. Finally, the circuit emulator of the chaotic satellite system investigated in this contribution has been designed in the PSpice environment. The investigation of the circuit perfectly supports the numerical results obtained during the control of multistability.

Automated summary

In this study, we investigate the dynamical behavior of a chaotic satellite system and demonstrate the occurrence of subcritical Hopf bifurcation as well as the coexistence of multiple orbits. We successfully apply a control strategy based on linear augmentation scheme to drive the satellite from chaotic orbit to periodic orbit.