The Hopf Bifurcation Of The Coupled Oscillator

Coupled oscillator system is a common and important dynamical system in natural science,which has been widely used in many fields,such as quantum thermodynamics,network engineering etc.Many scholars have put forward different coupled oscillator models,for example,S.P.Kuznetsov et al proposed a network model composed of five global coupled oscillators and gave the continuous birth cycle Hopf bifurcation of the increasing dimension ring.Moreover they discussed the conditions for proving the ensemble of interacting oscillators and the Landau-Hopf case for the continuous production of the multi-frequency quasiperiodic motion,O.V.Astakhov et al proposed a multi-circuit oscillator model with universal control scheme that drives self-sustained oscillations,they investigated the proposed oscillator by both experiment and numeric and confirmed the possibility of excited multibranching and quantum quasi-cycles with several positive Lyapunov exponents,chaotic and hyperchaotic oscillations,etc.A.P.Kuznetsov et al.[Physica D,2019]presented a coupled generation of quasi-periodic oscillations.Based on numerical simulation and calculating Lyapunov exponents,they found many dynamical behaviors of model,such as Hopf bifurcation,Neimark-Sacker bifurcation,invariant tori,etc.In this paper,we study the Hopf bifurcation and the Hopf-Hopf bifurcation of the model from the perspective of bifurcation theory,and theoretically confirm some numerical simulations in A.P.Kuznetsov et al.:the coupled oscillator has continuous periodic and quasi-periodic oscillations.The Hopf bifurcation is obtained by taking the dissipative coupling coefficient Mc as a bifurcation parameter,fixing other parameters,transforming independent variables,diagonalizating the coefficient matrix,restricting it to the central manifold,eliminating non-resonance terms and so on.We prove that there is a supercritical Hopf bifurcation behavior and a stable limit loop at the branch point Mc=3/2+25/156?2 and the only equilibrium point in the model.In studying the Hopf-Hopf bifurcation,the positive feedback strength parameters ?1,2 are taken as bifurcation parameters.By the center manifold theorm,normal form and bifurcation theory,the Hopf-Hopf bifurcation of the coupled oscillators is obtained at the branch point(?1,?2)=(3,3)and the only equilibrium point,and we can obtain the corresponding asymptotically stable two-dimensional invariant torus.