Non-resnonant Double Hopf Bifurcation In An Ikeda Model And A Van Der Pol-duffing Oscillator With Time Delay

Time delay is ubiquitous in natural world and humankind society. The phenomenon of time delay means that the current development tendency of the system depends greatly on its past history. It exists extensively in the fields of ecology, life science, neural network, laser, information technology, mechanical engineering, aerospace engineering, security communication and economics, etc. In this paper, the time delay of the Ikeda model we investigate in Chapter2 is the transmission of voltage signal and that of the van der Pol-Duffing oscillator in Chapter3 denotes the time delay occuring in the feedback path. The effects of delay on the dynamics of the systems usually are essential. It does not only effect the stability of the dynamical systems, but also makes the systems lose stability, and at the same time Hopf bifurcations and double Hopf bifurcations arise. Thus rich and complex dynamical behaviors such as stable equilibrium, periodic motions, quasi-periodic motions, multi-stability motions and chaotic motions appear. The double Hopf bifurcation is an important dynamical phenomenon induced by delay. In fact there are very rich dynamical phenomena near the double Hopf bifurcation points, such as periodic motions, quasi-periodic motions, bi-stability motions, three-dimensional tori motions and chaotic motions, and every dynamical pattern has significance in the physical background. Supported by the National Natural Science Foundation of China under Grant NO. 10702065, some problems are considered in this dissertation and some results are obtained as following:(1) The bidirectionally linearly coupled identical Ikeda model with time delay is reduced onto the four-dimensional center manifold by the center manifold Theorem, then its normal forms is obtained by the normal form method. The unfolding and classifications of the nonresonant double Hopf bifurcations are obtained by analyzing the bifurcations of the abovementioned normal forms. The unfolding belongs to Case Ib of the unfolding results and we find rich dynamical behaviors such as the stable periodic solutions and the co-existing bi-stability. Then The the numerical simulation is executed by employing the nonlinear dynamical software WinPP and the correctness the theoretical analysis is verified.(2) The van der Pol-Duffing oscillator with velocity feedback is reduced onto the four-dimensional center manifold by the center manifold Theorem, then its normal forms is obtained by the normal form method. The unfolding and classifications of the non-resonant double Hopf bifurcations are obtained by analyzing the bifurcations of the above-mentioned normal forms. The unfolding belongs to Case VIa of the unfolding results and we find rich dynamical behaviors such as the stable periodic solutions and the quasi-periodic solutions. Then The the numerical simulation is executed by employing the nonlinear dynamical software WinPP and the correctness the theoretical analysis is verified.