Homotopy Theory Of Nonlinear Dynamical Systems In Applied Research

A new homotopy technique based on the parameter expansion (so-called PE-HAM) is proposed to study strongly nonlinear oscillation. By means of constructing a new homotopy mapping and introducing the technique of parameter expansion with the theory of homotopy, we transform the original non-linear dynamical system into a set of linear differential equations which can be solved easily. Theoretically, one canobtain the m th-order approximation solutions u^k(t) and the more accurateapproximate solution can be found using Taylor expansion with the increase of m . The contents of this paper are:In chapter 1, we give a summarization of nonlinear dynamics system. The overview of deterministic and stochastic nonlinear dynamics system is made and the current situation and problems of strongly nonlinear dynamics system under harmonic or random excitation is investigated. Finally, a simple introduce of homotopy is employed.In chapter 2, a new homotopy technique which is based on the parameter expansion, that is PE-HAM, is proposed firstly to study strongly nonlinear oscillations. The new method is valid for problems which are independent of small parameters .A typical system in the form of conservative Duffing oscillator is employed to show the basic idea of this method. The zero-th and the first approximate solutions of the Duffing oscillator are obtained which are good agreements with the numerical solutions by means of 4-th Runge-Kutta routine. When α is not a small parameter, we proof that the relative error of period exceeds no more than 2.17 percent compared the exact period with the zero-th approximate one. The numerical simulation shows the accuracy, too.In chapter 3. the PE-HAM method is proposed to study the strongly non-linear dissipated Duffing's oscillator with harmonic excitation of the form u + βω₀u + (ω₀)²u + αu³ = λcosΩt to obtain the approximate solution. Also, by means of 4-th Runge-Kutta routine, the numerical simulation is carried out to obtain thenumerical solution and excellent agreement between the theoretical result and the numerical one is found.In chapter 4, the purpose of this section is to continue our investigation into obtaining analytic approximate solution of strongly nonlinear oscillations by the PE-HAM method. The PE-HAM method is proposed to investigate the strongly nonlinear dissipated oscillator with harmonic excitation and stochastic excitation. A strongly nonlinear dissipated Duffing's oscillator subjected to harmonic excitation and Gaussian white noise is studied using this method, and its approximate analytic solution process and steady-state probability density are obtained. Comparing the results attained by the present method with the ones given by 4-th Runge-Kutta routine and Wu and Y.K.Lin, 1984, excellent agreements can be found.In chapter 5, the dynamics of a particle in a triple well φ⁶ potential subject toharmonic excitation and stochastic excitation are investigated. The PE-HAM method is proposed to obtain the approximate solution. By constructing an appropriate homotopy mapping, the stochastic solution process and steady-state probability density are achieved successfully. Then, the chaotic behaviors are discussed in detail. Following Melnikov theory, conditions for the existence of transverse intersection on the surface of homoclinic or heteroclinic orbits for triple potential well case arederived, which are complemented by the numerical simulations in section 5.3.Finally the concluding remarks are employed to close this paper.