The perturbation-incremental method for nonlinear dynamical systems
Student thesis: Doctoral Thesis
Related Research Unit(s)
In this thesis, we will study the application of the Perturbation-Incremental (PI) method in nonlinear dynamical systems. The thesis consists of three parts. The first part is about a novel construction of homoclinic/heteroclinic orbits (HOs) in nonlinear oscillators by the Perturbation-Incremental method. Consider strongly nonlinear oscillators of the form x + g(x) = Ɛf(x, x, μ), (0.1) where g and f are arbitrary nonlinear functions of their arguments, Ɛ and μ are parameters of arbitrary magnitude. Accurate analytical solution of a HO for small perturbation can be obtained in terms of trigonometric functions. An advantage of the present construction is that it gives an accurate approximate solution of a HO for large parametric value in relatively few harmonic terms while other analytical methods such as the Lindstedt-Poincare method and the multiple scales method fail to do so. In the second part, we describe the application of the PI method to one-dimensional complex Ginzburg-Landau equation. We first consider the cubic complex Ginzburg-Landau equation as the form ∂tA=μA+β|A|2A+D∂xxA, (0.2) where β=βr+iβi,D=Dr+iDi Є C and μ Є R. Stationary pulse solution and hole solution are expressed in the form of A = ei(0(ᶓ)+ωt) μ(ᶓ) where μ(ᶓ) and θ(ᶓ) are real functions with ᶓ = x - vt, ω Є R. From the harmonic balance (HB) method with a nonlinear time transform φ, we obtain some exact stationary coherent structures including pulse and hole solutions, as well as traveling solutions. Then we consider the cubic-quintic complex Ginzburg-Landau equation (QCGLE) without regard to nonlinear gradient terms ∂tA=μA+β|A|2A+γ|A|4A+D∂xxA, (0.3) where β = βr + iβi, γ = γr + iγi,D=Dr+iDi Є C and μ Є R. Exact stationary hole solutions are found by the HB method with a nonlinear time transform φ. Some numerical solutions are studied by the PI method. In the third part, a novel approach of using HB method with a nonlinear time transform is presented to find front, soliton and hole solutions of a modified complex Ginzburg-Landau equation in the form of given by iut+1/2uxx+1/2(β-iF)uyy+(1-iδ)|u|2u=iγu, (0.4) where β, F, δ and γ are real constants. Exact stationary and traveling solutions in the form of u = ei(θ(ᶓ)+ωt)V(ᶓ) are studied, where v(ᶓ) and θ(ᶓ) are real functions with ᶓ= p1x + p2y + p3t, and p1, p2, p3 are constants to be determined. Three families of exact solutions are obtained, one of which contains two parameters while the others one parameter. The HB method is an efficient technique in finding limit cycles of dynamical systems. In this thesis, the method is extended to obtain HOs and then coherent structures. It provides a systematic approach in the computation as various methods may be needed to obtain the same families of solutions.
- Nonlinear theories, Mathematical models, Dynamics