The perturbationincremental method for nonlinear dynamical systems
攝動增量法在非線性動力系統中的應用
Student thesis: Doctoral Thesis
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Award date  3 Oct 2012 
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Permanent Link  https://scholars.cityu.edu.hk/en/theses/theses(4687d925fa624ec9aa980c4db609a309).html 

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Abstract
In this thesis, we will study the application of the PerturbationIncremental (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 PerturbationIncremental 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 LindstedtPoincare method and the multiple scales method fail to do so.
In the second part, we describe the application of the PI method to onedimensional
complex GinzburgLandau equation. We first consider the cubic complex GinzburgLandau
equation as the form ∂tA=μA+βA2A+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 cubicquintic complex GinzburgLandau equation (QCGLE) without
regard to nonlinear gradient terms ∂tA=μA+βA2A+γA4A+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
GinzburgLandau equation in the form of given by iut+1/2uxx+1/2(βiF)uyy+(1iδ)u2u=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