A Study of Homoclinic/Heteroclinic Bifurcations, HomoclinicDoubling Bifurcations and Cascades of Dynamical Systems Using a PerturbationIncremental Method
攝動增量法對動力系統中同宿/異宿分岔，同宿倍增分岔和序列的研究
Student thesis: Doctoral Thesis
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Award date  25 Aug 2017 
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Permanent Link  https://scholars.cityu.edu.hk/en/theses/theses(c31b5c8b858c4a85ba65f3dbb5ca7191).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 an analytical approximation of heteroclinic connections in the 1:3 resonance problem by the method of nonlinear time transformation. The analysis was carried out considering the slow flow of a selfexcited nonlinear Mathieu oscillator corresponding to the normal form near the 1:3 strong resonance. Two types of heteroclinic bifurcation of dimension one occur near the 1:3 resonance: the triangle and clover heteroclinic connections. Using the Hamiltonian system of slow flow of the oscillator, the unperturbed zeroorder approximation of the heteroclinic connections is established. Conditions of persistence of connections in the perturbed firstorder approximation of the heteroclinic connections provide close analytical approximations of the triangle and clover heteroclinic bifurcation curves, simultaneously. The analytical predictions are compared to the results obtained by numerical simulations for validation.
In the second part, homoclinicdoubling bifurcation and homoclinicdoubling cascade are studied. The normal form near the TakensBogdanov bifurcation from a convection problem of a square container is used as the illustrative example. Usually, a geometric approach is employed to investigate the existence of homoclinicdoubling cascades. However, this approach is not able to find analytical expression of homoclinic orbits and construct the bifurcation curves arisen from homoclinic doubling. Here, by using the nonlinear time transformation method, the algebraic approach is able to predict homoclinicdoubling bifurcation and homoclinicdoubling cascade. It has the advantage that the usual geometric consideration such as orbit flip, inclination flip, resonant homoclinic orbit and twisting is avoided. The theoretical prediction is crucial in finding the appropriate parametric values for homoclinicdoubling bifurcation analysis. In the numerical simulation, the interesting scenarios of splitting and merging of several homoclinic orbits are observed. The theoretical results agree very well with numerical simulation.
In the third part, the bifurcations of homoclinic orbits emanating from the merging of a pair of heteroclinic/homoclinic orbits are studied by the PerturbationIncremental(PI) method. Consider strongly nonlinear selfexcited oscillators
x''+g(x)=ε(μ+λ x+δ+c_{20}x^{2}+c_{11}xx'+c_{02}x'^{2})x',
where g(x) is an odd nonlinear function of its argument, ε is perturbation parameter, μ, λ, δ, c_{20}, c_{11} and c_{02 }are system parameters. The pair of asymmetric heteroclinic/homoclinic orbits are investigated using the nonlinear time transformation in the perturbation step when λ and δ are nonzero. We predict analytically the bifurcation point at which two branches of the pair of asymmetric orbits coexists. At this point, two new homoclinic orbits appear due to the merging of two heteroclinic/homoclinic orbits. The bifurcation curves of the newborn homoclinic orbits emanating from the bifurcation point are found by using the harmonic balance method in the incremental step. The results agree well with the numerical simulation performed by MatCont.
The first part is about an analytical approximation of heteroclinic connections in the 1:3 resonance problem by the method of nonlinear time transformation. The analysis was carried out considering the slow flow of a selfexcited nonlinear Mathieu oscillator corresponding to the normal form near the 1:3 strong resonance. Two types of heteroclinic bifurcation of dimension one occur near the 1:3 resonance: the triangle and clover heteroclinic connections. Using the Hamiltonian system of slow flow of the oscillator, the unperturbed zeroorder approximation of the heteroclinic connections is established. Conditions of persistence of connections in the perturbed firstorder approximation of the heteroclinic connections provide close analytical approximations of the triangle and clover heteroclinic bifurcation curves, simultaneously. The analytical predictions are compared to the results obtained by numerical simulations for validation.
In the second part, homoclinicdoubling bifurcation and homoclinicdoubling cascade are studied. The normal form near the TakensBogdanov bifurcation from a convection problem of a square container is used as the illustrative example. Usually, a geometric approach is employed to investigate the existence of homoclinicdoubling cascades. However, this approach is not able to find analytical expression of homoclinic orbits and construct the bifurcation curves arisen from homoclinic doubling. Here, by using the nonlinear time transformation method, the algebraic approach is able to predict homoclinicdoubling bifurcation and homoclinicdoubling cascade. It has the advantage that the usual geometric consideration such as orbit flip, inclination flip, resonant homoclinic orbit and twisting is avoided. The theoretical prediction is crucial in finding the appropriate parametric values for homoclinicdoubling bifurcation analysis. In the numerical simulation, the interesting scenarios of splitting and merging of several homoclinic orbits are observed. The theoretical results agree very well with numerical simulation.
In the third part, the bifurcations of homoclinic orbits emanating from the merging of a pair of heteroclinic/homoclinic orbits are studied by the PerturbationIncremental(PI) method. Consider strongly nonlinear selfexcited oscillators
x''+g(x)=ε(μ+λ x+δ+c_{20}x^{2}+c_{11}xx'+c_{02}x'^{2})x',
where g(x) is an odd nonlinear function of its argument, ε is perturbation parameter, μ, λ, δ, c_{20}, c_{11} and c_{02 }are system parameters. The pair of asymmetric heteroclinic/homoclinic orbits are investigated using the nonlinear time transformation in the perturbation step when λ and δ are nonzero. We predict analytically the bifurcation point at which two branches of the pair of asymmetric orbits coexists. At this point, two new homoclinic orbits appear due to the merging of two heteroclinic/homoclinic orbits. The bifurcation curves of the newborn homoclinic orbits emanating from the bifurcation point are found by using the harmonic balance method in the incremental step. The results agree well with the numerical simulation performed by MatCont.