Heteroclinic tangles and homoclinic snaking in the unfolding of a degenerate reversible Hamiltonian-Hopf bifurcation
Heteroclinic tangles and homoclinic snaking in the unfolding of a degenerate reversible Hamiltonian-Hopf bifurcation
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DOI：
10.1016/s0167-2789(98)00309-1
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发表时间：
1999-05-15
影响因子：
4
通讯作者：
Champneys, AR
中科院分区：
文献类型：
--
作者：
Woods, PD;Champneys, AR
本文研究了四维时间可逆常微分方程系统的退化可逆1-1共振（或hamilton - hopf）分岔的展开。当平衡特征值的复四重在虚轴上合并成为虚对时，就会发生这种分岔。简并是通过一个标准形式系数（q(2) = 0）的消失而发生的，该系数决定了分岔是超临界还是亚临界。特别值得关注的是平凡平衡和简单周期轨道之间的同斜和异斜连接的行为。在Dias和ioss （Eur）的研究中，已经出现了这种解决方案的部分展开。j .机械工程。B/ fluid, 15(1996) 367-393)，给出一个高阶项（q(4) < 0）的系数符号。在这里，他们的分析被推广到包括q4的另一个符号，由一个四阶ODE驱动，其解模型局部化了支柱的屈曲和广义Swift-Hohenberg方程的稳态。通过数值实验确定了同斜轨道到原点的全局可逆和哈密顿行为。显式计算了范式系数，并找到了一个参数空间区域，其中q(4) > 0和-1
This paper considers an unfolding of a degenerate reversible 1-1 resonance (or Hamiltonian-Hopf) bifurcation for four-dimensional systems of time reversible ordinary differential equations (ODEs). This bifurcation occurs when a complex quadruple of eigenvalues of an equilibrium coalesce on the imaginary axis to become imaginary pairs. The degeneracy occurs via the vanishing of a normal form coefficient (q(2) = 0) that determines whether the bifurcation is super- or subcritical. Of particular concern is the behaviour of homoclinic and heteroclinic connections between the trivial equilibrium and simple periodic orbits. A partial unfolding of such solutions already occurs in the work of Dias and Iooss (Eur. J. Mech. B/Fluids, 15 (1996) 367-393), given a sign of the coefficient of a higher-order term (q(4) < 0). Here, their analysis is generalised to include the other sign of q4, motivated by a fourth-order ODE whose solutions model localised buckling of struts and steady states of the generalised Swift-Hohenberg equation. Numerical experiments are undertaken to determine the global behaviour of homoclinic orbits to the origin in the example which is both reversible and Hamiltonian. The normal form coefficients are calculated explicitly and a region of parameter space found where q(4) > 0 and -1