无参数分叉：一些ODE和PDE例子
Bifurcations without parameters: some ODE and PDE examples
摘要：
标准分歧理论涉及向量场族$dx/dt = f(x,λ)$、$x \in \R^n$，涉及一个或多个常数实参数$λ$。作为一个微分方程对$(x,λ)$，我们观察到一个叶状的总相空间常数$λ$。通常，还强加了平凡的固定解$x=0$的存在：$0 = f(0,λ)$。与此相反，没有参数的分叉通过一个常数参数$λ$丢弃叶理。相反，我们考虑系统$dx/dt = f(x,y), dy/dt = g(x,y)$。标准分歧理论则对应于特殊情况$y=λ, g=0$。为了只保留平凡解$x=0$，我们只需要所有$y$的$0 = f(0,y) = g(0,y)$。一个丰富的动力学现象出现时，正常双曲平凡固定流形$x=0$失败，由于零或纯虚特征值的雅可比矩阵$f_x(0,y)$。具体来说，我们解决的情况下，由于一个简单的特征值零，一个简单的纯虚对（没有参数的Hopf分叉），一个双特征值零（没有参数的Takens-Bogdanov分叉），并由于一个双特征值零与额外的时间反演对称性的正常双曲失败。这一结果是与安德烈·阿芬迪科夫、詹姆斯·C.亚历山大和斯特凡·利布舍尔
英文摘要：
Standard bifurcation theory is concerned with families of vector fields $dx/dt = f(x,λ)$, $x \in \R^n$, involving one or several constant real parameters $λ$. Viewed as a differential equation for the pair $(x,λ)$, we observe a foliation of the total phase space by constant $λ$. Frequently, the presence of a trivial stationary solution $x=0$ is also imposed: $0 = f(0,λ)$. Bifurcation without parameters, in contrast, discards the foliation by a constant parameter $λ$. Instead, we consider systems $dx/dt = f(x,y), dy/dt = g(x,y)$. Standard bifurcation theory then corresponds to the special case $y=λ, g=0$. To preserve only the trivial solution $x=0$, instead, we only require $0 = f(0,y) = g(0,y)$ for all $y$. A rich dynamic phenomenology arises, when normal hyperbolicity of the trivial stationary manifold $x=0$ fails, due to zero or purely imaginary eigenvalues of the Jacobian $f_x(0,y)$. Specifically, we address the cases of failure of normal hyperbolicity due to a simple eigenvalue zero, a simple purely imaginary pair (Hopf bifurcation without parameters), a double eigenvalue zero (Takens-Bogdanov bifurcation without parameters), and due to a double eigenvalue zero with additional time reversal symmetries. The results are joint work with Andrei Afendikov, James C. Alexander, and Stefan Liebscher.
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