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\centerline{{\sc Problem set 7}}
\centerline{Math 207A, Fall 2018}
\centerline{Due: Fri., Nov. 30}


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{\bf 1.}
The normal form for a Takens-Bogdanov bifurcation is the following $2\times 2$ system
depending on two parameters $(\mu,\nu)$:
\[
\dot{x} = y,\qquad \dot{y} = \mu + \nu y + x^2 + xy.
\]
(a)  What is the algebraic  structure of the linearized system at $(x,y;\mu,\nu) = (0,0;0,0)$? Find the equilibria, classify them, and determine what bifurcations of equilibria occur as $(\mu,\nu)$ vary over $(\mu,\nu)\ne (0,0)$.

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(b) Compare your results with the global phase portraits shown in Figure~8.14 of the text by Meiss.

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{\bf 2.}  (a) Consider a Hamiltonian system for $x,y\in\Rl^n$
\[
\dot{x} = \frac{\partial H}{\partial y},\qquad \dot{y} = -\frac{\partial H}{\partial x}
\]
with Hamiltonian $H(x,y)\in \Rl$. Show that if $\lambda$ is an eigenvalue of the linearization at an equilibrium,
then so is $-\lambda$. What are the possible forms of the eigenvalues for a $2\times 2$ system?
\textsc{Hint.} Show that the matrix of the linearization has the form $JA$ where $A$ is symmetric and
\[
J = \left(\begin{array}{cc}  0 & I \\ -I &0\end{array}\right)
\]
(b) Suppose $x,y\in \Rl$ and
\[
H(x,y,\mu) = \frac{1}{2} y^2 + V(x,\mu),\qquad V(x,\mu) = \frac{1}{4} x^4 - \frac{1}{2}\mu x^2,
\]
where $\mu\in \Rl$ is a parameter.
Determine the equilibria of the corresponding Hamiltonian system as a function of $\mu$, classify them. Sketch the resulting
phase planes and the bifurcation diagram.

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