• # A1.6 B1.17

(i) State Liapunov's First Theorem and La Salle's Invariance Principle. Use these results to show that the system

$\ddot{x}+k \dot{x}+\sin x=0, \quad k>0$

has an asymptotically stable fixed point at the origin.

(ii) Define the basin of attraction of an invariant set of a dynamical system.

Consider the equations

$\dot{x}=-x+\beta x y^{2}+x^{3}, \quad \dot{y}=-y+\beta y x^{2}+y^{3}, \quad \beta>2$

(a) Find the fixed points of the system and determine their type.

(b) Show that the basin of attraction of the origin includes the union over $\alpha$ of the regions

$x^{2}+\alpha^{2} y^{2}<\frac{4 \alpha^{2}\left(1+\alpha^{2}\right)(\beta-1)}{\beta^{2}\left(1+\alpha^{2}\right)^{2}-4 \alpha^{2}} .$

Sketch these regions for $\alpha^{2}=1,1 / 2,2$ in the case $\beta=3$.

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• # A2.6 B2.17

(i) A linear system in $\mathbb{R}^{2}$ takes the form $\dot{\mathbf{x}}=\mathrm{Ax}$. Explain (without detailed calculation but by giving examples) how to classify the dynamics of the system in terms of the determinant and the trace of A. Show your classification graphically, and describe the dynamics that occurs on the boundaries of the different regions on your diagram.

(ii) A nonlinear system in $\mathbb{R}^{2}$ has the form $\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x}), \mathbf{f}(0)=0$. The Jacobian (linearization) $A$ of $\mathbf{f}$ at the origin is non-hyperbolic, with one eigenvalue of $A$ in the left-hand half-plane. Define the centre manifold for this system, and explain (stating carefully any results you use) how the dynamics near the origin may be reduced to a one-dimensional system on the centre manifold.

A dynamical system of this type has the form

\begin{aligned} &\dot{x}=a x^{3}+b x y+c x^{5}+d x^{3} y+e x y^{2}+f x^{7}+g x^{5} y \\ &\dot{y}=-y+x^{2}-x^{4} \end{aligned}

Find the coefficients for the expansion of the centre manifold correct up to and including terms of order $x^{6}$, and write down in terms of these coefficients the equation for the dynamics on the centre manifold up to order $x^{7}$. Using this reduced equation, give a complete set of conditions on the coefficients $a, b, c, \ldots$ that guarantee that the origin is stable.

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• # A3.6 B3.17

(i) Consider a system in $\mathbb{R}^{2}$ that is almost Hamiltonian:

$\dot{x}=\frac{\partial H}{\partial y}+\epsilon g_{1}(x, y), \quad \dot{y}=-\frac{\partial H}{\partial x}+\epsilon g_{2}(x, y)$

where $H=H(x, y)$ and $|\epsilon| \ll 1$. Show that if the system has a periodic orbit $\mathcal{C}$ then $\oint_{\mathcal{C}} g_{2} d x-g_{1} d y=0$, and explain how to evaluate this orbit approximately for small $\epsilon$. Illustrate your method by means of the system

$\dot{x}=y+\epsilon x\left(1-x^{2}\right), \quad \dot{y}=-x .$

(ii) Consider the system

$\dot{x}=y, \quad \dot{y}=x-x^{3}+\epsilon y\left(1-\alpha x^{2}\right)$

(a) Show that when $\epsilon=0$ the system is Hamiltonian, and find the Hamiltonian. Sketch the trajectories in the case $\epsilon=0$. Identify the value $H_{c}$ of $H$ for which there is a homoclinic orbit.

(b) Suppose $\epsilon>0$. Show that the small change $\Delta H$ in $H$ around an orbit of the Hamiltonian system can be expressed to leading order as an integral of the form

$\int_{x_{1}}^{x_{2}} \mathcal{F}(x, H) d x$

where $x_{1}, x_{2}$ are the extrema of the $x$-coordinates of the orbits of the Hamiltonian system, distinguishing between the cases $HH_{c}$.

(c) Find the value of $\alpha$, correct to leading order in $\epsilon$, at which the system has a homoclinic orbit.

(d) By examining the eigenvalues of the Jacobian at the origin, determine the stability of the homoclinic orbit, being careful to state clearly any standard results that you use.

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• # A4.6 B4.17

(a) Consider the map $G_{1}(x)=f(x+a)$, defined on $0 \leqslant x<1$, where $f(x)=x[\bmod 1]$, $0 \leqslant f<1$, and the constant $a$ satisfies $0 \leqslant a<1$. Give, with reasons, the values of $a$ (if any) for which the map has (i) a fixed point, (ii) a cycle of least period $n$, (iii) an aperiodic orbit. Does the map exhibit sensitive dependence on initial conditions?

Show (graphically if you wish) that if the map has an $n$-cycle then it has an infinite number of such cycles. Is this still true if $G_{1}$ is replaced by $f(c x+a), 0

(b) Consider the map

$G_{2}(x)=f\left(x+a+\frac{b}{2 \pi} \sin 2 \pi x\right)$

where $f(x)$ and $a$ are defined as in Part (a), and $b>0$ is a parameter.

Find the regions of the $(a, b)$ plane for which the map has (i) no fixed points, (ii) exactly two fixed points.

Now consider the possible existence of a 2-cycle of the map $G_{2}$ when $b \ll 1$, and suppose the elements of the cycle are $X, Y$ with $X<\frac{1}{2}$. By expanding $X, Y, a$ in powers of $b$, so that $X=X_{0}+b X_{1}+b^{2} X_{2}+O\left(b^{3}\right)$, and similarly for $Y$ and $a$, show that

$a=\frac{1}{2}+\frac{b^{2}}{8 \pi} \sin 4 \pi X_{0}+O\left(b^{3}\right)$

Use this result to sketch the region of the $(a, b)$ plane in which 2-cycles exist. How many distinct cycles are there for each value of $a$ in this region?

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