• # Paper 1, Section II, A

(a) State the properties defining a Lyapunov function for a dynamical system $\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x})$. State Lyapunov's first theorem and La Salle's invariance principle.

(b) Consider the system

\begin{aligned} &\dot{x}=y \\ &\dot{y}=-\frac{2 x\left(1-x^{2}\right)}{\left(1+x^{2}\right)^{3}}-k y \end{aligned}

Show that for $k>0$ the origin is asymptotically stable, stating clearly any arguments that you use.

$\left[\text { Hint: } \frac{d}{d x} \frac{x^{2}}{\left(1+x^{2}\right)^{2}}=\frac{2 x\left(1-x^{2}\right)}{\left(1+x^{2}\right)^{3}} \cdot\right]$

(c) Sketch the phase plane, (i) for $k=0$ and (ii) for $0, giving brief details of any reasoning and identifying the fixed points. Include the domain of stability of the origin in your sketch for case (ii).

(d) For $k>0$ show that the trajectory $\mathbf{x}(t)$ with $\mathbf{x}(0)=\left(1, y_{0}\right)$, where $y_{0}>0$, satisfies $0 for $t>0$. Show also that, for any $\epsilon>0$, the trajectory cannot remain outside the region $0.

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• # Paper 2, Section II, A

Consider a modified van der Pol system defined by

\begin{aligned} &\dot{x}=y-\mu\left(\frac{1}{3} x^{3}-x\right) \\ &\dot{y}=-x+F \end{aligned}

where $\mu>0$ and $F$ are constants.

(a) A parallelogram PQRS of width $2 L$ is defined by

$\begin{array}{ll} P=(L, \mu f(L)), & Q=(L, 2 L-\mu f(L)) \\ R=(-L,-\mu f(L)), & S=(-L, \mu f(L)-2 L) \end{array}$

where $f(L)=\frac{1}{3} L^{3}-L$. Show that if $L$ is sufficiently large then trajectories never leave the region inside the parallelogram.

Hence show that if $F^{2}<1$ there must be a periodic orbit. Explain your reasoning carefully.

(b) Use the energy-balance method to analyse the behaviour of the system for $\mu \ll 1$, identifying the difference in behaviours between $F^{2}<1$ and $F^{2}>1$.

(c) Describe the behaviour of the system for $\mu \gg 1$, using sketches of the phase plane to illustrate your arguments for the cases $0 and $F>1$.

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• # Paper 3, Section II, A

Consider the system

\begin{aligned} &\dot{x}=\mu y+\beta x y+y^{2}, \\ &\dot{y}=x-y-x^{2} \end{aligned}

where $\mu$ and $\beta$ are constants with $\beta>0$.

(a) Find the fixed points, and classify those on $y=0$. State how the number of fixed points depends on $\mu$ and $\beta$. Hence, or otherwise, deduce the values of $\mu$ at which stationary bifurcations occur for fixed $\beta>0$.

(b) Sketch bifurcation diagrams in the $(\mu, x)$-plane for the cases $0<\beta<1, \beta=1$ and $\beta>1$, indicating the stability of the fixed points and the type of the bifurcations in each case. [You are not required to prove that the stabilities or bifurcation types are as you indicate.]

(c) For the case $\beta=1$, analyse the bifurcation at $\mu=-1$ using extended centre manifold theory and verify that the evolution equation on the centre manifold matches the behaviour you deduced from the bifurcation diagram in part (b).

(d) For $0<\mu+1 \ll 1$, sketch the phase plane in the immediate neighbourhood of where the bifurcation of part (c) occurs.

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• # Paper 4, Section II, A

(a) A continuous map $F$ of an interval into itself has a periodic orbit of period 3 . Prove that $F$ also has periodic orbits of period $n$ for all positive integers $n$.

(b) What is the minimum number of distinct orbits of $F$ of periods 2,4 and 5 ? Explain your reasoning with a directed graph. [Formal proof is not required.]

(c) Consider the piecewise linear map $F:[0,1] \rightarrow[0,1]$ defined by linear segments between $F(0)=\frac{1}{2}, F\left(\frac{1}{2}\right)=1$ and $F(1)=0$. Calculate the orbits of periods 2,4 and 5 that are obtained from the directed graph in part (b).

[In part (a) you may assume without proof:

(i) If $U$ and $V$ are non-empty closed bounded intervals such that $V \subseteq F(U)$ then there is a closed bounded interval $K \subseteq U$ such that $F(K)=V$.

(ii) The Intermediate Value Theorem.]

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• # Paper 1, Section II, 32E

(i) For the dynamical system

$\dot{x}=-x\left(x^{2}-2 \mu\right)\left(x^{2}-\mu+a\right),$

sketch the bifurcation diagram in the $(\mu, x)$ plane for the three cases $a<0, a=0$ and $a>0$. Describe the bifurcation points that occur in each case.

(ii) For the case when $a<0$ only, confirm the types of bifurcation by finding the system to leading order near each of the bifurcations.

(iii) Explore the structural stability of these bifurcations by adding a small positive constant $\epsilon$ to the right-hand side of $(\uparrow)$ and by sketching the bifurcation diagrams, for the three cases $a<0, a=0$ and $a>0$. Which of the original bifurcations are structurally stable?

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• # Paper 2, Section II, E

(a) State and prove Dulac's criterion. State clearly the Poincaré-Bendixson theorem.

(b) For $(x, y) \in \mathbb{R}^{2}$ and $k>0$, consider the dynamical system

\begin{aligned} &\dot{x}=k x-5 y-(3 x+y)\left(5 x^{2}-6 x y+5 y^{2}\right) \\ &\dot{y}=5 x+(k-6) y-(x+3 y)\left(5 x^{2}-6 x y+5 y^{2}\right) \end{aligned}

(i) Use Dulac's criterion to find a range of $k$ for which this system does not have any periodic orbit.

(ii) Find a suitable $f(k)>0$ such that trajectories enter the disc $x^{2}+y^{2} \leqslant f(k)$ and do not leave it.

(iii) Given that the system has no fixed points apart from the origin for $k<10$, give a range of $k$ for which there will exist at least one periodic orbit.

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• # Paper 3, Section II, E

(a) A dynamical system $\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x})$ has a fixed point at the origin. Define the terms asymptotic stability, Lyapunov function and domain of stability of the fixed point $\mathbf{x}=\mathbf{0}$. State and prove Lyapunov's first theorem and state (without proof) La Salle's invariance principle.

(b) Consider the system

\begin{aligned} \dot{x} &=-2 x+x^{3}+\sin (2 y), \\ \dot{y} &=-x-y^{3} \end{aligned}

(i) Show that trajectories cannot leave the square $S=\{(x, y):|x|<1,|y|<1\}$. Show also that there are no fixed points in $S$ other than the origin. Is this enough to deduce that $S$ is in the domain of stability of the origin?

(ii) Construct a Lyapunov function of the form $V=x^{2} / 2+g(y)$. Deduce that the origin is asymptotically stable.

(iii) Find the largest rectangle of the form $|x| on which $V$ is a strict Lyapunov function. Is this enough to deduce that this region is in the domain of stability of the origin?

(iv) Purely from using the Lyapunov function $V$, what is the most that can be deduced about the domain of stability of the origin?

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• # Paper 4, Section II, E

(a) Let $F: I \rightarrow I$ be a continuous map defined on an interval $I \subset \mathbb{R}$. Define what it means (i) for $F$ to have a horseshoe and (ii) for $F$ to be chaotic. [Glendinning's definition should be used throughout this question.]

(b) Consider the map defined on the interval $[-1,1]$ by

$F(x)=1-\mu|x|$

with $0<\mu \leqslant 2$.

(i) Sketch $F(x)$ and $F^{2}(x)$ for a case when $0<\mu<1$ and a case when $1<\mu<2$.

(ii) Describe fully the long term dynamics for $0<\mu<1$. What happens for $\mu=1$ ?

(iii) When does $F$ have a horseshoe? When does $F^{2}$ have a horseshoe?

(iv) For what values of $\mu$ is the map $F$ chaotic?

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• # Paper 1, Section II, E

For a dynamical system of the form $\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x})$, give the definition of the alpha-limit set $\alpha(\mathbf{x})$ and the omega-limit set $\omega(\mathbf{x})$ of a point $\mathbf{x}$.

Consider the dynamical system

\begin{aligned} &\dot{x}=x^{2}-1, \\ &\dot{y}=k x y, \end{aligned}

where $\mathbf{x}=(x, y) \in \mathbb{R}^{2}$ and $k$ is a real constant. Answer the following for all values of $k$, taking care over boundary cases (both in $k$ and in $\mathbf{x}$ ).

(i) What symmetries does this system have?

(ii) Find and classify the fixed points of this system.

(iii) Does this system have any periodic orbits?

(iv) Give $\alpha(\mathbf{x})$ and $\omega(\mathbf{x})$ (considering all $\mathbf{x} \in \mathbb{R}^{2}$ ).

(v) For $\mathbf{x}_{0}=\left(0, y_{0}\right)$, give the orbit of $\mathbf{x}_{0}$ (considering all $y_{0} \in \mathbb{R}$ ). You should give your answer in the form $y=y\left(x, y_{0}, k\right)$, and specify the range of $x$.

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• # Paper 2, Section II, E

For a map $F: \Lambda \rightarrow \Lambda$ give the definitions of chaos according to (i) Devaney (Dchaos) and (ii) Glendinning (G-chaos).

Consider the dynamical system

$F(x)=a x \quad(\bmod 1)$

on $\Lambda=[0,1)$, for $a>1$ (note that $a$ is not necessarily an integer). For both definitions of chaos, show that this system is chaotic.

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• # Paper 3, Section II, E

Consider a dynamical system of the form

\begin{aligned} &\dot{x}=x(1-y+a x) \\ &\dot{y}=r y(-1+x-b y) \end{aligned}

on $\Lambda=\{(x, y): x>0$ and $y>0\}$, where $a, b$ and $r$ are real constants and $r>0$.

(a) For $a=b=0$, by considering a function of the form $V(x, y)=f(x)+g(y)$, show that all trajectories in $\Lambda$ are either periodic orbits or a fixed point.

(b) Using the same $V$, show that no periodic orbits in $\Lambda$ persist for small $a$ and $b$ if $a b<0$.

[Hint: for $a=b=0$ on the periodic orbits with period $T$, show that $\int_{0}^{T}(1-x) d t=0$ and hence that $\int_{0}^{T} x(1-x) d t=\int_{0}^{T}\left[-(1-x)^{2}+(1-x)\right] d t<0$.]

(c) By considering Dulac's criterion with $\phi=1 /(x y)$, show that there are no periodic orbits in $\Lambda$ if $a b<0$.

(d) Purely by consideration of the existence of fixed points in $\Lambda$ and their Poincaré indices, determine those $(a, b)$ for which the possibility of periodic orbits can be excluded.

(e) Combining the results above, sketch the $a-b$ plane showing where periodic orbits in $\Lambda$ might still be possible.

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• # Paper 4, Section II, E

Consider the dynamical system

\begin{aligned} &\dot{x}=x+y^{2}-a \\ &\dot{y}=y\left(4 x-x^{2}-a\right) \end{aligned}

for $(x, y) \in \mathbb{R}^{2}, a \in \mathbb{R}$.

Find all fixed points of this system. Find the three different values of $a$ at which bifurcations appear. For each such value give the location $(x, y)$ of all bifurcations. For each of these, what types of bifurcation are suggested from this analysis?

Use centre manifold theory to analyse these bifurcations. In particular, for each bifurcation derive an equation for the dynamics on the extended centre manifold and hence classify the bifurcation.

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• # Paper 1, Section II, E

Consider the system

$\dot{x}=-2 a x+2 x y, \quad \dot{y}=1-x^{2}-y^{2}$

where $a$ is a constant.

(a) Find and classify the fixed points of the system. For $a=0$ show that the linear classification of the non-hyperbolic fixed points is nonlinearly correct. For $a \neq 0$ show that there are no periodic orbits. [Standard results for periodic orbits may be quoted without proof.]

(b) Sketch the phase plane for the cases (i) $a=0$, (ii) $a=\frac{1}{2}$, and (iii) $a=\frac{3}{2}$, showing any separatrices clearly.

(c) For what values of a do stationary bifurcations occur? Consider the bifurcation with $a>0$. Let $y_{0}, a_{0}$ be the values of $y, a$ at which the bifurcation occurs, and define $Y=y-y_{0}, \mu=a-a_{0}$. Assuming that $\mu=O\left(x^{2}\right)$, find the extended centre manifold $Y=Y(x, \mu)$ to leading order. Further, determine the evolution equation on the centre manifold to leading order. Hence identify the type of bifurcation.

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• # Paper 2, Section II, 32E

Consider the system

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

where $\alpha$ and $\epsilon$ are real constants, and $0 \leqslant \epsilon \ll 1$. Find and classify the fixed points.

Show that when $\epsilon=0$ the system is Hamiltonian and find $H$. Sketch the phase plane for this case.

Suppose now that $0<\epsilon \ll 1$. Show that the small change in $H$ following a trajectory of the perturbed system around an orbit $H=H_{0}$ of the unperturbed system is given to leading order by an equation of the form

$\Delta H=\epsilon \int_{x_{1}}^{x_{2}} F\left(x ; \alpha, H_{0}\right) d x$

where $F$ should be found explicitly, and where $x_{1}$ and $x_{2}$ are the minimum and maximum values of $x$ on the unperturbed orbit.

Use the energy-balance method to find the value of $\alpha$, correct to leading order in $\epsilon$, for which the system has a homoclinic orbit. [Hint: The substitution $u=1-\frac{1}{2} x^{2}$ may prove useful.]

Over what range of $\alpha$ would you expect there to be periodic solutions that enclose only one of the fixed points?

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• # Paper 3, Section II, 32E

Consider the system

$\dot{x}=y, \quad \dot{y}=\mu_{1} x+\mu_{2} y-(x+y)^{3}$

where $\mu_{1}$ and $\mu_{2}$ are parameters.

By considering a function of the form $V(x, y)=f(x+y)+\frac{1}{2} y^{2}$, show that when $\mu_{1}=\mu_{2}=0$ the origin is globally asymptotically stable. Sketch the phase plane for this case.

Find the fixed points for the general case. Find the values of $\mu_{1}$ and $\mu_{2}$ for which the fixed points have (i) a stationary bifurcation and (ii) oscillatory (Hopf) bifurcations. Sketch these bifurcation values in the $\left(\mu_{1}, \mu_{2}\right)$-plane.

For the case $\mu_{2}=-1$, find the leading-order approximation to the extended centre manifold of the bifurcation as $\mu_{1}$ varies, assuming that $\mu_{1}=O\left(x^{2}\right)$. Find also the evolution equation on the extended centre manifold to leading order. Deduce the type of bifurcation, and sketch the bifurcation diagram in the $\left(\mu_{1}, x\right)$-plane.

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• # Paper 4, Section II, E

Let $F: I \rightarrow I$ be a continuous one-dimensional map of an interval $I \subset \mathbb{R}$. Define what it means (i) for $F$ to have a horseshoe (ii) for $F$ to be chaotic. [Glendinning's definition should be used throughout this question.]

Prove that if $F$ has a 3 -cycle $x_{1} then $F$ is chaotic. [You may assume the intermediate value theorem and any corollaries of it.]

State Sharkovsky's theorem.

Use the above results to deduce that if $F$ has an $N$-cycle, where $N$ is any integer that is not a power of 2 , then $F$ is chaotic.

Explain briefly why if $F$ is chaotic then $F$ has $N$-cycles for many values of $N$ that are not powers of 2. [You may assume that a map with a horseshoe acts on some set $\Lambda$ like the Bernoulli shift map acts on $[0,1)$.]

The logistic map is not chaotic when $\mu<\mu_{\infty} \approx 3.57$ and it has 3 -cycles when $\mu>1+\sqrt{8} \approx 3.84$. What can be deduced from these statements about the values of $\mu$ for which the logistic map has a 10-cycle?

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• # Paper 1, Section II, A

Consider the dynamical system

\begin{aligned} &\dot{x}=-x+x^{3}+\beta x y^{2} \\ &\dot{y}=-y+\beta x^{2} y+y^{3} \end{aligned}

where $\beta>-1$ is a constant.

(a) Find the fixed points of the system, and classify them for $\beta \neq 1$.

Sketch the phase plane for each of the cases (i) $\beta=\frac{1}{2}$ (ii) $\beta=2$ and (iii) $\beta=1$.

(b) Given $\beta>2$, show that the domain of stability of the origin includes the union over $k \in \mathbb{R}$ of the regions

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

By considering $k \gg 1$, or otherwise, show that more information is obtained from the union over $k$ than considering only the case $k=1$.

$\left[\right.$ Hint: If $B>A, C$ then $\left.\max _{u \in[0,1]}\left\{A u^{2}+2 B u(1-u)+C(1-u)^{2}\right\}=\frac{B^{2}-A C}{2 B-A-C} \cdot\right]$

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• # Paper 2, Section II, A

(a) State Liapunov's first theorem and La Salle's invariance principle. Use these results to show that the fixed point at the origin of the system

$\ddot{x}+k \dot{x}+\sin ^{3} x=0, \quad k>0,$

is asymptotically stable.

(b) State the Poincaré-Bendixson theorem. Show that the forced damped pendulum

$\dot{\theta}=p, \quad \dot{p}=-k p-\sin \theta+F, \quad k>0,$

with $F>1$, has a periodic orbit that encircles the cylindrical phase space $(\theta, p) \in \mathbb{R}[\bmod 2 \pi] \times \mathbb{R}$, and that it is unique.

[You may assume that the Poincaré-Bendixson theorem also holds on a cylinder, and comment, without proof, on the use of any other standard results.]

(c) Now consider $(*)$ for $F, k=O(\epsilon)$, where $\epsilon \ll 1$. Use the energy-balance method to show that there is a homoclinic orbit in $p \geqslant 0$ if $F=F_{h}(k)$, where $F_{h} \approx 4 k / \pi>0$.

Explain briefly why there is no homoclinic orbit in $p \leqslant 0$ for $F>0$.

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• # Paper 3, Section II, A

State, without proof, the centre manifold theorem. Show that the fixed point at the origin of the system

\begin{aligned} &\dot{x}=y-x+a x^{3} \\ &\dot{y}=r x-y-y z \\ &\dot{z}=x y-z \end{aligned}

where $a \neq 1$ is a constant, is nonhyperbolic at $r=1$. What are the dimensions of the linear stable and (non-extended) centre subspaces at this point?

Make the substitutions $2 u=x+y, 2 v=x-y$ and $\mu=r-1$ and derive the resultant equations for $\dot{u}, \dot{v}$ and $\dot{z}$.

The extended centre manifold is given by

$v=V(u, \mu), \quad z=Z(u, \mu),$

where $V$ and $Z$ can be expanded as power series about $u=\mu=0$. What is known about $V$ and $Z$ from the centre manifold theorem? Assuming that $\mu=O\left(u^{2}\right)$, determine $Z$ to $O\left(u^{2}\right)$ and $V$ to $O\left(u^{3}\right)$. Hence obtain the evolution equation on the centre manifold correct to $O\left(u^{3}\right)$, and identify the type of bifurcation distinguishing between the cases $a>1$ and $a<1$.

If now $a=1$, assume that $\mu=O\left(u^{4}\right)$ and extend your calculations of $Z$ to $O\left(u^{4}\right)$ and of the dynamics on the centre manifold to $O\left(u^{5}\right)$. Hence sketch the bifurcation diagram in the neighbourhood of $u=\mu=0$.

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• # Paper 4, Section II, A

Consider the one-dimensional map $F: \mathbb{R} \rightarrow \mathbb{R}$ defined by

$x_{i+1}=F\left(x_{i} ; \mu\right)=x_{i}\left(a x_{i}^{2}+b x_{i}+\mu\right),$

where $a$ and $b$ are constants, $\mu$ is a parameter and $a \neq 0$.

(a) Find the fixed points of $F$ and determine the linear stability of $x=0$. Hence show that there are bifurcations at $\mu=1$, at $\mu=-1$ and, if $b \neq 0$, at $\mu=1+b^{2} /(4 a)$.

Sketch the bifurcation diagram for each of the cases:

$\text { (i) } a>b=0, \quad \text { (ii) } a, b>0 \text { and (iii) } a, b<0 \text {. }$

In each case show the locus and stability of the fixed points in the $(\mu, x)$-plane, and state the type of each bifurcation. [Assume that there are no further bifurcations in the region sketched.]

(b) For the case $F(x)=x\left(\mu-x^{2}\right)$ (i.e. $\left.a=-1, b=0\right)$, you may assume that

$F^{2}(x)=x+x\left(\mu-1-x^{2}\right)\left(\mu+1-x^{2}\right)\left(1-\mu x^{2}+x^{4}\right) .$

Show that there are at most three 2-cycles and determine when they exist. By considering $F^{\prime}\left(x_{i}\right) F^{\prime}\left(x_{i+1}\right)$, or otherwise, show further that one 2-cycle is always unstable when it exists and that the others are unstable when $\mu>\sqrt{5}$. Sketch the bifurcation diagram showing the locus and stability of the fixed points and 2-cycles. State briefly what you would expect to occur for $\mu>\sqrt{5}$.

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• # Paper 1, Section II, E

Consider the dynamical system

\begin{aligned} &\dot{x}=x(y-a), \\ &\dot{y}=1-x-y^{2}, \end{aligned}

where $-1. Find and classify the fixed points of the system.

Use Dulac's criterion with a weighting function of the form $\phi=x^{p}$ and a suitable choice of $p$ to show that there are no periodic orbits for $a \neq 0$. For the case $a=0$ use the same weighting function to find a function $V(x, y)$ which is constant on trajectories. [Hint: $\phi \dot{\mathbf{x}}$ is Hamiltonian.]

Calculate the stable manifold at $(0,-1)$ correct to quadratic order in $x$.

Sketch the phase plane for the cases (i) $a=0$ and (ii) $a=\frac{1}{2}$.

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• # Paper 2, Section II, E

Consider the nonlinear oscillator

\begin{aligned} \dot{x} &=y-\mu x\left(\frac{1}{2}|x|-1\right), \\ \dot{y} &=-x \end{aligned}

(a) Use the Hamiltonian for $\mu=0$ to find a constraint on the size of the domain of stability of the origin when $\mu<0$.

(b) Assume that given $\mu>0$ there exists an $R$ such that all trajectories eventually remain within the region $|\mathbf{x}| \leqslant R$. Show that there must be a limit cycle, stating carefully any result that you use. [You need not show that there is only one periodic orbit.]

(c) Use the energy-balance method to find the approximate amplitude of the limit cycle for $0<\mu \ll 1$.

(d) Find the approximate shape of the limit cycle for $\mu \gg 1$, and calculate the leading-order approximation to its period.

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• # Paper 3, Section II, E

Consider the dependence of the system

\begin{aligned} \dot{x} &=\left(a-x^{2}\right)\left(a^{2}-y\right) \\ \dot{y} &=x-y \end{aligned}

on the parameter $a$. Find the fixed points and plot their location in the $(a, x)$-plane. Hence, or otherwise, deduce that there are bifurcations at $a=0$ and $a=1$.

Investigate the bifurcation at $a=1$ by making the substitutions $u=x-1, v=y-1$ and $\mu=a-1$. Find the extended centre manifold in the form $v(u, \mu)$ correct to second order. Find the evolution equation on the extended centre manifold to second order, and determine the stability of its fixed points.

Use a plot to show which branches of fixed points in the $(a, x)$-plane are stable and which are unstable, and state, without calculation, the type of bifurcation at $a=0$. Hence sketch the structure of the $(x, y)$ phase plane very close to the origin for $|a| \ll 1$ in the cases (i) $a<0$ and (ii) $a>0$.

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• # Paper 4, Section II, E

Consider the map defined on $\mathbb{R}$ by

$F(x)= \begin{cases}3 x & x \leqslant \frac{1}{2} \\ 3(1-x) & x \geqslant \frac{1}{2}\end{cases}$

and let $I$ be the open interval $(0,1)$. Explain what it means for $F$ to have a horseshoe on $I$ by identifying the relevant intervals in the definition.

Let $\Lambda=\left\{x: F^{n}(x) \in I, \forall n \geqslant 0\right\}$. Show that $F(\Lambda)=\Lambda$.

Find the sets $\Lambda_{1}=\{x: F(x) \in I\}$ and $\Lambda_{2}=\left\{x: F^{2}(x) \in I\right\}$.

Consider the ternary (base-3) representation $x=0 \cdot x_{1} x_{2} x_{3} \ldots$ of numbers in $I$. Show that

$F\left(0 \cdot x_{1} x_{2} x_{3} \ldots\right)=\left\{\begin{array}{ll} x_{1} \cdot x_{2} x_{3} x_{4} \ldots & x \leqslant \frac{1}{2} \\ \sigma\left(x_{1}\right) \cdot \sigma\left(x_{2}\right) \sigma\left(x_{3}\right) \sigma\left(x_{4}\right) \ldots & x \geqslant \frac{1}{2} \end{array},\right.$

where the function $\sigma\left(x_{i}\right)$ of the ternary digits should be identified. What is the ternary representation of the non-zero fixed point? What do the ternary representations of elements of $\Lambda$ have in common?

Show that $F$ has sensitive dependence on initial conditions on $\Lambda$, that $F$ is topologically transitive on $\Lambda$, and that periodic points are dense in $\Lambda$. [Hint: You may assume that $F^{n}\left(0 \cdot x_{1} \ldots x_{n-1} 0 x_{n+1} x_{n+2} \ldots\right)=0 \cdot x_{n+1} x_{n+2} \ldots$ for $x \in \Lambda$.]

Briefly state the relevance of this example to the relationship between Glendinning's and Devaney's definitions of chaos.

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• # Paper 1, Section II, 28B

(a) What is a Lyapunov function?

Consider the dynamical system for $\mathbf{x}(t)=(x(t), y(t))$ given by

\begin{aligned} &\dot{x}=-x+y+x\left(x^{2}+y^{2}\right) \\ &\dot{y}=-y-2 x+y\left(x^{2}+y^{2}\right) \end{aligned}

Prove that the origin is asymptotically stable (quoting carefully any standard results that you use).

Show that the domain of attraction of the origin includes the region $x^{2}+y^{2} where the maximum possible value of $r_{1}$ is to be determined.

Show also that there is a region $E=\left\{\mathbf{x} \mid x^{2}+y^{2}>r_{2}^{2}\right\}$ such that $\mathbf{x}(0) \in E$ implies that $|\mathbf{x}(t)|$ increases without bound. Explain your reasoning carefully. Find the smallest possible value of $r_{2}$.

(b) Now consider the dynamical system

\begin{aligned} \dot{x} &=x-y-x\left(x^{2}+y^{2}\right) \\ \dot{y} &=y+2 x-y\left(x^{2}+y^{2}\right) \end{aligned}

Prove that this system has a periodic solution (again, quoting carefully any standard results that you use).

Demonstrate that this periodic solution is unique.

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• # Paper 2, Section II, B

(a) An autonomous dynamical system $\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x})$ in $\mathbb{R}^{2}$ has a periodic orbit $\mathbf{x}=\mathbf{X}(t)$ with period $T$. The linearized evolution of a small perturbation $\mathbf{x}=\mathbf{X}(t)+\boldsymbol{\eta}(t)$ is given by $\eta_{i}(t)=\Phi_{i j}(t) \eta_{j}(0)$. Obtain the differential equation and initial condition satisfied by the matrix $\Phi(t)$.

Define the Floquet multipliers of the orbit. Explain why one of the multipliers is always unity and give a brief argument to show that the other is given by

$\exp \left(\int_{0}^{T} \nabla \cdot \mathbf{f}(\mathbf{X}(t)) d t\right)$

(b) Use the energy-balance method for nearly Hamiltonian systems to find leading-order approximations to the two limit cycles of the equation

$\ddot{x}+\epsilon\left(2 \dot{x}^{3}+2 x^{3}-4 x^{4} \dot{x}-\dot{x}\right)+x=0$

where $0<\epsilon \ll 1$.

Determine the stability of each limit cycle, giving reasoning where necessary.

[You may assume that $\int_{0}^{2 \pi} \cos ^{4} \theta d \theta=3 \pi / 4$ and $\int_{0}^{2 \pi} \cos ^{6} \theta d \theta=5 \pi / 8$.]

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• # Paper 3, Section II, B

Consider the dynamical system

\begin{aligned} &\dot{x}=-\mu+x^{2}-y \\ &\dot{y}=y(a-x) \end{aligned}

where $a$ is to be regarded as a fixed real constant and $\mu$ as a real parameter.

Find the fixed points of the system and determine the stability of the system linearized about the fixed points. Hence identify the values of $\mu$ at given $a$ where bifurcations occur.

Describe informally the concepts of centre manifold theory and apply it to analyse the bifurcations that occur in the above system with $a=1$. In particular, for each bifurcation derive an equation for the dynamics on the extended centre manifold and hence classify the bifurcation.

What can you say, without further detailed calculation, about the case $a=0$ ?

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• # Paper 4, Section II, B

Let $f: I \rightarrow I$ be a continuous one-dimensional map of an interval $I \subset \mathbb{R}$. Explain what is meant by the statements (i) that $f$ has a horseshoe and (ii) that $f$ is chaotic (according to Glendinning's definition).

Assume that $f$ has a 3-cycle $\left\{x_{0}, x_{1}, x_{2}\right\}$ with $x_{1}=f\left(x_{0}\right), x_{2}=f\left(x_{1}\right), x_{0}=f\left(x_{2}\right)$ and, without loss of generality, $x_{0}. Prove that $f^{2}$ has a horseshoe. [You may assume the intermediate value theorem.]

Represent the effect of $f$ on the intervals $I_{a}=\left[x_{0}, x_{1}\right]$ and $I_{b}=\left[x_{1}, x_{2}\right]$ by means of a directed graph, explaining carefully how the graph is constructed. Explain what feature of the graph implies the existence of a 3-cycle.

The map $g: I \rightarrow I$ has a 5-cycle $\left\{x_{0}, x_{1}, x_{2}, x_{3}, x_{4}\right\}$ with $x_{i+1}=g\left(x_{i}\right), 0 \leqslant i \leqslant 3$ and $x_{0}=g\left(x_{4}\right)$, and $x_{0}. For which $n, 1 \leqslant n \leqslant 4$, is an $n$-cycle of $g$ guaranteed to exist? Is $g$ guaranteed to be chaotic? Is $g$ guaranteed to have a horseshoe? Justify your answers. [You may use a suitable directed graph as part of your arguments.]

How do your answers to the above change if instead $x_{4} ?

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• # Paper 1, Section I, D

Consider the system

\begin{aligned} &\dot{x}=y+x y \\ &\dot{y}=x-\frac{3}{2} y+x^{2} \end{aligned}

Show that the origin is a hyperbolic fixed point and find the stable and unstable invariant subspaces of the linearised system.

Calculate the stable and unstable manifolds correct to quadratic order, expressing $y$ as a function of $x$ for each.

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• # Paper 2, Section I, D

Consider the system

\begin{aligned} \dot{x} &=-x+y+y^{2} \\ \dot{y} &=\mu-x y \end{aligned}

Show that when $\mu=0$ the fixed point at the origin has a stationary bifurcation.

Find the centre subspace of the extended system linearised about $(x, y, \mu)=(0,0,0)$.

Find an approximation to the centre manifold giving $y$ as a function of $x$ and $\mu$, including terms up to quadratic order.

Hence deduce an expression for $\dot{x}$ on the centre manifold, and identify the type of bifurcation at $\mu=0$.

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• # Paper 3, Section $I$, D

Define the Poincaré index of a closed curve $\mathcal{C}$ for a vector field $\mathbf{f}(\mathbf{x}), \mathbf{x} \in \mathbb{R}^{2}$.

Explain carefully why the index of $\mathcal{C}$ is fully determined by the fixed points of the dynamical system $\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x})$ that lie within $\mathcal{C}$.

What is the Poincaré index for a closed curve $\mathcal{C}$ if it (a) encloses only a saddle point, (b) encloses only a focus and (c) encloses only a node?

What is the Poincaré index for a closed curve $\mathcal{C}$ that is a periodic trajectory of the dynamical system?

A dynamical system in $\mathbb{R}^{2}$ has 2 saddle points, 1 focus and 1 node. What is the maximum number of different periodic orbits? [For the purposes of this question, two orbits are said to be different if they enclose different sets of fixed points.]

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• # Paper 3, Section II, D

Let $f: I \rightarrow I$ be a continuous one-dimensional map of an interval $I \subset \mathbb{R}$. Explain what is meant by saying that $f$ has a horseshoe.

A map $g$ on the interval $[a, b]$ is a tent map if

(i) $g(a)=a$ and $g(b)=a$;

(ii) for some $c$ with $a is linear and increasing on the interval $[a, c]$, linear and decreasing on the interval $[c, b]$, and continuous at $c$.

Consider the tent map defined on the interval $[0,1]$ by

$f(x)= \begin{cases}\mu x & 0 \leqslant x \leqslant \frac{1}{2} \\ \mu(1-x) & \frac{1}{2} \leqslant x \leqslant 1\end{cases}$

with $1<\mu \leqslant 2$. Find the corresponding expressions for $f^{2}(x)=f(f(x))$.

Find the non-zero fixed point $x_{0}$ and the points $x_{-1}<\frac{1}{2} that satisfy

$f^{2}\left(x_{-2}\right)=f\left(x_{-1}\right)=x_{0}=f\left(x_{0}\right) .$

Sketch graphs of $f$ and $f^{2}$ showing the points corresponding to $x_{-2}, x_{-1}$ and $x_{0}$. Indicate the values of $f$ and $f^{2}$ at their maxima and minima and also the gradients of each piece of their graphs.

Identify a subinterval of $[0,1]$ on which $f^{2}$ is a tent map. Hence demonstrate that $f^{2}$ has a horseshoe if $\mu \geqslant 2^{1 / 2}$.

Explain briefly why $f^{4}$ has a horseshoe when $\mu \geqslant 2^{1 / 4}$.

Why are there periodic points of $f$ arbitrarily close to $x_{0}$ for $\mu \geqslant 2^{1 / 2}$, but no such points for $2^{1 / 4} \leqslant \mu<2^{1 / 2}$ ? Explain carefully any results or terms that you use.

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• # Paper 4, Section I, D

Consider the map $x_{n+1}=\lambda x_{n}\left(1-x_{n}^{2}\right)$ for $-1 \leqslant x_{n} \leqslant 1$. What is the maximum value, $\lambda_{\max }$, for which the interval $[-1,1]$ is mapped into itself?

Analyse the first two bifurcations that occur as $\lambda$ increases from 0 towards $\lambda_{\max }$, including an identification of the values of $\lambda$ at which the bifurcation occurs and the type of bifurcation.

What type of bifurcation do you expect as the third bifurcation? Briefly give your reasoning.

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• # Paper 4, Section II, D

A dynamical system $\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x})$ has a fixed point at the origin. Define the terms Lyapunov stability, asymptotic stability and Lyapunov function with respect to this fixed point. State and prove Lyapunov's first theorem and state (without proof) La Salle's invariance principle.

(a) Consider the system

\begin{aligned} &\dot{x}=y \\ &\dot{y}=-y-x^{3}+x^{5} \end{aligned}

Construct a Lyapunov function of the form $V=f(x)+g(y)$. Deduce that the origin is asymptotically stable, explaining your reasoning carefully. Find the greatest value of $y_{0}$ such that use of this Lyapunov function guarantees that the trajectory through $\left(0, y_{0}\right)$ approaches the origin as $t \rightarrow \infty$.

(b) Consider the system

\begin{aligned} &\dot{x}=x+4 y+x^{2}+2 y^{2}, \\ &\dot{y}=-3 x-3 y . \end{aligned}

Show that the origin is asymptotically stable and that the basin of attraction of the origin includes the region $x^{2}+x y+y^{2}<\frac{1}{4}$.

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• # Paper 1, Section I, C

Consider the dynamical system $\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x})$ in $\mathbb{R}^{n}$ which has a hyperbolic fixed point at the origin.

Define the stable and unstable invariant subspaces of the system linearised about the origin. Give a constraint on the dimensions of these two subspaces.

Define the local stable and unstable manifolds of the origin for the system. How are these related to the invariant subspaces of the linearised system?

For the system

\begin{aligned} &\dot{x}=-x+x^{2}+y^{2} \\ &\dot{y}=y+y^{2}-x^{2} \end{aligned}

calculate the stable and unstable manifolds of the origin, each correct up to and including cubic order.

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• # Paper 2, Section I, $7 \mathrm{C}$

Let $\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x})$ be a two-dimensional dynamical system with a fixed point at $\mathbf{x}=\mathbf{0}$. Define a Lyapunov function $V(\mathbf{x})$ and explain what it means for $\mathbf{x}=\mathbf{0}$ to be Lyapunov stable.

For the system

\begin{aligned} &\dot{x}=-x-2 y+x^{3} \\ &\dot{y}=-y+x+\frac{1}{2} y^{3}+x^{2} y \end{aligned}

determine the values of $C$ for which $V=x^{2}+C y^{2}$ is a Lyapunov function in a sufficiently small neighbourhood of the origin.

For the case $C=2$, find $V_{1}$ and $V_{2}$ such that $V(\mathbf{x}) at $t=0$ implies that $V \rightarrow 0$ as $t \rightarrow \infty$ and $V(\mathbf{x})>V_{2}$ at $t=0$ implies that $V \rightarrow \infty$ as $t \rightarrow \infty$

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• # Paper 3, Section I, C

A one-dimensional map is defined by

$x_{n+1}=F\left(x_{n}, \mu\right)$

where $\mu$ is a parameter. What is the condition for a bifurcation of a fixed point $x_{*}$ of $F$ ?

Let $F(x, \mu)=x\left(x^{2}-2 x+\mu\right)$. Find the fixed points and show that bifurcations occur when $\mu=-1, \mu=1$ and $\mu=2$. Sketch the bifurcation diagram, showing the locus and stability of the fixed points in the $(x, \mu)$ plane and indicating the type of each bifurcation.

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• # Paper 3, Section II, C

Let $f: I \rightarrow I$ be a continuous map of an interval $I \subset \mathbb{R}$. Explain what is meant by the statements (a) $f$ has a horseshoe and (b) $f$ is chaotic according to Glendinning's definition of chaos.

Assume that $f$ has a 3-cycle $\left\{x_{0}, x_{1}, x_{2}\right\}$ with $x_{1}=f\left(x_{0}\right), x_{2}=f\left(x_{1}\right), x_{0}=f\left(x_{2}\right)$, $x_{0}. Prove that $f^{2}$ has a horseshoe. [You may assume the Intermediate Value Theorem.]

Represent the effect of $f$ on the intervals $I_{a}=\left[x_{0}, x_{1}\right]$ and $I_{b}=\left[x_{1}, x_{2}\right]$ by means of a directed graph. Explain how the existence of the 3 -cycle corresponds to this graph.

The map $g: I \rightarrow I$ has a 4-cycle $\left\{x_{0}, x_{1}, x_{2}, x_{3}\right\}$ with $x_{1}=g\left(x_{0}\right), x_{2}=g\left(x_{1}\right)$, $x_{3}=g\left(x_{2}\right)$ and $x_{0}=g\left(x_{3}\right)$. If $x_{0} is $g$ necessarily chaotic? [You may use a suitable directed graph as part of your argument.]

How does your answer change if $x_{0} ?

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• # Paper 4, Section I, C

Consider the system

\begin{aligned} &\dot{x}=y+a x+b x^{3} \\ &\dot{y}=-x \end{aligned}

What is the Poincaré index of the single fixed point? If there is a closed orbit, why must it enclose the origin?

By writing $\dot{x}=\partial H / \partial y+g(x)$ and $\dot{y}=-\partial H / \partial x$ for suitable functions $H(x, y)$ and $g(x)$, show that if there is a closed orbit $\mathcal{C}$ then

$\oint_{\mathcal{C}}\left(a x+b x^{3}\right) x d t=0$

Deduce that there is no closed orbit when $a b>0$.

If $a b<0$ and $a$ and $b$ are both $O(\epsilon)$, where $\epsilon$ is a small parameter, then there is a single closed orbit that is to within $O(\epsilon)$ a circle of radius $R$ centred on the origin. Deduce a relation between $a, b$ and $R$.

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• # Paper 4, Section II, C

Consider the dynamical system

\begin{aligned} &\dot{x}=(x+y+a)(x-y+a) \\ &\dot{y}=y-x^{2}-b \end{aligned}

where $a>0$.

Find the fixed points of the dynamical system. Show that for any fixed value of $a$ there exist three values $b_{1}>b_{2} \geqslant b_{3}$ of $b$ where a bifurcation occurs. Show that $b_{2}=b_{3}$ when $a=1 / 2$.

In the remainder of this question set $a=1 / 2$.

(i) Being careful to explain your reasoning, show that the extended centre manifold for the bifurcation at $b=b_{1}$ can be written in the form $X=\alpha Y+\beta \mu+p(Y, \mu)$, where $X$ and $Y$ denote the departures from the values of $x$ and $y$ at the fixed point, $b=b_{1}+\mu, \alpha$ and $\beta$ are suitable constants (to be determined) and $p$ is quadratic to leading order. Derive a suitable approximate form for $p$, and deduce the nature of the bifurcation and the stability of the different branches of the steady state solution near the bifurcation.

(ii) Repeat the calculations of part (i) for the bifurcation at $b=b_{2}$.

(iii) Sketch the $x$ values of the fixed points as functions of $b$, indicating the nature of the bifurcations and where each branch is stable.

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• # Paper 1, Section I, $7 \mathrm{D}$

State the Poincaré-Bendixson theorem.

A model of a chemical process obeys the second-order system

$\dot{x}=1-x(1+a)+x^{2} y, \quad \dot{y}=a x-x^{2} y$

where $a>0$. Show that there is a unique fixed point at $(x, y)=(1, a)$ and that it is unstable if $a>2$. Show that trajectories enter the region bounded by the lines $x=1 / q$, $y=0, y=a q$ and $x+y=1+a q$, provided $q>(1+a)$. Deduce that there is a periodic orbit when $a>2$.

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• # Paper 2, Section I, D

Consider the dynamical system

$\dot{x}=\mu x+x^{3}-a x y, \quad \dot{y}=\mu-x^{2}-y,$

where $a$ is a constant.

(a) Show that there is a bifurcation from the fixed point $(0, \mu)$ at $\mu=0$.

(b) Find the extended centre manifold at leading non-trivial order in $x$. Hence find the type of bifurcation, paying particular attention to the special values $a=1$ and $a=-1$. [Hint. At leading order, the extended centre manifold is of the form $y=\mu+\alpha x^{2}+\beta \mu x^{2}+\gamma x^{4}$, where $\alpha, \beta, \gamma$ are constants to be determined.]

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• # Paper 3, Section I, D

State without proof Lyapunov's first theorem, carefully defining all the terms that you use.

Consider the dynamical system

\begin{aligned} &\dot{x}=-2 x+y-x y+3 y^{2}-x y^{2}+x^{3} \\ &\dot{y}=-2 y-x-y^{2}-3 x y+2 x^{2} y \end{aligned}

By choosing a Lyapunov function $V(x, y)=x^{2}+y^{2}$, prove that the origin is asymptotically stable.

By factorising the expression for $\dot{V}$, or otherwise, show that the basin of attraction of the origin includes the set $V<7 / 4$.

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• # Paper 3, Section II, D

Consider the dynamical system

$\ddot{x}-(a-b x) \dot{x}+x-x^{2}=0, \quad a, b>0 .$

(a) Show that the fixed point at the origin is an unstable node or focus, and that the fixed point at $x=1$ is a saddle point.

(b) By considering the phase plane $(x, \dot{x})$, or otherwise, show graphically that the maximum value of $x$ for any periodic orbit is less than one.

(c) By writing the system in terms of the variables $x$ and $z=\dot{x}-\left(a x-b x^{2} / 2\right)$, or otherwise, show that for any periodic orbit $\mathcal{C}$

$\oint_{\mathcal{C}}\left(x-x^{2}\right)\left(2 a x-b x^{2}\right) d t=0$

Deduce that if $a / b>1 / 2$ there are no periodic orbits.

(d) If $a=b=0$ the system (1) is Hamiltonian and has homoclinic orbit

$X(t)=\frac{1}{2}\left(3 \tanh ^{2}\left(\frac{t}{2}\right)-1\right)$

which approaches $X=1$ as $t \rightarrow \pm \infty$. Now suppose that $a, b$ are very small and that we seek the value of $a / b$ corresponding to a periodic orbit very close to $X(t)$. By using equation (3) in equation (2), find an approximation to the largest value of $a / b$ for a periodic orbit when $a, b$ are very small.

[Hint. You may use the fact that $\left.(1-X)=\frac{3}{2} \operatorname{sech}^{2}\left(\frac{t}{2}\right)=3 \frac{d}{d t}\left(\tanh \left(\frac{t}{2}\right)\right)\right]$

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• # Paper 4, Section I, D

Describe the different types of bifurcation from steady states of a one-dimensional map of the form $x_{n+1}=f\left(x_{n}\right)$, and give examples of simple equations exhibiting each type.

Consider the map $x_{n+1}=\alpha x_{n}^{2}\left(1-x_{n}\right), 0. What is the maximum value of $\alpha$ for which the interval is mapped into itself?

Show that as $\alpha$ increases from zero to its maximum value there is a saddle-node bifurcation and a period-doubling bifurcation, and determine the values of $\alpha$ for which they occur.

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• # Paper 4, Section II, D

What is meant by the statement that a continuous map of an interval $I$ into itself has a horseshoe? State without proof the properties of such a map.

Define the property of chaos of such a map according to Glendinning.

A continuous map $f: I \rightarrow I$ has a periodic orbit of period 5 , in which the elements $x_{j}, j=1, \ldots, 5$ satisfy $x_{j} and the points are visited in the order $x_{1} \rightarrow x_{3} \rightarrow x_{4} \rightarrow x_{2} \rightarrow x_{5} \rightarrow x_{1}$. Show that the map is chaotic. [The Intermediate Value theorem can be used without proof.]

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• # Paper 1, Section I, C

Find the fixed points of the dynamical system (with $\mu \neq 0$ )

\begin{aligned} &\dot{x}=\mu^{2} x-x y \\ &\dot{y}=-y+x^{2} \end{aligned}

and determine their type as a function of $\mu$.

Find the stable and unstable manifolds of the origin correct to order $4 .$

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• # Paper 2, Section I, C

State the Poincaré-Bendixson theorem for two-dimensional dynamical systems.

A dynamical system can be written in polar coordinates $(r, \theta)$ as

\begin{aligned} &\dot{r}=r-r^{3}(1+\alpha \cos \theta) \\ &\dot{\theta}=1-r^{2} \beta \cos \theta \end{aligned}

where $\alpha$ and $\beta$ are constants with $0<\alpha<1$.

Show that trajectories enter the annulus $(1+\alpha)^{-1 / 2}.

Show that if there is a fixed point $\left(r_{0}, \theta_{0}\right)$ inside the annulus then $r_{0}^{2}=(\beta-\alpha) / \beta$ and $\cos \theta_{0}=1 /(\beta-\alpha)$.

Use the Poincaré-Bendixson theorem to derive conditions on $\beta$ that guarantee the existence of a periodic orbit.

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• # Paper 3, Section I, C

For the map $x_{n+1}=\lambda x_{n}\left(1-x_{n}^{2}\right)$, with $\lambda>0$, show the following:

(i) If $\lambda<1$, then the origin is the only fixed point and is stable.

(ii) If $\lambda>1$, then the origin is unstable. There are two further fixed points which are stable for $1<\lambda<2$ and unstable for $\lambda>2$.

(iii) If $\lambda<3 \sqrt{3} / 2$, then $x_{n}$ has the same sign as the starting value $x_{0}$ if $\left|x_{0}\right|<1$.

(iv) If $\lambda<3$, then $\left|x_{n+1}\right|<2 \sqrt{3} / 3$ when $\left|x_{n}\right|<2 \sqrt{3} / 3$. Deduce that iterates starting sufficiently close to the origin remain bounded, though they may change sign.

[Hint: For (iii) and (iv) a graphical representation may be helpful.]

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• # Paper 3, Section II, C

Explain what is meant by a steady-state bifurcation of a fixed point $\mathbf{x}_{0}(\mu)$ of a dynamical system $\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x}, \mu)$ in $\mathbb{R}^{n}$, where $\mu$ is a real parameter.

Consider the system in $x \geqslant 0, y \geqslant 0$, with $\mu>0$,

\begin{aligned} \dot{x} &=x\left(1-y^{2}-x^{2}\right), \\ \dot{y} &=y\left(\mu-y-x^{2}\right) . \end{aligned}