Phase Plane Analysis

Consider the nonlinear system in \(\mathbb {R}^3\): \[ \begin {cases} \dot {x}=y-x^2,\\ \dot {y}=-z+xy,\\ \dot {z}=-\lambda z+x^2, \end {cases} \]

where \(\lambda > 0\) is a parameter.

1. Find all critical points of the system and classify them when \(\lambda > 0\).
2. Analyze the stability of the origin \((0,0,0)\) using the center manifold theorem. Compute the reduced system on the center manifold up to quadratic terms.
3. Discuss how the stability of the origin changes as \(\lambda \to 0^+\). What happens if \(\lambda =0\)?

Consider the ODE system

1. Define Lyapunov stability. Are the stationary points at \((\pm 1,0)\) Lyapunov stable? Justify.
2. Find all fixed points, classify their stability (with justification), and make an accurate drawing of the phase plane including any heteroclinic / homoclinic orbits.
3. Define the notion of asymptotically stable stationary point. Are any of the stationary points asymptotically stable?

[25] Consider the system

Determine the stability at the origin by constructing an approximate center manifold of the form \(y=ax^2+bx^3+\cdots \) and analyzing the flow on the approximate center manifold. Sketch the phase portrait showing the center and stable manifolds.

Determine whether the given equilibrium point for the given system is Liapunov stable, is attractive, is asymptotically stable, or is not stable. Give a brief explanation.

1. \((x,y,z)=(0,0,0)\) for \[ \begin {cases} \dot {x}=y-x,\\ \dot {y}=-y-xz,\\ \dot {z}=xy-z. \end {cases} \]
2. \((x,y)=(0,0)\) for \[ \begin {cases} \dot {x}=-\dfrac {\partial V}{\partial x}(x,y),\\ \dot {y}=-\dfrac {\partial V}{\partial y}(x,y), \end {cases} \] where \(V(x,y)=x^4-x^2+y^2\).
3. The zero solution \(x(t)=0\) for \(\ddot {x}+\dot {x}+g(x)=0\) where \(g\in C^1(\mathbb {R})\) and \(xg(x)>0\) for all \(x\ne 0\).

Find the approximate center manifold at the origin. Use it to determine the stability type of the origin in the system

Consider the differential equation \(\ddot {x}+\eta \dot {x}+x^3-x=0\), with \(0<\eta <2\sqrt {2}\) a constant.

1. Sketch a phase portrait for this equation. Identify all fixed points and their type.
2. Find a first integral of the motion if \(\eta = 0\). Make a sketch of these integral curves in the \(x-\dot {x}\) plane.
3. Suppose that at time \(t=0\), \(\frac {\dot {x}^2}{2}+\frac {1}{4}x^4-\frac {1}{2}x^2<0\), and \(0<x<\sqrt {2}\). Determine \(\lim _{t\to \infty }x(t)\), and explain how to prove this.

Consider the differential operator \(Lu=-u''\) with boundary conditions \(u'(0)=u'(1)=0\).

1. What Hilbert space and inner product is used for the analysis of this operator?
2. Is this operator symmetric? Verify your answer.
3. Is this operator invertible? Why or why not?
4. What are the eigenvalues and eigenfunctions for this operator?
5. Are these eigenfunctions complete on the domain of the operator? If yes, how do you know (what theorem can you invoke?) and if no, produce a function in the domain of L that is orthogonal to all of the eigenfunctions.
6. Is \(f(x)=\cos ^2(\pi x)\) in the domain of the operator? Find a representation of \(f(x)\) in terms of the eigenfunctions of \(L\), if it exists?
7. Is \(f(x)=\cos ^2(\pi x)\) in the range of \(L\)? Why or why not?

Find the stable/center/unstable manifolds of the system

The simple pendulum consists of a point particle of mass \(m\) suspended from a fixed point by a massless rod of length \(L\), which is allowed to swing in a vertical plane. If friction is ignored then the equation of motion is \[ \ddot {x}+\omega ^2\sin x=0,\quad \omega ^2=\frac {g}{L}, \] where \(x\) is the angle of inclination of the rod with respect to the downward vertical and \(g\) is the gravitational constant.

1. Using conservation of energy, show that the angular velocity of the pendulum satisfies \[ \dot {x}=\pm \sqrt {2}(C+\omega ^2\cos x)^{1/2}, \] where \(C\) is an arbitrary constant. Express \(C\) in terms of the total energy of the system.
2. Sketch the phase diagram of the pendulum equation in the \((x, \dot {x})-plane\). Illustrate the one-parameter family of curves given by part (a) for different values of \(C\). Take \(-3\pi \le x \le 3\pi \). Indicate the fixed points of the system and the separatrices - curves linking the fixed points. Give a physical interpretation of the underlying trajectories in the two distinct dynamical regimes \(|C|<\omega ^2\) and \(|C|>\omega ^2\).
3. Show that in the regime \(|C|<\omega ^2\), the period of oscillations is \[ T=4\sqrt {\frac {L}{g}}\,K(\sin (x_0/2)), \] where \(\dot {x}=0\) when \(x=x_0\) and \(K\) is the complete elliptic integral of the first kind, which is defined by \[ K(\alpha )=\int _0^{\pi /2}\frac {1}{\sqrt {1-\alpha ^2\sin ^2 u}}\,du. \] HINT: Derive an integral expression for \(T\) and then perform a change of variables \[ \sin u=\frac {\sin (x/2)}{\sin (x_0/2)}. \]
4. For small amplitude oscillations, the pendulum equation can be approximated by the linear equation \[ \ddot {x}+\omega ^2x=0. \] Solve this equation for the initial conditions \(x(0)=A\), \(\dot {x}(0)=0\) and sketch the phase-plane for different values of \(A\). Compare with the phase-plane for the full nonlinear equation in part (b).

Suppose that \(u(x)\in C^2(D)\) is a harmonic function on the smooth bounded domain \(D\subset \mathbb {R}^n\) so that \(\nabla ^2u=0\) in \(D\).

1. prove the mean value property: if \(x\in D\) and \(r>0\) is chosen so that the ball of radius \(r\) and center \(x\) satisfies \(B_r(x)\subset D\) then \[ u(x)=\frac {1}{\omega _nr^{n-1}}\int _{\partial B_r(x)}u(y)\dd y \] where \(\omega _n\) is the \(n-1\) dimensional area of \(\partial B_1\). Hence show that \[ u(x)=\frac {n}{\omega _nr^n}\int _{B_r(x)}u(z)\dd z. \]
2. Assuming that \(D\) is connected, prove that \(u\) can attain its maximum value at an interior point \(x\in D\) only if \(u\) is constant.

Consider Rayleigh’s equation \[ \ddot {x}+\dot {x}^3-2\mu \dot {x}+x=0. \] What kind of bifurcations occur to the equilibrium point of system as the parameter \(\mu \) is varied? Determine the bifurcation values and describe how the solutions change near the bifurcation point.

Suppose that \(u(x)\) is a \(C^2\) harmonic function in the domain \(\Omega \subset \mathbb {R}^n\), so \(\Delta u=0\) in \(\Omega \).

1. Prove the mean value property: if \(x\in \Omega \) and \(r>0\) is chosen such that \(B_r(x)\subset \Omega \) (ball of radius \(r\) centered at \(x\)) then \[ u(x)=\frac {1}{\omega _nr^{n-1}}\int _{\partial B_r(x)}u(s)\dd s, \] where \(\omega _n\) is the measure of \(\partial B_1\). Hence show that \[ u(x)\le \frac {n}{\omega _nr^n}\int _{B_r(x)}u(y)\dd y. \]
2. Assuming \(\Omega \) is connected, prove that \(u\) can attain its maximum value at an interior point \(x\in \Omega \), only if \(u\) is constant.

The displacement \(x\) of a spring-mounted mass under the action of dry friction is assumed to satisfy \[ m\ddot {x}+kx=-F_0\sgn (\dot {x}) \] An example would be a mass \(m\) connected to a fixed support by a spring of stiffness \(k\) and resting on a surface with frictional force \(F_0\), \(F_0>0\). Set \(m=k=1\) for convenience and let \(y=\dot {x}\).

1. Consider the initial conditions \(x=x_0>0\), \(\dot {x}=0\) at \(t=0\). Show that the phase path will spiral exactly \(n\) times before reaching equilibrium if \[ (4n-1)F_0<x_0<(4n+1)F_0. \]
2. Suppose the initial conditions at \(t=0\) are \(x=x_0>3F_0\) and \(\dot {x}=0\). Subsequently, whenever \(x=-\alpha \), where \(2F_0-x_0<-\alpha <0\) and \(\dot {x}>0\), a trigger operates to increase suddenly the forward velocity so that the kinetic energy increases by a constant amount \(E\). Show that if \(E>8F_0^2\) then a periodic motion is approached, and show that the largest value of \(x\) in the periodic motion is equal to \(F_0+E/(4F_0)\).

The simple pendulum consists of a point particle of mass \(m\) suspended from a fixed point by a massless rod of length \(L\), which is allowed to swing in a vertical plane. If friction is ignored then the equation of motion is \[ \ddot {x}+\omega ^2\sin x=0,\qquad \omega ^2=\frac {g}{L}, \] where \(x\) is the angle of inclination of the rod with respect to the downward vertical and \(g\) is the gravitational constant.

1. Using conservation of energy, show that the angular velocity of the pendulum satisfies \[ \dot {x}=\pm \sqrt {2}(C+\omega ^2\cos x)^{1/2}, \] where \(C\) is an arbitrary constant. Express \(C\) in terms of the total energy of the system.
2. Sketch the phase diagram of the pendulum equation in the \((x,\dot {x})\)-plane. Illustrate the one-parameter family of curves given by part (a) for different values of \(C\). Take \(-3\pi \le x\le 3\pi \). Indicate the fixed points of the system and the separatrices - curves linking the fixed points. Give a physical interpretation of the underlying trajectories in the two distinct dynamical regimes \(\abs {C}<\omega ^2\) and \(\abs {C}>\omega ^2\).
3. Show that in the regime \(\abs {C}<\omega ^2\), the period of oscillations is \[ T=4\sqrt {\frac {L}{g}}K(\sin x_0/2), \] where \(\dot {x}=0\) when \(x=x_0\) and \(K\) is the complete elliptic integral of the first kind, which is defined by \[ K(\alpha )=\int _0^{\pi /2}\frac {1}{\sqrt {1-\alpha ^2\sin ^2u}}d u. \]
4. For small amplitude oscillations, the pendulum equation can be approximated by the linear equation \[ \ddot {x}+\omega ^2x=0. \] Solve this equation for the initial conditions \(x(0)=A\), \(\dot {x}(0)=0\) and sketch the phase-plane for different values of \(A\). Compare with the phase-plane for the full nonlinear equation in part (b).

Consider a linear chain of \(2N\) atoms consisting of two different masses \(m\), \(M\) \(M>m\), placed alternately. The atoms are equally spaced with lattice spacing \(a\) with nearest neighbor interactions represented by Hookean springs with spring constant \(\beta \). Label the light atoms by even integers \(2n\), \(n=0,\ldots ,N-1\) and the heavy atoms by odd integers \(2n-1\), \(n=1,\ldots ,N\). Denoting their displacements from equilibrium by the variables \(U_{2n}\) and \(V_{2n-1}\) respectively, Newton’s law of motion gives \[ \begin {aligned} m\ddot {U}_{2n} &= \beta [V_{2n-1}+V_{2n+1}-2U_{2n}],\\ M\ddot {V}_{2n-1} &= \beta [U_{2n}+U_{2n-2}-2V_{2n-1}] \end {aligned} \] Assume periodic boundary conditions \(U_0=U_{2N}\) and \(V_1=V_{2N+1}\).

1. Sketch the configuration of atoms and briefly explain how the dynamical equations arise from Newton’s law of motion.
2. Assuming a solution of the form \[ U_{2n}=\Phi e^{2inka}e^{-iwt},\qquad V_{2n+1}=\Psi e^{i(2n+1)ka}e^{-iwt}, \] derive an eigenvalue equation for the amplitudes \((\Phi ,\Psi )\) and determine the eigenvalues.
3. Using part (b), show that there are two branches of solution and determine the speed \(w/k\) on the two branches for small \(k\).

Suppose that \(u(x)\) is a \(C^2\) harmonic function in the domain \(\Omega \subset \mathbb {R}^n\), so \(\Delta u=0\) in \(\Omega \).

1. Prove the mean value property: if \(x\in \Omega \) and \(r>0\) is chosen such that \(B_r(x)\subset \Omega \) (ball of radius \(r\) centered at \(x\)) then \[ u(x)=\frac {1}{\omega _n r^{n-1}}\int _{\partial B_r(x)}u(s)\dd s, \] where \(\omega _n\) is the measure of \(\partial B_1\). Hence show that \[ u(x)\le \frac {n}{\omega _n r^n}\int _{B_r(x)}u(y)\dd y. \]
2. Assuming \(\Omega \) is connected, prove that \(u\) can attain its maximum value at an interior point \(x\in \Omega \), only if \(u\) is constant.

Consider the equation \[ \frac {\partial u}{\partial t} +\frac {\partial }{\partial x}\left (\frac {u^2}{2}\right ) =\epsilon \frac {\partial ^2u}{\partial x^2}, \] for \(-\infty <x<\infty \) and \(t>0\).

1. Look for a traveling wave solution \(u(x,t)=U(z)\), \(z=(x-Vt)/\epsilon \), with velocity \(V>0\) and \(U(z)\to U_\pm \) as \(z\to \pm \infty \). Solve the resulting ODE for \(U(z)\) and deduce that \[ V=\frac {[U^2/2]_{-\infty }^{\infty }}{[U]_{-\infty }^{\infty }} \]
2. Discuss how the traveling wave solution relates to shock solutions of the quasilinear equation obtained by setting \(\epsilon =0\).
3. Using phase-plane analysis, show that the wave profile \(U\) is monotonically decreasing, that is, \(dU/dz<0\).

Suppose that \(u(x)\) is a \(C^2\) harmonic function in the domain \(\Omega \subset \mathbb {R}^n\), so \(\Delta u=0\) in \(\Omega \).

1. Prove the mean value property: if \(x\in \Omega \) and \(r>0\) is chosen such that \(B_r(x)\subset \Omega \) (ball of radius \(r\) centered at \(x\)) then \[ u(x)=\frac {1}{\omega _n r^{n-1}}\int _{\partial B_r(x)}u(s)\dd s, \] where \(\omega _n\) is the measure of \(\partial B_1\). Hence show that \[ u(x)\le \frac {n}{\omega _n r^n}\int _{B_r(x)}u(y)\dd y. \]
2. Assuming \(\Omega \) is connected, prove that \(u\) can attain its maximum value at an interior point \(x\in \Omega \), only if \(u\) is constant.

Consider the system \[ \begin {aligned} \dot {x} &= \mu +x^2+y^2,\\ \dot {y} &= -y+x^2 \end {aligned} \]

1. State the Center Manifold Theorem for rest points. Briefly explain its importance in bifurcation theory.
2. Using an approximate center manifold, determine the nature of the bifurcation at the origin at \(\mu =0\).

Suppose that \(u(\mathbf {x})\) is a \(C^2\) harmonic function in the domain \(\Omega \subset \mathbb {R}^n\), so \(\Delta u=0\) in \(\Omega \).

1. Prove the mean value property: if \(\mathbf {x}\in \Omega \) and \(r>0\) is chosen such that \(B_r(\mathbf {x})\subset \Omega \) (ball of radius \(r\) centered at \(\mathbf {x}\)) then \[ u(\mathbf {x})=\frac {1}{\omega _n r^{n-1}}\int _{\partial B_r(\mathbf {x})}u(s)\dd s, \] where \(\omega _n\) is the measure of \(\partial B_1\). Hence show that \[ u(\mathbf {x})\le \frac {n}{\omega _n r^n}\int _{B_r(\mathbf {x})}u(\mathbf {y})\dd y. \]
2. Assuming \(\Omega \) is connected, prove that \(u\) can attain its maximum value at an interior point \(x\in \Omega \), only if \(u\) is constant.

1. State the Center Manifold Theorem for rest points. Briefly explain its importance in bifurcation theory.
2. Using an approximate center manifold, determine the nature of the bifurcation from \((x,\dot {x},\epsilon )=(0,0,0)\) for the quadratic Duffing’s Equation \[ \ddot {x}+\dot {x}+\epsilon x+x^2=0. \]

Suppose that \(u(\mathbf {x})\) is a \(C^2\) harmonic function in the domain \(\Omega \subset \mathbb {R}^n\), so \(\Delta u=0\) in \(\Omega \).

1. Prove the mean value property: if \(\mathbf {x}\in \Omega \) and \(r>0\) is chosen such that \(B_r(\mathbf {x})\subset \Omega \) (ball of radius \(r\) centered at \(\mathbf {x}\)) then \[ u(\mathbf {x})=\frac {1}{\omega _n r^{n-1}}\int _{\partial B_r(\mathbf {x})}u(s)\dd s, \] where \(\omega _n\) is the measure of \(\partial B_1\). Hence show that \[ u(\mathbf {x})\le \frac {n}{\omega _n r^n}\int _{B_r(\mathbf {x})}u(\mathbf {y})\dd y. \]
2. Assuming \(\Omega \) is connected, prove that \(u\) can attain its maximum value at an interior point \(x\in \Omega \), only if \(u\) is constant.

1. State the Center Manifold Theorem for rest points. Briefly explain its importance in bifurcation theory.
2. Using an approximate center manifold, determine the nature of the bifurcation at the origin when \(\epsilon =0\). \[ \begin {aligned} \dot {x} &= \epsilon x-x^3+y^2,\\ \dot {y} &= -y+x^2. \end {aligned} \]

Suppose that \(u(x_1,x_2)\) is a \(C^2\) harmonic function in the domain \(\Omega \subset \mathbb {R}^2\), so \(\Delta u=0\) in \(\Omega \).

1. Prove the mean value property: if \(y=(y_1,y_2)\in \Omega \) and \(r>0\) is chosen such that \(B_r(y)\subset \Omega \) (ball of radius \(r\) centered at \(y\)) then \[ u(y_1,y_2)=\frac {1}{2\pi }\int _0^{2\pi }u(y_1+r\cos \theta ,y_2+r\sin \theta )\dd \theta . \]
2. Assuming \(\Omega \) is connected, prove that \(u\) can attain its maximum value at an interior point \(x\in \Omega \), only if \(u\) is constant.

1. State the Center Manifold Theorem for rest points. Briefly explain its importance in bifurcation theory.
2. Construct an approximation to the center manifold at the origin and use it to determine the local behavior of solutions. \[ \begin {aligned} \dot {x} &= -xy,\\ \dot {y} &= -y+x^2-2y^2. \end {aligned} \]

1. State the Center Manifold Theorem for rest points. Briefly explain its importance in bifurcation theory.
2. Construct an approximation to the center manifold at the origin and use it to determine the local behavior of solutions. \[ \begin {aligned} \dot {x} &= -xy,\\ \dot {y} &= -y+x^2-2y^2. \end {aligned} \]

1. State the Center Manifold Theorem for rest points. Briefly explain its importance in bifurcation theory.
2. Construct an approximation to the center manifold at the origin and use it to determine the local behavior of solutions. \[ \begin {aligned} \dot {x} &= -y-xz,\\ \dot {y} &= x-y^3,\\ \dot {z} &= -z-2xy-2x^4+x^2. \end {aligned} \]

1. State the Center Manifold Theorem for rest points. Briefly explain its importance in bifurcation theory.
2. Construct an approximation to the center manifold at the origin and use it to determine the local behavior of solutions. \[ \begin {aligned} \dot {x} &= xy+x^3,\\ \dot {y} &= -y-x^2y. \end {aligned} \]

Suppose that \(u(x_1,x_2)\) is a \(C^2\) harmonic function in the domain \(\Omega \subset \mathbb {R}^2\), so \(\Delta u=0\) in \(\Omega \).

1. Prove the mean value property: if \(y=(y_1,y_2)\in \Omega \) and \(r>0\) is chosen such that \(B_r(y)\subset \Omega \) (ball of radius \(r\) centered at \(y\)) then \[ u(y_1,y_2)=\frac {1}{2\pi }\int _0^{2\pi }u(y_1+r\cos \theta ,y_2+r\sin \theta )\dd \theta . \]
2. Assuming \(\Omega \) is connected, prove that \(u\) can attain its maximum value at an interior point \(x\in \Omega \), only if \(u\) is constant.

Suppose that \(u(x_1,x_2)\) is a \(C^2\) harmonic function in the domain \(\Omega \subset \mathbb {R}^2\), so \(\Delta u=0\) in \(\Omega \).

1. Prove the mean value property: if \(y=(y_1,y_2)\in \Omega \) and \(r>0\) is chosen such that \(B_r(y)\) (the ball of radius \(r\), centered at \(y\)) satisfies \(B_r(y)\subset \Omega \), then \[ u(y_1,y_2)=\frac {1}{2\pi }\int _0^{2\pi } u(y_1+r\cos \theta ,y_2+r\sin \theta )\dd \theta . \]
2. Assuming \(\Omega \) is connected, prove that \(u\) can attain a maximum value at an interior point \(x\in \Omega \), only if \(u\) is constant.

The displacement \(x\) of a spring–mounted mass under the action of dry friction is assumed to satisfy \[ \ddot {x}+x=F_0\operatorname {sgn}(v_0-\dot {x}) \]

1. Calculate the phase paths in the \((x,y)\) plane and draw the phase diagram. Deduce that the system ultimately converges into a limit cycle oscillation. What happens when \(v_0=0\)?
2. Suppose \(v_0=0\) and the initial conditons at \(t=0\) are \(x=x_0>3F_0\) and \(\dot {x}=0\). Subsequently, whenever \(x=-\alpha \), where \(2F_0-x_0<-\alpha <0\) and \(\dot {x}>0\), a trigger operates to increase suddenly the forward velocity so that the kinetic energy increases by a constant amount \(E\). Show that if \(E>8F_0^2\) then a periodic motion is approached, and show that the largest value of \(x\) in the periodic motion is equal to \(F_0+E/(4F_0)\).

State the center manifold theorem and explain its importance in bifurcation theory. Consider the system \[ \dot {x}=-xy,\quad \dot {y}=-y-x^2. \] Construct an approximation to the center manifold at the origin and hence determine the local behavior of solutions.

The response of a certain biological oscillator, \((x,y)\), \(x\ge 0\), \(y\ge 0\), to a stimulus of size \(b\) satisfies the differential equation \[ \begin {aligned} \dot {x}&=x-ay+b,\quad &\dot {y}&=x-cy && \text {for } x\ge 0,\ y\ge 0;\\ &&\dot {y}&=-cy && \text {for } x=0, \end {aligned} \] with \(a,b,c>0\).

1. Using phase-plane analysis show that there exists a limit cycle, part of which lies on the \(y\)-axis, when \(c<1\) and \(4a>(1+c)^2\). [Hint: existence of limit cycle depends on existence of an unstable spiral].
2. Show that the period of the orbit is independent of \(b\).

State the center manifold theorem and briefly explain its importance in bifurcation theory. Consider the system \[ \dot {x}=-2x+y-x^2,\quad \dot {y}=x(y-x). \] Construct an approximation to the center manifold at the origin and hence determine the local behavior of solutions. [Hint: first perform a change of variables in order to diagonalize the linear part of the system].

Problem. Consider the nonlinear system \[ \begin {aligned} x' &= 2\sin y,\\ y' &= 3\sin x-\sin y. \end {aligned} \] Provide a complete phase plane analysis of this system and give justifications for your reasoning.

Two Species Competition Model. Consider the simplified competition model for two populations \(x(t)\ge 0\) and \(y(t)\ge 0\), \[ \begin {cases} \dot {x}=x(1-x-3y),\\ \dot {y}=3y(1-y-2x). \end {cases} \] Its equilibrium points are \((0,0)\), \((0,1)\), \((1,0)\) and \((.4,.2)\). The equilibria \((0,1)\) and \((1,0)\) are stable, but \((0,0)\) is unstable. This system has a seperatrix, a curve which divides the first quadrant of the plane into two basins of attraction corresponding to the two stable equilibria. Thus one species or the other wins out, depending which has the starting advantage. As a first step in showing that \((.4,.2)\) lies on the seperatrix, prove that in an open neighborhood \(O\) of \((.4,.2)\) there is a \(C^1\) curve \(\sigma \) through \((.4,.2)\) with the property that trajectories starting on one side of the \(\sigma \) in \(O\), stay on that side of \(\sigma \) while in \(O\).

Consider the system in the plane \[ \begin {aligned} u'&=-v+uw,\\ v'&=u+vw, \end {aligned} \tag {5} \] where \[ r^2=u^2+v^2,\qquad w=r^4-6r^2+8. \] Show that (5) has precisely two nontrivial periodic orbits and provide a “complete” phase plane analysis of the system.

Consider \[ \begin {aligned} \frac {dx}{dt} &= Ax+f(x),\\ x(0) &= x_0 \end {aligned} \tag {2.1} \] where \(x\in \mathbb {R}^n\) and \(A\) is a constant \(n\times n\) matrix. Suppose that \(A\) is semisimple, that \(f\in C^2\) and that \(f(0)=0\), \(D_xf(0)=0\).

1. Define what is meant by the stable manifold, the center manifold and the unstable manifold at the origin for (2.1).
2. Give a representation of the solution of (2.1) when \(f=0\), in terms of the eigenvalues of \(A\) and the associated projections and nilpotents. Indicate how to compute the projections.
3. Discuss the asymptotic behavior of the solution in (b) for \(t\to \infty \). Characterize the manifolds defined in (a) as precisely as possible.
4. Give a representation of the solution of (2.1) for \(f\ne 0\). Under what conditions, if any, on \(A\) and the initial data can one conclude that this solution exists for all \(t\in \mathbb {R}^+\)? You need not prove all the assertions for the last part, but should at least state your reasons.
5. State the Hartman-Grobman theorem for \[ x'=F(x) \] and state clearly what further conditions, if any, are needed on (2.1) to apply the Hartman-Grobman theorem to (2.1).

Write a short essay discussing the basic concepts of autonomous dynamical systems. Describe flows, trajectories, equilibrium points, stability, asymptotic stability and local linearizations, etc.