Stability of Linear Systems

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 nonlinear system on \(\mathbb {R}^2\): \[ \begin {cases} \dot {x}=-y+x(1-x^2-y^2),\\ \dot {y}=x+y(1-x^2-y^2). \end {cases} \]

1. Show that the unit circle \(S^1=\{(x,y):x^2+y^2=1\}\) is an invariant set for the flow.
2. Prove that every trajectory starting near \(S^1\) spirals toward it as \(t\to \infty \).
3. Define a Poincaré section \(\Sigma =\{(x,y):y=0,\ x>0\}\) and compute the Poincaré map \(P:\Sigma \to \Sigma \) for points near the unit circle.
4. Linearize the Poincaré map near the fixed point corresponding to the periodic orbit on \(S^1\), and determine its stability.

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?

Consider the system

1. Plot the sets \(\{\dot {x}=0\}\) and \(\{\dot {y}=0\}\) in the \((x,y)\)-plane, find the fixed points of the system and determine their stability.
2. Show that the annular region \(A=\{1<|(x,y)|<\sqrt {2}\}\) is forward invariant.
3. State the Poincaré-Bendixson theorem.
4. Show that the system has at least one non-constant periodic orbit.

[25] Let \(B(t)\) be a continuous real \(d\times d\) matrix function defined for \(t\in [0,\infty )\) such that \[ \int _0^\infty \|B(s)\|\,ds<\infty . \] Let \(A\) be a real \(d\times d\) matrix such that \(\operatorname {Re}\lambda <0\) for all eigenvalues of \(A\). Show that the initial value problem

(1)

has a bounded solution on \([0,\infty )\). [Provide the complete proof. Do not just quote a theorem. Hint: Use the variation of parameters formula and Gronwall’s Inequality. Gronwall’s Inequality says that if \(c_0\ge 0\) and if \(f(t)\), \(\phi (t)\) are nonnegative continuous functions on the interval \([0,T]\) which satisfy \[ f(t)\le c_0+\int _0^t \phi (s)f(s)\,ds \] for all \(t\in [0,T]\), then \[ f(t)\le c_0\exp \left (\int _0^t \phi (s)\,ds\right ) \] for all \(t\in [0,T]\).]

Consider Griffith’s model for a genetic control system, where \(x\) and \(y\) are nonnegative concentrations of a protein and messenger RNA, and \(a\) and \(\mu \) are positive constants.

\[ \dot {y}=a+\frac {x^2}{1+x^2}-y,\quad y(0)=y_0. \tag {2} \]

1. Show that the positive quadrant is positively invariant. Show that there is a rectangle in the positive quadrant that is positively invariant.
2. Show graphically that there is a region of the \(a-\mu \) parameter space for which there is a unique stable steady state solution, and a region of parameter space in which there are three steady state solutions. Identify the stability of the solutions. Give some identifying features of these regions.
3. What is the bifurcation structure of the solutions as a function of \(\mu \)?

[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.

Let \(A\) be a real \(d\times d\) matrix. Consider the initial value problem

(3)

where \(h(t)\in \mathbb {R}^d\) is a real \(T\)-periodic continuous function.

1. Define the Poincare Map in this context and give an explicit formula for the Poincare Map in terms of \(A\), \(h\) and \(x_0\).
2. Under what conditions on \(A\) and \(h\) is there is a unique \(T\)-periodic solution of (3)?
3. Compute the derivative of the Poincare map at the periodic solution. What stability property do you conclude for the periodic solution?

Let \(B(t)\) be a \(C^1\) real \(d\times d\) matrix function defined for \(t\in [0,\infty )\) such that

Let \(A\) be a real \(d\times d\) matrix such that \(\operatorname {Re}\lambda <0\) for all eigenvalues of \(A\). Show that there is \(\delta >0\) such that if \(|x_0|\le \delta \), then the initial value problem

(2)

has a solution such that \(x(t)\to 0\) as \(t\to \infty \). [Provide the complete proof. Do not just quote a theorem.]

[Hint: Use the variation of parameters formula and Gronwall’s Inequality, which says that for \(f(t)\), \(\phi (t)\) nonnegative continuous functions on the interval \([0,T]\) and for \(c_0\ge 0\), if \[ f(t)\le c_0+\int _0^t \phi (s)f(s)\,ds \] for all \(t\in [0,T]\), then \[ f(t)\le c_0\exp \left (\int _0^t \phi (s)\,ds\right ) \] for all \(t\in [0,T]\). You might first show that there is a constant \(k\) such that if \(\epsilon \) is small enough and \(|x_0|<\epsilon \) then the solution remains bounded \(|x(t)|\le k|x_0|\) for all \(t\in [0,T]\).]

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

Let \(A\) be a real \(d\times d\) matrix such that \(\operatorname {Re}\lambda <0\) for all eigenvalues of \(A\). Consider the system

where \(h(t)\in \mathbb {R}^d\) is a \(T\)-periodic continuous function.

1. Define the Poincare Map in this context and give an explicit formula for the Poincare Map in terms of \(A\) and \(h\)
2. Show that there is a unique \(T\)-periodic solution of (3). [Hint: there is again an explicit equation in terms of \(A\) and \(h\).]
3. What stability property does the periodic orbit have? Provide a justification of your answer with explicit calculation/proof.
4. How could the requirements stipulated on the eigenvalues of \(A\) be generalized to still maintain the existence and uniqueness results for the periodic orbit from (b)?

For a system of equations \(\frac {dx}{dt}=A(t)x\), where \(x\in \mathbb {R}^n\) and \(A(t)\) is a real \(n\times n\) matrix function which is smooth in \(t\),

1. What is Abel’s formula? (State without proof).
2. Suppose \(A(t)\) is periodic, \(A(t+T)=A(t)\), \(T>0\). Define the monodromy matrix \(M\).
3. Show that the eigenvalues \(\mu _i\), \(i=1,\cdots ,n\), of the monodromy matrix satisfy \[ \prod _{i=1}^n \mu _i = \exp \left (\int _0^T \operatorname {trace}(A(t))\,dt\right ). \]
4. Suppose \(\left |\prod _{i=1}^n \mu _i\right |>1\). Prove that the zero solution of \(\frac {dx}{dt}=A(t)x\) is unstable. If \(\left |\prod _{i=1}^n \mu _i\right |<1\), does that guarantee that the zero solution of \(\frac {dx}{dt}=A(t)x\) is stable? Why or why not?

5. Suppose \[ A(t)= \begin {pmatrix} \lambda (t) & \omega (t)\\ -\omega (t) & \lambda (t) \end {pmatrix}, \]

   where both \(\lambda (t)\) and \(\omega (t)\) are periodic functions of \(t\) with period \(T=1\). Find a condition on \(\lambda (t)\) that determines stability or instability of the zero solution of \(\frac {dx}{dt}=A(t)x\).

   Hint: Examine \(\frac {d}{dt}(x_1^2+x_2^2)\).

Consider the system of equations

with \(-\frac {\pi }{2}\le x\le \frac {\pi }{2}\).

1. Sketch a phase portrait for this system of equations. Identify all fixed points and their stability.
2. Show that the rectangle with vertices \((x_0,y_0)\), \((-x_0,y_0)\), \((-x_0,-y_0)\), \((x_0,-y_0)\), where \(y_0=\tan (x_0)\), is positively invariant for any \(x_0\), \(0<x_0<\frac {\pi }{2}\).
3. What is \(\omega _+(x,y)\) for \(-\frac {\pi }{2}<x<\frac {\pi }{2}\)? Are there any periodic orbits?
4. What is \(\omega _+(x,y)\) for \(x=\frac {\pi }{2}\)? What is \(\omega _+(x,y)\) for \(x=-\frac {\pi }{2}\)?

Let \(A\) be a real matrix such that \(\operatorname {Re}\lambda <0\) for all eigenvalues of \(A\).

1. Show that there is \(\delta >0\) such that if \(|x_0|\le \delta \), then the initial value problem

   \[ \dot {x}=Ax+e^{-t}|x|x \]

   (1)

   \[ x(0)=x_0 \]

   has a bounded solution on \([0,\infty )\). [Provide the complete proof. Do not just quote a theorem.]

2. Is the zero solution of (1) asymptotically stable? Explain. [Hint: let \(f(t)\), \(\phi (t)\) be nonnegative continuous functions on the interval \(J=(\alpha ,\beta )\) containing \(t_0\). Let \(c_0\ge 0\). Gronwall’s Lemma says that if \[ f(t)\le c_0+\left |\int _{t_0}^t \phi (s)f(s)\,ds\right | \] for all \(t\in J\), then \[ f(t)\le c_0\exp \left |\int _{t_0}^t \phi (s)\,ds\right | \] for all \(t\in J\).]

Let \(f(x)\in C^1(\mathbb {R}^n,\mathbb {R}^n)\) have a non-constant, \(T>0\) periodic trajectory \(\gamma (t)\) satisfying

1. Define: the periodic solution \(\gamma (t)\) is orbitally stable.
2. Define: the Poincare Map for the orbit \(\gamma \).
3. \([17]\) \(\gamma (t)=(2\cos 2t,\sin 2t)\) is a periodic solution to \[ \begin {cases} \dot {x}=-4y+x\left (1-\dfrac {x^2}{4}-y^2\right ),\\ \dot {y}=x+y\left (1-\dfrac {x^2}{4}-y^2\right ). \end {cases} \] Determine the orbital stability of \(\gamma (t)\) by computing the derivative of its Poincare Map.

Consider the differential equation

Suppose that \(U(x) \to \infty \) as \(x \to \pm \infty \). Find a Lyapunov function for this equation and use this to show that

1. there are no periodic solutions.
2. minima of \(U(x)\) are asymptotically stable steady solutions.
3. Under what conditions on \(U(x)\) is a minimum of \(U(x)\) exponentially stable?

1. For a general dynamical system, give the definitions of an invariant set, attracting set and \(\omega \)-limit sets.

2. For the system

   \[ \dot {x}=x-y-x^3,\quad \dot {y}=x+y-y^3, \]

   identify all critical points and their stability, and use the Poincare-Bendixson theorem to show that this system has a periodic orbit in the annular region \(1<x^2+y^2<2\).

Consider the \(T\)-periodic non-autonomous linear differential equation \[ \dot {x}=A(t)x,\quad x\in \mathbb {R}^n,\quad A(t)=A(t+T). \] Let \(\Phi (t)\) be a fundamental matrix with \(\Phi (0)=I\).

1. Show that there exists at least one nontrivial solution \(\chi (t)\) such that \[ \chi (t+T)=\mu \chi (t) \] where \(\mu \) is an eigenvalue of \(\Phi (T)\).
2. Suppose that \(\Phi (T)\) has \(n\) distinct eigenvalues \(\mu _i\), \(i=1,\ldots ,n\). Show that there are then \(n\) linearly independent solutions of the form \[ x_i=p_i(t)e^{\rho _i t} \] where the \(p_i(t)\) are \(T\)-periodic. How is \(\rho _i\) related to \(\mu _i\)?
3. Consider the equation \(\dot {x}=f(t)A_0x\), \(x\in \mathbb {R}^2\), with \(f(t)\) a scalar \(T\)-periodic function and \(A_0\) a constant matrix with real distinct eigenvalues. Determine the corresponding Floquet multipliers.
4. Suppose that the autonomous nonlinear equation \(\dot {x}=f(x)\), \(x\in \mathbb {R}^d\), exhibits a limit cycle. By linearizing about this solution, explain how Floquet theory can be used to determine the linear stability of the limit cycle.

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).

1. For a system of equations \(\frac {dx}{dt}=A(t)x\), where \(x\in \mathbb {R}^n\) and \(A(t)\) is a real \(n\times n\) matrix function which is smooth in \(t\), what is Abel’s formula? (State without proof).
2. For the linear differential equation \[ \frac {d^2x}{dt^2}-p(t)\frac {dx}{dt}+q(t)x=0 \tag {4} \] where \(p(t)>\alpha >0\) is a smooth function defined for all \(t\), prove there is at least one solution \(x(t)\) that is unbounded as \(t\to \infty \).

1. For a general dynamical system, give the definitions of an invariant set, attracting set and \(\omega \)-limit sets.
2. For the system \[ \dot {x}=-f(x,y)x-g(x,y)y,\qquad \dot {y}=g(x,y)x-f(x,y)y, \tag {6} \] where \(g(x,y)>0\) for all \(x,y\), and \(f(x,y)=x^2+y^2+2y-1\), identify all critical points and their stability, and show that this system has an invariant annular region which encloses the origin, and therefore has a nontrivial periodic solution.

1. For a general dynamical system, give the definitions of an invariant set, attracting set and \(\omega \)-limit sets.
2. For the system \[ \dot {x}=x-y-x^3,\qquad \dot {y}=x+y-y^3, \] identify all critical points and their stability, and use the Poincare-Bendixson theorem to show that this system has a periodic orbit in the annular region \(1<x^2+y^2<2\).

1. Let \(f(x)\in C^1(\mathbb {R}^n,\mathbb {R}^n)\) satisfy \(f(0)=0\). State the definitions: the zero solution of \[ \begin {cases} \dfrac {dx}{dt}=f(x),\\ x(0)=x_0, \end {cases} \] is Stable, Asymptotically Stable and Unstable.
2. Show that the zero solution is asymptotically stable. \[ \begin {cases} \dot {x}=xy-x^3\\ \dot {y}=-y+x^3y \end {cases} \]
3. Suppose that the real \(n\times n\) matrix \(A\) has an eigenvalue with \(\Reop \,\lambda >0\). Prove that the zero solution of \(\dot {x}=Ax\) is unstable.

Let \(f(x)\in C^1(\mathbb {R}^n,\mathbb {R}^n)\) have a non-constant, \(T>0\) periodic trajectory \(\gamma (t)\) satisfying \(\dot {\gamma }(t)=f(\gamma (t))\).

1. Define: the periodic solution \(\gamma (t)\) is orbitally stable.
2. Define: the Poincaré Map for the orbit \(\gamma \).
3. Find a periodic solution \(\gamma (t)\) to \[ \begin {cases} \dot {x}=x-y-x(x^2+y^2),\\ \dot {y}=x+y-y(x^2+y^2). \end {cases} \] Determine the orbital stability of \(\gamma (t)\) by computing the derivative of its Poincaré Map.

1. Let \(f(x)\in C^1(\mathbb {R}^n,\mathbb {R}^n)\) satisfy \(f(0)=0\). State the definitions: the zero solution of \[ \begin {cases} \dfrac {dx}{dt}=f(x),\\ x(0)=x_0, \end {cases} \] is stable in the sense of Liapunov and asymptotically stable.
2. Suppose that all eigenvalues of the real \(n\times n\) matrix \(A\) have negative real part \(\Reop \,\lambda _i<0\). State an estimate for solutions of \[ \begin {cases} \dfrac {dx}{dt}=Ax,\\ x(0)=x_0, \end {cases} \] and use it to prove that the zero solution is asymptotically stable.
3. Let \(A\) as in (b) and \(f\in C^1(\mathbb {R}^n,\mathbb {R}^n)\) be a smooth function such that for some \(L<\infty \), \(\abs {f(x)}\le L\abs {x}^2\) for all \(x\). Consider the initial value problem \[ \begin {cases} \dfrac {dx}{dt}=Ax+f(x),\\ x(0)=x_0. \end {cases} \tag {3} \] Prove that the zero solution of (3) is stable in the sense of Liapunov. [Don’t just quote a theorem. Hint: Suppose \(g(t)\) and \(u(t)\) are nonnegative functions and \(c_0\ge 0\) is a constant that satisfy \(u(t)\le c_0+\int _0^t g(s)u(s)\dd s\) for all \(t\ge 0\), then Gronwall’s Inequality implies \(u(t)\le c_0\exp \left (\int _0^t g(s)\dd s\right )\) for all \(t\ge 0\).]

Let \(f(x)\in C^1(\mathbb {R}^n,\mathbb {R}^n)\) have a non-constant, \(T>0\) periodic trajectory \(\gamma (t)\) satisfying \(\dot {\gamma }(t)=f(\gamma (t))\).

1. Define: the periodic solution \(\gamma (t)\) is orbitally stable.
2. Define: the Poincaré Map for the orbit \(\gamma \).
3. \(\gamma (t)=(2\cos 2t,\sin 2t)\) is a periodic solution to \[ \begin {cases} \dot {x}=-4y+x\left (1-\dfrac {x^2}{4}-y^2\right ),\\ \dot {y}=x+y\left (1-\dfrac {x^2}{4}-y^2\right ). \end {cases} \] Determine the orbital stability of \(\gamma (t)\) by computing the derivative of its Poincaré Map.

Consider the following linear equation for \(x\in \mathbb {R}^n\): \[ \dot {x}=A+B(t)x,\qquad x(t_0)=x_0. \] where \(A,B(t)\) are \(n\times n\) matrices. Suppose that all eigenvalues \(\lambda _j\), \(j=1,\ldots ,n\), of the matrix \(A\) satisfy \(\Reop (\lambda _j)<0\), and let \(B(t)\) be continuous for \(0\le t<\infty \) with \(\int _0^\infty \norm {B(t)}\dd t<\infty \).

1. Using the variation of constants formula show that there exist constants \(K,\sigma >0\) such that \[ \abs {x(t)} \le Ke^{-\sigma (t-t_0)}\abs {x_0} +K\int _{t_0}^t e^{-\sigma (t-s)}\abs {x(s)}\norm {B(s)}\dd s. \]
2. Let \(u(t)=e^{\sigma t}\abs {x(t)}\), \(v(t)=\norm {B(t)}\) and \(c=Ke^{\sigma t_0}\abs {x_0}\). Show that the inequality of part (a) can be rewritten as \[ u(t)\le c+\int _{t_0}^t v(s)u(s)\dd s. \]
3. From Gronwall’s inequality we have \[ u(t)\le c\exp \left (\int _{t_0}^t v(s)\dd s\right ). \] Use this to show that the zero solution of the IVP is asymptotically stable.

Consider a nonconstant \(T\)-periodic orbit \(\gamma (t)\) of the smooth autonomous differential equation \[ \dot {x}=f(x),\qquad x\in \mathbb {R}^n. \]

1. Describe the Poincaré Return Map associated to \(\gamma \).
2. A \(2\pi \)-periodic solution of the planar system \[ \begin {aligned} \dot {x} &= x-(x+y)\sqrt {x^2+y^2},\\ \dot {y} &= y+(x-y)\sqrt {x^2+y^2} \end {aligned} \] is given by \(\gamma (t)=(\cos t,\sin t)\). Find a Poincaré Return Map associated to \(\gamma \).
3. Find the derivative of the Poincaré Return Map from part (b) at \(\gamma \).
4. Using (c), determine whether the solution (b) is stable and define the sense of stability you mean.

Consider a Brusselator system for chemical reactants \(x,y\) given by \[ \begin {aligned} \dot {x} &= 1-4x+x^2y,\\ \dot {y} &= 3x-x^2y \end {aligned} \]

1. Show that the quadrilateral bounded by the lines \(x=0\), \(y=0\), \(y=8+x\) and \(y=10-x\) is a forward invariant subset. [Hint: need to show that if \(n\) is the outer normal to a given edge of the quadrilateral, then \(n\cdot (\dot {x},\dot {y})\le 0\).]
2. Find the fixed points and determine their stability.
3. Show that the system has at least one nonconstant periodic solution.

Consider a Brusselator system for chemical reactants \(x,y\) given by \[ \begin {aligned} \dot {x} &= 1-4x+x^2y,\\ \dot {y} &= 3x-x^2y \end {aligned} \]

1. Show that the quadrilateral bounded by the lines \(x=0\), \(y=0\), \(y=8+x\) and \(y=10-x\) is a forward invariant subset.
2. Find the fixed points and determine their stability.
3. Show that the system has at least one nonconstant periodic solution.

Consider a modified predator-prey system for populations \(x,y\) given by \[ \begin {aligned} \dot {x} &= x\cdot \left (1-\frac {x}{4}-\frac {y}{1+x}\right ),\\ \dot {y} &= y\cdot \left (\frac {x}{1+x}-\frac 12\right ) \end {aligned} \]

1. Show that the triangle with vertices \((0,0)\), \((5,0)\) and \((0,5)\) is a forward invariant subset.
2. Find the fixed points and determine their stability.
3. Show that the system has at least one nonconstant periodic solution.

Suppose that \(H:\mathbb {R}^2\to \mathbb {R}\) is a smooth function. Consider the Hamiltonian System \[ \begin {aligned} \dot {x} &= H_y(x,y),\\ \dot {y} &= -H_x(x,y). \end {aligned} \] Suppose that \(\Gamma _0\) is a nonconstant periodic orbit for this system. Show that \(\Gamma _0\) cannot be Asymptotically Orbitally Stable. (This is sometimes referred to as asymptotic stability of the periodic orbit \(\Gamma _0\) or asymptotic Poincarè stability.)

Consider the problem \[ \begin {aligned} u_t &= u_{xx}, && t\in (0,1),\ x\in (0,1),\\ u(0,t)=u(1,t) &= 0, && t\in (0,1),\\ u(x,1) &= f, && x\in (0,1). \end {aligned} \]

1. What are the least restrictive condition you can find on \(f\) which guarantee that the problem admits a solution \(u\), bounded in the norm \[ \norm {u}_B=\sup _{t\in [0,1]}\left (\int _0^1 u(x,t)^2d x\right )^{1/2}\ ? \]
2. Under what conditions on \(f\) does the stability estimate \[ \norm {u}_B\le C\norm {f}_{L^2(0,1)} \] hold?
3. Assuming \(u\in C^2(\overline {U_1})\), where \(U_1=(0,1)\times (0,1]\), is uniqueness of solutions guaranteed?

Let \(A\) be an \(n\times n\) real matrix whose eigenvalues \(\lambda _i\) satisfy \(\Re \lambda _i<0\) for all \(i=1,\ldots ,n\). Let \(g(x,t)\in C^\infty (\mathbb {R}^n\times \mathbb {R},\mathbb {R}^n)\) such that \(\lvert g(x,t)\rvert \le \lvert x\rvert ^2\) for all \(x\in \mathbb {R}^n\). Consider the ODE \[ y'=Ay+g(y,t). \]

1. Show that if the initial value \(\lVert x(0)\rVert \) is small enough, then the solution exists for all time and \(\lim _{t\to \infty }x(t)=0\). [Hint: Gronwall’s Inequality for nonnegative functions \(u(t)\) and \(v(t)\) that satisfy \(v(t)\le c_0+\int _0^t u(s)v(s)\dd s\) is \(v(t)\le c_0\exp \left (\int _0^t u(s)\dd s\right )\).]
2. Let \(z(t)=0\) for all \(t\) be the zero solution. Define what it means for \(z\) to be Stable (often called Liapunov Stable.) Discuss the stability of \(z\) for this equation.

Consider the linear non–autonomous first order system \[ \dot {x}=Ax+B(t)x,\qquad x\in \mathbb {R}^n \] with \(A\) non-singular and \(B(t)\) continuous for \(t\ge 0\). Further, assume that

• the eigenvalues of \(A\) have non–positive real parts and those with zero real part are non-degenerate
• \(\displaystyle \int _0^\infty \lVert B(t)\rVert \dd t=c_1\) with \(c_1\) a positive constant.

1. Let \(\Phi (t)\) be the fundamental matrix of the equation \(\dot {x}=Ax\) with \(\Phi (0)=I\). Derive the variation of constants formula \[ x(t)=\Phi (t)x(0)+\int _0^t \Phi (t-s)B(s)x(s)\dd s \]
2. Prove that the solution \(x(t)\) is bounded for all times \(t>0\). [Hint: Use part (a) and Gronwall’s lemma in the following form: \(v(t)\le v_0+\int _0^t u(s)v(s)\dd s\) implies that \(v(t)\le v_0\exp (\int _0^t u(s)\dd s)\) for \(t>0\)]. What does this imply about the stability of the origin?

Consider the \(T\)-periodic non-autonomous linear differential equation \[ \dot {x}=A(t)x,\quad x\in \mathbb {R}^n,\quad A(t)=A(t+T). \] Let \(\Phi (t)\) be a fundamental matrix with \(\Phi (0)=I\).

1. Show that there exists at least one nontrivial solution \(x=\chi (t)\) such that \[ \chi (t+T)=\mu \chi (t) \] where \(\mu \) is an eigenvalue of \(\Phi (T)\).
2. Suppose that \(\Phi (T)\) has \(n\) distinct eigenvalues \(\mu _i\), \(i=1,\ldots ,n\). Show that there are then \(n\) linearly independent solutions of the form \[ x_i=p_i(t)e^{\rho _i t} \] where the \(p_i(t)\) are \(T\)-periodic. How is \(\rho _i\) related to \(\mu _i\)?
3. Suppose that the autonomous nonlinear equation \(\dot {x}=f(x)\) exhibits a limit cycle. By linearizing about this solution, explain how Floquet theory can be used to determine the linear stability of the limit cycle.

Consider a second order, linear autonomous system \(\dot {x}=Ax\), \(x\in \mathbb {R}^2\).

1. Suppose that \(A\) has a pair of complex conjugate eigenvalues. By performing an appropriate change of coordinates, convert the equation into Jordan normal form and thus obtain the general solution.
2. Under what conditions is the fixed point at the origin hyperbolic? Explain the significance of hyperbolic fixed points in the theory of nonlinear ODEs.

Consider the \(T\)-periodic non-autonomous linear differential equation \[ \dot {x}=A(t)x,\quad x\in \mathbb {R}^n,\quad A(t)=A(t+T). \] Let \(\Phi (t)\) be a fundamental matrix with \(\Phi (0)=I\).

1. Show that there exists at least one nontrivial solution \(x=\chi (t)\) such that \[ \chi (t+T)=\mu \chi (t) \] where \(\mu \) is an eigenvalue of \(\Phi (T)\).
2. Suppose that \(\Phi (T)\) has \(n\) distinct eigenvalues \(\mu _i\), \(i=1,\ldots ,n\). Show that there are then \(n\) linearly independent solutions of the form \[ x_i=p_i(t)e^{\rho _i t} \] where the \(p_i(t)\) are \(T\)-periodic. How is \(\rho _i\) related to \(\mu _i\)?
3. Explain how Floquet theory can be used to determine the linear stability of a limit cycle.

1. Give the definitions of Poincare and Liapunov stability. Show that solutions of the system \(\dot {x}=y\), \(\dot {y}=0\) are Poincare but not Liapunov stable.
2. Consider the system \[ \begin {aligned} \dot {x} &= x-y-x(x^2+y^2)+\frac {xy}{\sqrt {x^2+y^2}},\\ \dot {y} &= x+y-y(x^2+y^2)-\frac {x^2}{\sqrt {x^2+y^2}}. \end {aligned} \] Show that the above pair of equations can be rewritten in polar coordinates as \[ \dot {r}=r(1-r^2),\quad \dot {\theta }=2\sin ^2(\theta /2) \] and sketch the phase portrait. Determine whether or not the fixed point \((1,0)\) is Liapunov stable.

Linear Systems. Consider the linear initial value problem \[ \begin {aligned} \dot {x} &= Ax,\\ x(0) &= x_0, \end {aligned} \] where \(A\in \mathbb {R}^{n\times n}\) is constant, and \(x_0\in \mathbb {R}^n\) is given.

1. Prove that the matrix exponential \(e^{At}\) exists and satisfies \(\lVert e^{At}\rVert \le e^{\lVert A\rVert t}\), where \(\lVert \cdot \rVert \) is the operator norm.
2. Prove directly that \(x(t)=e^{At}x_0\) is the unique solution to the initial value problem above.

Floquet Theory. Consider the system \[ \dot {x}(t)=A(t)x(t), \] where \(A:\mathbb {R}\to \mathbb {R}^{n\times n}\) is continuous and \(T\)-periodic, i.e., \(A(t+T)=A(t)\) for all \(t\in \mathbb {R}\). Let \(\Phi (t)\) be the fundamental matrix solution for this system. Assume that there exists a complex-valued matrix \(B\) such that \(e^{BT}=\Phi (T)\).

1. Explain in detail how solutions \(x(t)\) to the periodic system above are related to solutions \(y(t)\) of the constant-coefficient problem \(\dot {y}=By\), and prove your assertions.
2. Explain what the eigenvalues of \(A(t)\) or \(B\) have to do with stability of the solutions \(x(t)\).

Linear Systems. Suppose \(A\) is a real \(n\times n\) matrix whose eigenvalues \(\lambda _i\) satisfy \(\operatorname {Re}\lambda _i\le 0\). Show that there are constants \(C<\infty \) and \(k\le n-1\) so that any solution \(x(t)\in \mathbb {R}^n\) of the initial value problem \[ \begin {aligned} \frac {dx}{dt} &= Ax,\\ x(0) &= x_0, \end {aligned} \] satisfies \[ \lvert x(t)\rvert \le C(1+\lvert t\rvert ^k)\lvert x_0\rvert \] for all \(t\in \mathbb {R}\) and \(x_0\in \mathbb {R}^n\).

Linearization. Assume \(g\in C^\infty (\mathbb {R}^n)\) has a strict local minimum at the origin. Consider the second-order system \[ \ddot {x}+B\dot {x}+\nabla g(x)=0, \] where \(x(t)\in \mathbb {R}^n\) and \(B\in \mathbb {R}^{n\times n}\) is a symmetric positive definite matrix.

1. Rewrite the system as a first-order system.
2. Write the linearization of the system at the origin.
3. Prove that the origin is an asymptotically stable fixed point.

Constant Coefficients. Suppose that \(A\) is a real \(n\times n\) matrix all of whose eigenvalues \(\lambda \) have strictly negative real part \(\operatorname {Re}\lambda <0\). Suppose \[ \frac {dx}{dt}=Ax,\qquad x(0)=x_0. \tag {1.} \]

1. Show that there is a decreasing function \(\delta (t)>0\) such that \(\delta (t)\to 0\) as \(t\to \infty \) and such that every solution of (1.) satisfies \[ \lvert x(t)\rvert \le \delta (t)\lvert x_0\rvert \qquad \text {as }t\to \infty . \]
2. Define: \(z\) is Liapunov Stable. Define: \(z\) is Asymptotically Stable. Show that the zero solution \(z(t)=0\) of the autonomous system is asymptotically stable.

Stability of Solutions. The following questions have to do with the stability of the zero solution \(z(t)=0\) to a system of ODE’s \[ y'=f(t,y), \] where \(f(t,x)\in C^\infty (\mathbb {R}\times \mathbb {R}^n,\mathbb {R}^n)\) such that \(f(t,0)=0\) for all \(t\).

1. Define: \(z\) is Liapunov Stable. Define: \(z\) is asymptotically stable.
2. Give an example of an ODE system such that the zero solution is Liapunov stable but not asymptotically stable.
3. Let \(A\) be an \(n\times n\) real matrix whose eigenvalues \(\lambda _i\) satisfy \(\operatorname {Re}\lambda _i<0\) for all \(i=1,\ldots ,n\). Let \(g(x)\in C^\infty (\mathbb {R}^n,\mathbb {R}^n)\) such that there are constants \(1<\beta ,c<\infty \) so that \(\lvert g(x)\rvert \le c\lvert x\rvert ^\beta \) for all \(x\in \mathbb {R}^n\). Show that the zero solution \(z(t)\) is asymptotically stable in the equation \[ y'=Ay+\sin (t)g(y). \]

Let \(A\) be a real \(N\times N\) matrix, all of whose eigenvalues have negative real parts. Let \[ h:\mathbb {R}\to \mathbb {R}^N \] be a continuous function which is periodic of period \(T\). Let \[ \begin {aligned} (t,v)&\longmapsto u(t,v),\\ \mathbb {R}\times \mathbb {R}^N&\longrightarrow \mathbb {R}^N \end {aligned} \] denote the unique solution of \[ u'=Au+h(t),\qquad u(0)=v. \tag {3} \] For given \(T>0\), define \[ S_T:\mathbb {R}^N\to \mathbb {R}^N \] by \[ S_T(v)=u(T,v). \]

1. Use the variation of constants formula (and what you know about stability theory for the linear system \(u'=Au\)) to show that \(S_{nT}\) is a contraction mapping, for some positive integer \(n\), sufficiently large.
2. Show that \[ S_{nT}=\underbrace {S_T\circ \cdots \circ S_T}_{n}, \] the \(n\)-fold composition of \(S_T\) with itself.
3. Show that if \(S_{nT}\) has a unique fixed point, then so does \(S_T\).
4. Apply this to show that if \(h\) is a continuous periodic function of period \(T\), then (3) has a unique \(T\)-periodic solution.

Consider \(\dot {x}=\mathbf {f}(x)\) where \(x\in \mathbb {R}^n\) and \(\mathbf {f}\in C^1(\mathbb {R}^n,\mathbb {R}^n)\). Suppose that the solution through any point \(x_0\in \mathbb {R}^n\) exists for all \(t\in \mathbb {R}\).

1. Define the \(\alpha \)- and \(\omega \)-limit sets of a point \(x_0\in \mathbb {R}^n\).
2. Consider the nonlinear spring equation \[ x''+\mu x'+g(x)=0 \] where \(g(x)\) is an odd function of \(x\), \(g'(x)>0\) and \(\mu \ge 0\). Translate this equation to the \(\dot {x}=\mathbf {f}(x)\) form and linearize about the trivial equilibrium. What conclusions can be made about the stability of this equilibrium?
3. Define what is meant by a Lyapunov function for the above equation; derive one and verify that it is a Lyapunov function. Describe the \(\alpha \)- and \(\omega \)-limit sets of a point \(x_0\in \mathbb {R}^n\) for the case \(\mu =0\) and for the case \(\mu >0\). Make a conclusion about the stability of the trivial equilibrium for each of these two cases.

Consider the system \[ x'=f(x) \tag {4.1} \] where \(f\in C^1(\mathbb {R}^n,\mathbb {R}^n)\), and suppose that (4.1) has a periodic solution \(\gamma (t)\) of least period \(T\).

1. Define what it means to say that \(\gamma (t)\) is orbitally asymptotically stable with asymptotic phase.
2. Let \[ \xi '=A(t)\xi \tag {4.2} \] be the variational equation (the linearization of (4.1)) relative to \(\gamma (t)\). State the Floquet representation for the fundamental matrix solution of (4.2) and discuss the stability of \(\gamma (t)\) in terms of properties of the fundamental matrix solution.
3. Consider the system \[ \begin {aligned} x' &= x-y-x(x^2+y^2),\\ y' &= x+y-y(x^2+y^2). \end {aligned} \] Find a nontrivial periodic solution of this equation, and prove (in the easiest way possible) that this periodic solution is orbitally asymptotically stable with asymptotic phase. Compute the asymptotic phase explicitly.
4. Compute the Poincare map associated with the periodic orbit you found in (c).

Assume that \(A\) is a real matrix whose eigenvalues have negative real part \((\Reals \,\sigma (A)<0)\) and \(B(t)\) is continuous for all \(t\in \mathbb {R}\). Show that there is an \(\epsilon >0\) depending on \(A\) so that if \(\norm {B}\le \epsilon \) then the zero solution of (2) with \(f(x,t)=Ax+B(t)x\) is uniformly Liapunov stable.

1. Suppose \(A\) is a constant \(n\times n\) matrix and consider the equation \[ \dot {x}(t)=Ax(t). \] Show that if all eigenvalues have negative real parts then the trivial solution \(x(t)\equiv 0\) is asymptotically stable; that is, every solution \(x(t)\to 0\) as \(t\to \infty \).
2. Show that if one eigenvalue of \(A\) has positive real part, then the trivial solution of \(A\) is not stable.
3. Under what circumstances is the trivial solution stable if \(A\) has no eigenvalue with positive real part, but does have eigenvalues with zero real part? (State, don’t prove)
4. Do any of these results carry over if \(A\) is not a constant matrix? (State, don’t prove.)

(Concrete cases from 2.) Change to vector-matrix form and discuss the stability of the trivial solution of the following equations.

1. \(u''-u=0\)
2. \(u''+2ku'+u=0\) with \(k>0\)
3. \(u^{(4)}+2u''+u=0\)
4. \(y'=\Theta y\) where \(\Theta \) is the matrix of all zeros.

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

Assume that the eigenvalues of a real \(n\times n\) matrix \(A\) have negative real part strictly less than \(-5\). Prove that there exists a positive constant \(M\) such that for all \(x\in \mathbb {R}^n\) and \(t\ge 0\) \[ \left \lVert e^{At}x\right \rVert \le M\lVert x\rVert e^{-5t}. \]

Let \(A\) be a constant \(n\times n\) matrix, and define the matrix exponential by \[ e^A=\sum _{k=0}^{\infty }A^k/k!. \]

1. Prove that \(e^{At}\) is differentiable (you may use known theorems about scalar power series), and that it satisfies the initial value problem \(X'=AX\), \(X(0)=I\) (the identity matrix). Justify your steps.
2. Let \(A\) have distinct eigenvalues. Prove that for the origin to be a globally asymptotically stable point for \(X'=AX\) it is sufficient that all of the eigenvalues of \(A\) have negative real parts.
3. Is the condition in b) that \(A\) have distinct eigenvalues required in order to prove the result? (no proof necessary)
4. Is the sufficient condition in b) also necessary? Justify your answer.

DE Qualifying Examination 1992

1. What does Floquet’s Theorem say about solutions of linear \(n\times n\) systems \(X'=A(t)X\) (\(A\) is a continuous \(n\times n\) matrix)?
2. Show that, given any \(n\times n\) matrices \(B\) (constant) and \(A(t)\) (continuous), there is a change of variables \(Y=P(t)X\) which reduces the system \(X'=A(t)X\) to the system \(Y'=BY\).
3. Show that there is a particular choice of \(P(t)\) and \(B\) that is related to Floquet’s Theorem. Why might you be interested in that choice?

1. Show that the only equilibrium point of the system \[ x'=-2y-4x^3-4xy^2,\qquad y'=x-2x^2y-2y^3 \] is the origin.
2. Draw what conclusions you can about stability, asymptotic stability, and regions of attraction, using first i) linearization and then ii) Lyapunov functions. Make clear the reasons for your conclusions.
3. What if the signs of the nonlinear terms in a) were changed?
4. What kinds of functions \(f\) and \(g\) would give similar results for systems of the form \(x'=-2y+f(x,y)\), \(y'=x+g(x,y)\)?