Linear Systems and Matrix Exponentials

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 \(T\)-periodic non-autonomous linear differential equation

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 \(n\) 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 eigen- values. Determine the corresponding Floquet multipliers.

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

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

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 \(t\to \infty \).

Suppose \(A(t)\) is a real \(n\times n\) matrix function which is smooth in \(t\) and periodic of period \(T>0\). Consider the linear differential equation in \(\mathbb {R}^n\) \[ \begin {cases} \dfrac {dx}{dt}=A(t)x,\\ x(0)=x_0. \end {cases} \tag {2} \] Let \(\Phi (t)\) be the fundamental solution matrix with \(\Phi (0)=I\).

1. 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 \(\mu _i\)?
2. What is the fundamental solution matrix and what are the Floquet exponents for the system \[ A(t)= \begin {pmatrix} a & 1\\ 0 & \dfrac {\dot {p}(t)}{p(t)} \end {pmatrix}, \tag {3} \] where \(p(t)\) is a \(T\)-periodic function?

Show that a real symmetric linear operator has a full set of real orthogonal eigenvectors (even if it has multiple eigenvalues). [Assume that the operator is a matrix acting in \(\mathbb {R}^n\).]

1. Let \(A(t)\) be a continuous, non-constant, real matrix-valued, \(T>0\) periodic function \(A(t+T)=A(t)\). For the system \[ \dot {x}=A(t)x \tag {2} \] define: Monodromy Matrix, Floquet Multipliers and Floquet Exponents.
2. If \(e^{T\gamma }=\rho \) is a Floquet multiplier, then there is a solution of (2) of the form \[ x(t)=e^{\gamma t}p(t) \] where \(p(t)=\pm p(t+T)\).
3. Let \(a,b\in \mathbb {R}\). Suppose that a Floquet multiplier is \(\mu _1=-1\). What can you say about the boundedness of all solutions on \([0,\infty )\)? \[ \begin {pmatrix}\dot {x}\\ \dot {y}\end {pmatrix} = \begin {pmatrix} -1 & a\\ b & \sin (\pi t) \end {pmatrix} \begin {pmatrix}x\\ y\end {pmatrix} \]

1. Let \(A(t)\) be a continuous, non-constant, real matrix-valued, \(T>0\) periodic function \(A(t+T)=A(t)\). For the system \[ \dot {x}=A(t)x \tag {4} \] define: Monodromy Matrix, Floquet Multipliers and Floquet Exponents.
2. If \(e^{T\gamma }=\rho \) is a Floquet multiplier, then there is a solution of (4) of the form \[ x(t)=e^{\gamma t}p(t) \] where \(p(t)=\pm p(t+T)\).
3. Prove that this equation does not have a fundamental set of bounded solutions. \[ \ddot {x}-(\cos ^2t)\dot {u}+(\sin ^2t)u=0 \]

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 the \(T\)-periodic non-autonomous linear differential equation \[ \dot {x}=A(t)x,\qquad x\in \mathbb {R}^n,\qquad A(t)=A(t+T) \] 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. By determining the time derivative of the Wronskian \(W(t)=\det [\Phi (t)]\), prove that the product of the Floquet multipliers \(\mu _i\) satisfies \[ \prod _{i=1}^n\mu _i =\exp \left (\int _0^T\operatorname {tr}[A(s)]\dd s\right ) \]
4. Using part (c) show that \(\mu _1\mu _2=1\) in the case of Mathieu’s equation for a parametric oscillator: \[ \ddot {x}+(\alpha +\beta \cos t)x=0. \]

Consider the \(T\)-periodic non-autonomous linear differential equation \[ \dot {x}=A(t)x,\qquad x\in \mathbb {R}^n,\qquad A(t)=A(t+T) \] 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.

Let \(A\) be an \(n\times n\) real matrix. Consider the linear initial value problem \[ \begin {cases} \dfrac {dx}{dt}=Ax,\\ x(0)=x_0. \end {cases} \] Give an algebraic condition on the matrix \(A\) which holds if and only if all solutions of the initial value problem are bounded for all \(t\in \mathbb {R}\). Prove the equivalence.

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 \[ \dot {x}=A(t)x,\qquad x\in \mathbb {R}^n,\qquad A(t)=A(t+T) \] 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.

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.

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].

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 \(x(t)\).

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.

Problem. Let \(A\) be an \(N\times N\) matrix, none of whose eigenvalues have zero real part. Show that if \(b:(-\infty ,\infty )\to \mathbb {R}^N\) is a continuous function which is periodic of period 1, \((b(t+1)=b(t),\ t\in \mathbb {R})\), then the equation \[ x'=Ax+b(t), \] has a unique periodic solution of period 1.

Problem. In an isolated region of the Canadian Northwest Territories, a population of white wolves, \(x\), and one of silver foxes, \(y\), compete for survival (for each population one unit represents 100 individuals). They have a common, limited food supply, which consists mainly of mice. A mathematical biologist models the dynamics of the competing species by the nonlinear system \[ \begin {aligned} x' &= x-x^2-xy,\\ y' &= \frac {3}{4}y-y^2-\frac {1}{2}xy. \end {aligned} \] Can the two species survive together according to this model?

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.

Stable and Unstable Manifolds. Find the stable and unstable manifolds in the neighborhood of the origin for the system \[ \begin {aligned} x' &= -x,\\ y' &= y+x^2,\\ z' &= z+y^2. \end {aligned} \] How are the stable and unstable manifolds for the linearization of this system related to those of the nonlinear system?

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.

Let \(f\) have the special form \[ f(t,x)=Ax+h(t), \tag {2.10} \] where \(h:\mathbb {R}\to \mathbb {R}^N\) is a continuous function of period \(T\) and \(A\) is a constant \(N\times N\) matrix which is positive (negative) definite (i.e. \(Ax\cdot x\ge \mu \lvert x\rvert ^2\), resp. \(Ax\cdot x\le -\mu \lvert x\rvert ^2\), \(\mu >0\)). Use the result proved above to show that the equation \[ x'=Ax+h(t) \tag {2.11} \] has a unique \(T\)-periodic solution.

1. Suppose \(A\) is an \(n\times n\) matrix whose elements \(a_{ij}(t)\) are each of period \(T\). You may assume existence and uniqueness of a fundamental matrix \(\Phi (t)\) for the equation \[ \dot {x}(t)=A(t)x(t). \] Show that \(\Phi (t+T)\) is also a fundamental matrix for the equation, hence \[ \Phi (t+T)=\Phi (t)\cdot C \] for some non-singular matrix \(C\).

2. You may assume that a non-singular matrix \(C\) has a logarithm, i.e., \(C=e^B\) for some \(n\times n\) matrix \(B\).

   Show that there exists a non-singular matrix \(P(t)\) of period \(T\) such that \[ \Phi (t)=P(t)e^{Bt} \qquad \text {Floquet's Theorem.} \]

   Hint: Solve for \(P(t)\) and check.

3. Make the change of variables \(y=P(t)u(t)\) and change the periodic system \(y'=A(t)y\) into the system \[ u'=Bu. \]

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?