Asymptotic and Exponential Stability

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.

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

1. Suppose an initial profile \(u_0(x)\) is monotone with \(\lim _{x\to -\infty }u_0(x, )=u_L\), \(\lim _{x\to \infty }u_0(x)=u_R\). Under what conditions on \(u_L\) and \(u_R\) does the solution of Burger’s equation \[ \frac {\partial u}{\partial t} +u\frac {\partial u}{\partial x}=0, \] evolve into a shock, and what is its asymptotic speed?
2. Consider the viscous Burger’s equation \[ \frac {\partial u}{\partial t} +u\frac {\partial u}{\partial x} =\epsilon \frac {\partial ^2u}{\partial x^2}. \] with \(\epsilon >0\). Under what conditions on \(u_L\) and \(u_R\), where \(\lim _{x\to -\infty }u(x,t)=u_L\), \(\lim _{x\to \infty }u(x,t)=u_R\) does this equation have a travelling wave solution, and what is its speed? Verify your result.

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.)

The time-dependent Schrödinger equation can be written \[ i u_t=-\Delta u+Vu, \] where \(u(x,t)\), \(x\in \mathbb {R}^n\), \(t\in \mathbb {R}\), is the complex-valued “wave function,” and \(V(x)\ge 0\) is a given continuous real-valued potential function.

1. Using complex exponential plane waves, describe the dispersion relation for this equation in case \(V\equiv 1\). Is this problem dispersive?
2. Assuming \(V\) is bounded, and the problem is formulated over the smooth bounded connected spatial domain \(U\subset \mathbb {R}^n\), with homogeneous Dirichlet boundary conditions, prove that the solutions to the time dependent problem can be expressed as a discrete sum involving solutions to the time-independent problem \[ E_k\phi _k=\Delta \phi _k+V\phi _k,\qquad \text {in }U \] where \(0<E_1\le E_2\le \cdots \).

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 scalar equation \[ \ddot {x}+\dot {x}=-\epsilon (x^2-x),\qquad 0<\epsilon \ll 1 \] Using the method of multiple scales show that the \(O(1)\) solution is \[ x_0(t,\tau )=A(\tau )+B(\tau )e^{-t}, \] where \(\tau =\epsilon t\), and identify any resonant terms at \(O(\epsilon )\). Show that the non–resonance condition for the amplitude \(A\) is \[ A_\tau =A-A^2 \] and hence determine the asymptotic behavior of \(x_0\). Comment on the domain of validity of the asymptotic expansion.

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.

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 \[ y'=Ay+\sin (t)g(y). \]

(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.

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 \(x'=-2y+f(x,y)\), \(y'=x+g(x,y)\)?