Equilibrium points, Lyapunov stability, linearization, and asymptotic behavior of solutions of ordinary differential equations.
These practice problems ask whether a solution of an ordinary differential equation stays near an equilibrium, and in what sense. You will test stability from the definition, linearize a nonlinear equation at a steady state, and build a Lyapunov function when the linear test is inconclusive. The set is shorter than the other differential-equation topics, and it is the harder one. Solutions spell out the definition that is being checked.
Equilibrium Points asks you to find steady states and to argue stability directly from the definition, often on a scalar equation you can solve. Linearization replaces a nonlinear system by its derivative at the equilibrium and reads the conclusion off that linear system. Lyapunov Stability uses an energy-like function on a neighborhood of the equilibrium when eigenvalues do not settle the question.
Starting from the definition of stability, examine the stability of the solution of the differential equation $$\frac{dy}{dx}=-a^2 y,\quad a\neq0,\quad y(x_0)=y_0.$$
Starting from the definition of stability, examine the stability of the solution of the differential equation $$\frac{dy}{dt}=a^2 y,\quad a\neq0,\quad y(t_0)=y_0.$$
Starting from the definition of stability, examine the stability of the zero solution of the system $$\frac{dx}{dt}=4y,\qquad \frac{dy}{dt}=-x,\qquad x(0)=y(0)=0.$$
For which values of the parameter $\alpha$ is the rest point $(0,0,0)$ of the system $$\dot{x}=\alpha x-y,\qquad\dot{y}=\alpha y-z,\qquad\dot{z}=\alpha z-x$$ stable?
Prove that if the singular point of the equation $$(ax+by)\,dx+(mx+ny)\,dy=0$$ is a centre, then the equation is an exact differential. Show that the converse is not true.
Given the equation $$y'=\frac{ax+by+f(x,y)}{cx+dy+\varphi(x,y)},$$ where $f$ and $\varphi$ are defined and continuous in a neighbourhood of $(0,0)$ and vanish together with their first partial derivatives at $(0,0)$, prove that if the equation is unchanged under the substitution $y\mapsto-y$, and the roots of the characteristic equation $$\begin{vmatrix}c-\lambda & d\\ a & b-\lambda\end{vmatrix}=0$$ are purely imaginary, then the singular point $(0,0)$ is a centre.
Examine the stability of the rest point $(0,0)$ of the system $$\dot{x}=-x+y+2x^4-y^6,\qquad\dot{y}=x-3y+11y^4,$$ by the method of first approximation (linearisation).
Examine the stability of the rest point $(0,0)$ of the system $$\dot{x}=-x+2y-3x^2,\qquad\dot{y}=3x-2y+2x^2+y^4,$$ by the method of first approximation (linearisation).
Examine the stability of the rest point $(0,0)$ of the system $$\dot{x}=-\sin x+3y+x^5,\qquad\dot{y}=\tfrac{1}{4}x-2y-\tfrac{1}{6}y^3,$$ by the method of first approximation (linearisation).
Examine the stability of the rest point $(0,0)$ of the system $$\dot{x}=-3x+4y+\sin^3 x-y^2,\qquad \dot{y}=-2x+\sin y+e^y x^2,$$ by the method of first approximation (linearisation).
Examine the stability of the rest point of the system $$\dot{x}=e^{x+2y}-\cos3x,\qquad\dot{y}=\sqrt{4+8x}-2e^y,$$ by the method of first approximation (linearisation).
Examine the stability of the rest point of the system $$\dot{x}=\ln(4y+e^{-3x}),\qquad\dot{y}=2y-1+\sqrt{1-6x},$$ by the method of first approximation (linearisation).
Examine the stability of the rest point of the system $$\dot{x}=2x+8\sin y,\qquad\dot{y}=2-e^x-3y-\cos y,$$ by the method of first approximation (linearisation).
Examine the stability of the rest point $(0,0)$ of the system $$\dot{x}=-4x+\tfrac{7}{2}\sin y-3x^2,\qquad \dot{y}=-2x+x^2+y+y^3,$$ by the method of first approximation (linearisation).
Examine the stability of the rest point $(0,0)$ of the system $$\dot{x}=-4y-x^3,\qquad\dot{y}=3x-y^3,$$ by the method of first approximation (linearisation).
Examine the stability of the rest point $(0,0)$ of the system $$\dot{x}=10\sin x-29y+3y^3,\qquad\dot{y}=5x-14\sin y+y^2,$$ by the method of first approximation (linearisation).
Verify whether the trivial solution of a system of differential equations is stable in the sense of Lyapunov, given that the general solution of the system has the form $$x=\frac{c_1-c_2 t}{1+t^2},\qquad y=(c_1 t^3+c_2)\,e^{-t}.$$
Prove that the condition $$\lim_{t\to+\infty}\int_0^t a(s)\,ds<+\infty$$ is necessary and sufficient for the trivial solution of the equation $$\frac{dx}{dt}=a(t)\,x,$$ where $a(t)$ is a continuous function, to be stable in the sense of Lyapunov.
Prove that if some particular solution of a system of differential equations is stable in the sense of Lyapunov, then all solutions of that system are stable.
For which values of the parameter $a$ is the trivial solution $x=0,\;y=0,\;z=0$ of the system $$\frac{dx}{dt}=z,\qquad\frac{dy}{dt}=-3x,\qquad\frac{dz}{dt}=ax+2y-3z$$ asymptotically stable?