Instability of solutions to nonlinear systems

In this post, we propose two results in the instability of solutions to nonlinear systems. Here, we study ordinary differential equations. Well-know stability theorems are due to Liapunov, based on the linearization of the vector field.

Consider a continuous function $f:\Omega\subset \mathbb{R}^d\to\mathbb{R}^d$, and $x_0\in \Omega$. Let the Cauchy problem\begin{align*}\tag{Eq}\dot{u}(t)=f(u(t)),\quad u(0)=x_0.\end{align*}

The flow of an autonomous system

According to Peano’s theorem, the maximal solution of the differential equation $({\rm Eq})$ exists. We denote by $J_{x_0}$ the interval in which the maximal solution is well defined. We denote \begin{align*}D(f)=\bigcup_{x\in\Omega}\left(J_x\times\{x\}\right).\end{align*} The flow associated to the equation $({\rm Eq})$ is the following application \begin{align*} \Phi: D(f)\to \Omega,\quad (t,x)\mapsto \Phi(t,x)=u(t),\end{align*}where $u$ is the maximal solution associated to the initial condition $u(0)=x$. Also, we denote $\Phi_t(x)=\Phi(t,x)$.

We mention that $J_{\Phi_t(x)}=J_x-t$, and if $t_1+t_2\in J_x$, we have\begin{align*}\Phi_{t_2}\circ \Phi_{t_1} (x)= \Phi_{t_1+t_2}(x).\end{align*}

Assume that $f$ is locally Lipschitz on $\Omega$. Then $D(f)$ is an open set of $\mathbb{R}\times \Omega$. Moreover, the flow $\Phi$ is locally Lipschitz, in particular, it is continuous, on $D(f)$.

 Instability of solutions to nonlinear systems

An equilibrium point of $f:\Omega\to \mathbb{R}^d$ is an element $x_0\in \Omega$ such that $f(x_0)=0$. An immediate consequence is that the constant function equal to $x_0$ satisfies the differential equation $({\rm Eq})$.

The equilibrium point $x_0$ is stable if and only if for any neighborhood $W$ of $x_0$ in $\Omega,$ there exists a neighborhood $U$ of $x_0$ in $\Omega$ such that for all $x\in U,$ $\Phi_t(x)$ is defined for any $t\ge 0,$ i.e. global solution, and takes value in $W$.

The equilibrium $x_0$ is unstable if and only if it is not stable.

The equilibrium $x_0$ is asymptotically stable if and only if it is stable and there exists a neighborhood $W$ of $x_0$ in $\Omega$ such that for any $x\in W,$ $\Phi_t(x)$ is defined for any $t\ge 0$ and $\Phi_t(x)\to x_0$ as $t\to +\infty$.

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