Periodic perturbation of ODE

Question

Recently, I have read Khasminskii's book (Stochastic Stability of Differential Equations). In Section 3.5, the author mentioned that the following is well-known in the theory of ODEs.

If $x_0$ is an asymptotically stable equilibrium point of an autonomous system $\mathrm{d}x/\mathrm{d}t=F(x)$ and $f(t)$ is $\theta$-periodic, then for sufficiently small $\epsilon$ the system $\mathrm{d}x/\mathrm{d}t=F(x)+\epsilon f(t)$ has a $\theta$-periodic solution in a neighborhood of the equilibrium point.

But I didn't find references about the fact. Could you give me some references or hints? Thanks!

Answer 1

For simplicity, set the equilibrium to be $0$ and $A=DF(0)$. So $$ F(x) = Ax+O(x^2). $$ Thus the DE becomes $$ x'=Ax+\epsilon f(t)+ O(x^2).$$

Choose the constant $\delta$ to satisfy $$ \int_0^\theta e^{At}(f(t)-\delta)dt=0.$$

Note that the auxiliary DE is $$ y' =Ay+\epsilon f(t) $$ which has the solution $$\begin{eqnarray} y(t)&=&e^{At}y_0+\epsilon\int_0^t e^{A(t-s)}f(s)ds\\ &=&e^{At}y_0+\epsilon\int_0^t e^{As}(f(t-s)-\delta)ds+\epsilon\delta A^{-1}(e^{At}-1)\\ &=&\epsilon g(t)+e^{At}y_0+\epsilon\delta A^{-1}(e^{At}-1)=\epsilon g(t)+o(1) \end{eqnarray}$$ where $$ g(t)=\int_0^t e^{As}(f(t-s)-\delta)ds $$ for small $\epsilon>0$ and big $t$. Now we prove $g$ is periodic. In fact, $$\begin{eqnarray} g(\theta+t)&=&\int_0^{\theta+t} e^{As}(f(\theta+t-s)-\delta)ds\\ &=&\int_0^{\theta+t} e^{As}(f(t-s)-\delta)ds\\ &=&\int_0^{t} e^{As}(f(t-s)-\delta)ds+\int_t^{\theta+t} e^{As}(f(t-s)-\delta)ds\\ &=&g(t)+\int_0^{\theta} e^{A(s+t)}(f(s)-\delta)ds=g(t) \end{eqnarray}$$ So for small $\epsilon$ and big $t$, $y(t)$ is $\theta$-periodic and hence $x(t)$ is $\theta$-periodic solution in a neighborhood of the equilibrium point.

Answer 2

Actually, there is a much more general result. Assume that $F : \mathbb{R}^n \to \mathbb{R}^n$ and $G : \mathbb{R}^n \times \mathbb{R} \to \mathbb{R}^n$ are smooth vector valued functions with $G$ being $\theta-$periodic, i.e. $G(x, t + \theta) = G(x, t)$ any $x \in \mathbb{R}^n$ and for any $t \in \mathbb{R}$. Define the system of ODEs

$$\frac{dx}{dt} \,=\, F(x) + \varepsilon \,G(x, t)$$

where $\varepsilon$ is a small parameter. Without loss of generality assume that $F(0) = 0$. Denote by $DF(x)$ the $n \times n$ Jacobian matrix of $F(x)$ and by $DG(x,t)$ the $n \times n$ Jacobian matrix of $G(x, t)$ with respect to the variables $x$ only, while treating $t$ as a fixed variable. Set $DF(0) = A$.

Theorem. If $F(0) = 0$ and all eigenvalues of $A = DF(0)$ have non-zero real parts, then, for all small enough values of $\varepsilon$, there is a periodic solution of the system of ODEs $$\frac{dx}{dt} \,=\, F(x) + \varepsilon \,G(x, t)$$

Sketch of proof.

Given initial conditions $x \in \mathbb{R}^n$ and $\tau \in [0, \theta]$, let $$x(t) = \phi(t, \tau, x, \varepsilon)$$ be the unique solution to the initial value problem \begin{align} &\frac{dx}{dt} \,=\, F(x) + \varepsilon \,G(x, t)\\ &x(\tau) = x \end{align} i.e. $$\frac{\partial}{\partial t} \phi(t, \tau, x, \varepsilon) \,=\, F\big(\,\phi(t, \tau, x, \varepsilon)\,\big) + \varepsilon \,G\big(\,\phi(t, \tau, x, \varepsilon)\,, \,t\,\big)$$ and $$\phi(\tau, \tau, x, \varepsilon) = x$$

By the theorem on smooth dependence with respsect to parameters and initial conditions, the map \begin{align} &\phi \, : \, \mathbb{R}\times \mathbb{R} \times \mathbb{R}^n \times \mathbb{R} \, \to \, \mathbb{R}^n\\ &\phi \, : \, (t, \tau, x, \varepsilon) \,\mapsto \, \phi(t, \tau, x, \varepsilon) \end{align} exists, at least locally, and is differentiable. Since $x(t) = 0$ is a solution to the system, when $\varepsilon = 0$, then $\phi(t, \tau, 0, 0) = 0$ for all $t$ and $\tau$. Hence, there exist $a>0, \, b>0, $ and an open neighbourhood $U$ of $x=0$, such that for any $(\tau, x) \in (-b, b) \times U$ and for any $\varepsilon \in (-a, a)$ the solution $\phi(t, \tau, x, \varepsilon)$ of the initial value problem \begin{align} &\frac{dx}{dt} \,=\, F(x) + \varepsilon \,G(x, t)\\ &x(\tau) = x \end{align} exists for all $t \in (\tau-\delta, \tau+\theta + \delta)$ for some $\delta > 0$.

For a fixed $\tau$, define the map \begin{align} &P_{\tau} \, : \, U \times (-a, a) \to \mathbb{R}^n\\ &P_{\tau} \,:\, (x, \varepsilon) \,\mapsto\, P_{\tau}(x, \varepsilon) = \phi(\tau + \theta, \tau, x, \varepsilon) \end{align}

Then assume, that for a fixed $\varepsilon$ and for some $x \in U$, the solution $\phi(t, \tau, x, \varepsilon)$ is periodic, i.e. $\phi(t + \theta, \tau, x, \varepsilon) = \phi(t, \tau, x, \varepsilon)$ for all $t \in \mathbb{R}$. Then, for $t=\tau$ we have
$\phi(\tau + \theta, \tau, x, \varepsilon) = \phi(\tau, \tau, x, \varepsilon) = x$, which in the language of the map $P_{\tau}$ yields $P_{\tau}(x, \varepsilon) = x$ is a fixed point. Conversely, if for a fixed $\varepsilon$ and for some $x \in U$, we have $P_{\tau}$ yields $P_{\tau}(x, \varepsilon) = x$ which means $\phi(\tau + \theta, \tau, x, \varepsilon) = \phi(\tau, \tau, x, \varepsilon) = x$. Consider the solutions $\phi(t + \theta, \tau, x, \varepsilon)$ and $\phi(t, \tau, x, \varepsilon)$. Then, by the definition of $\phi$, \begin{align} &\frac{\partial}{\partial t} \phi(t, \tau, x, \varepsilon) \,=\, F\big(\,\phi(t, \tau, x, \varepsilon)\,\big) + \varepsilon \,G\big(\,\phi(t, \tau, x, \varepsilon)\,, \,t\,\big)\\ &\phi(\tau, \tau, x, \varepsilon) = x \end{align} and \begin{align} &\frac{\partial}{\partial t} \phi(t + \theta, \tau, x, \varepsilon) \,=\, F\big(\,\phi(t+\theta, \tau, x, \varepsilon)\,\big) + \varepsilon \,G\big(\,\phi(t+\theta, \tau, x, \varepsilon)\,, \,t + \theta\,\big) \\ &\phantom{\frac{\partial}{\partial t} \phi(t + \theta, \tau, x, \varepsilon) }\,=\, F\big(\,\phi(t+\theta, \tau, x, \varepsilon)\,\big) + \varepsilon \,G\big(\,\phi(t+\theta, \tau, x, \varepsilon)\,, \,t \big)\\ &\phi(\tau + \theta, \tau, x, \varepsilon) = \phi(\tau, \tau, x, \varepsilon) = x \end{align} Thus, we can see that both $\phi(t + \theta, \tau, x, \varepsilon)$ and $\phi(t, \tau, x, \varepsilon)$ solve the same initial value problem \begin{align} &\frac{dx}{dt} \,=\, F(x) + \varepsilon \,G(x, t)\\ &x(\tau) = x \end{align} (here the periodicity $G(x, t+\theta) = G(x, t)$ played a crucial role, without it, this statement is generally not true), which by the existence and uniqueness theorem of differentiable systems of ODEs implies that the two solutions coincide, i.e. $\phi(t + \theta, \tau, x, \varepsilon)\,=\,\phi(t, \tau, x, \varepsilon)$ for all $t \in \mathbb{R}.$

The bottom line is that the system has a periodic solution of period $\theta$ if and only if $P_{\tau}$ has a fixed point.

So our goal will be to prove that if $P_{\tau}(0, 0) = 0$ then there exists a continuous family of points $x(\varepsilon)$, for small enough $\varepsilon$ such that $P_{\tau}\big(x(\varepsilon), \varepsilon\big) = x(\varepsilon)$.

To that end, we will apply the implicit function theorem to the system of equations $$P_{\tau}(x,\,\varepsilon) - x \,=\, 0$$ Since by assumption $x=0$ is a solution to the system when $\varepsilon =0$, then $$P_{\tau}(0,\,0) - 0 \,=\, 0 - 0 = 0$$ To apply the implicit function theorem, one needs to prove that the matrix of derivatives $$DP_{\tau}(x, \varepsilon)_{|_{(x=0, \varepsilon=0)}} - I$$ with respect to the variables $x$ and evaluated at $x=0, \varepsilon=0$ is an invertиble matrix (matrix with no kernel). Since the map $P$ is directly linked to the solution map $\phi$, the derivatives of $P$ with respect to $x$ are directly linked to the derivatives of $\phi$ with respect to $x$. Differentiating a solution $\phi$ with respect to $x$ is achieved by differentiating the original system of ODEs with respect to $x$, which becomes \begin{align*} & \frac{d}{dt} D\phi \,=\, DF\big(\phi(t, \tau, x, \varepsilon)\big)\,D\phi \,+\, \varepsilon\,DG\big(\phi(t, \tau, x, \varepsilon), t\big)\, D\phi \end{align*} We need to set $x=0$ and $\varepsilon=0$, which gives us \begin{align*} & \frac{d}{dt} D\phi \,=\, DF\big(\phi(t, \tau, 0, 0)\big)\,D\phi \,+\, 0\,DG\big(\phi(t, \tau, 0, 0), t\big)\, D\phi \end{align*} Since by assumption $\phi(t, \tau, 0, 0) = 0$ for all $t$, we arrive at the linear system \begin{align*} & \frac{d}{dt} D\phi \,=\, DF(0)\,D\phi \end{align*} and since we denoted $DF(0)=A$, \begin{align*} & \frac{d}{dt} D\phi \,=\, A\,D\phi \end{align*} the solution of the latter linear system is $D\phi(t, \tau, 0, 0) \,=\, e^{(t-\tau)A}$ and consequently $$DP_{\tau}(0, 0) \,=\,D\phi(\tau + \theta, \tau, 0, 0) \,=\, e^{(\tau + \theta -\tau)A} \,=\, e^{\theta A}$$ Let us represent $A = U\,\Lambda\,U^{-1}$ where $\Lambda$ is the Jordan normal form of $A$. Then $$DP_{\tau}(x, \varepsilon)_{|_{(x=0, \varepsilon=0)}} - I\,=\, e^{\theta A} - I \,=\, e^{\theta U\,\Lambda\,U^{-1}} - U\,U^{-1} \,=\, U\,e^{\theta \Lambda}\,U^{-1} - U\,U^{-1} \,=\, U\big(\,e^{\theta \Lambda} - I\,\big)\,U^{-1}$$ Since $\Lambda$ is upper-triangular matrix, so is $e^{\theta\, \Lambda}$. By assumption, the eigenvalues of $A$ are all with non-zero real parts, which means that the same is true for $\Lambda$. Consequently, the eigenvalues, which are the diagonal elements of $e^{\theta\, \Lambda}$, are all with real parts not equal to $1$ and therefore the diagonal elements of $e^{\theta \Lambda} - I$ are all non-zero. Cоnsequently, $e^{\theta\, \Lambda} - I$ is an invertible matrix, which means that $A - I = U\big(\,e^{\theta \Lambda} - I\,\big)\,U^{-1}$ is also invertible matrix, as a product of invertible matrices. Hence, $DP_{\tau}(x, \varepsilon)_{|_{(x=0, \varepsilon=0)}} - I\,=\, e^{\theta A} - I$ is an invertible matrix, and therefore by the implicit function theorem, there is a small neighbourhood $(-a, a)$ such that for $\varepsilon \in (-a, a)$ there is solution $x(\varepsilon)$ such that $$P_{\tau}\big( x(\varepsilon), \, \varepsilon\big) - x(\varepsilon) \,=\, 0$$ and as discussed earlier the one parameter family of $\theta-$ periodic solutions $$\phi\big(\,t, \tau, x(\varepsilon), \varepsilon\,\big)$$ of the one parameter family of systems $$\frac{dx}{dt}\,=\,F(x) \,+\,\varepsilon\, G(x,\,t)$$