# Periodic perturbation of ODE

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!

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.

• Thus far you found a periodic solution for the perturbed (to linear) equation. Now you have to show that a periodic solution of the original equation is only a perturbation of the found solution away. Commented May 9 at 5:30

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