On the periodic solutions of perturbed 4D non-resonant systems

Abstract

We provide sufficient conditions for the existence of periodic solutions of a 4D non-resonant system perturbed by smooth or non-smooth functions. We apply these results to study the small amplitude periodic solutions of the non-linear planar double pendulum perturbed by smooth or non-smooth function.

Appendix: Basic concepts on Filippov systems

We say that a vector field $X:D\subset \mathbb {R}^n\rightarrow \mathbb {R}^n$ is piecewise continuous if its domain of definition D can be partitioned in a finite collection of connected, open and disjoint sets $D_i$, $i=1,\cdots ,k$, such that $\cup \overline{D_i}=D$, and the vector field $X\big |_{\overline{D}_i}$ is continuous for $i=1,\cdots ,k$.

We denote by $S_X\subset \partial D_1\cup \cdots \cup \partial D_k$ the set of points where the vector field X is discontinuous. By assumptions, the set $S_X$ has measure zero.

If $\Sigma \subset S_X$ is a manifold of codimension one, then $\Sigma$ can be decomposed as the union of the closure of the following three kind of regions (see Fig. 5):

$$\begin{aligned} \Sigma ^c= & {} \left\{ x\in \Sigma : (Xh)(Yh)(x)>0\right\} ;\\ \Sigma ^e= & {} \left\{ x\in \Sigma : (Xh)(x)>0 \text { e } (Yh)(x)<0\right\} ;\\ \Sigma ^s= & {} \left\{ x\in \Sigma : (Xh)(x)<0 \text { e } (Yh)(x)>0\right\} . \end{aligned}$$

Fig. 5

Crossing region $(\Sigma ^c)$, escaping region $(\Sigma ^e)$ and sliding region $(\Sigma ^s)$

For $p\in \Sigma ^e\cup \Sigma ^s$ we define the Sliding Vector Field as

$$\begin{aligned} Z_s(p)=\dfrac{1}{(Yh)(p)-(Xh)(p)}\left( (Yh)(Xh)(p)-(Xh)(Yh)(p)\right) . \end{aligned}$$

(35)

Consider the following equation

$$\begin{aligned} \dot{x}=X(x), \end{aligned}$$

(36)

where $X:D\subset \mathbb {R}^n\rightarrow \mathbb {R}^n$ is a piecewise continuous vector field. The local solution of the equation (36) passing through a point $p\in \Sigma$ is given by the Filippov convention:

• (i) for $p\in \Sigma ^c$ such that $(Xh)(p),(Yh)(p)>0$ and taking the origin of time at p, the trajectory is defined as $\varphi _Z(t,p)=\varphi _Y(t,p)$ for $t\in I_p\cap \{t<0\}$ and $\varphi _Z(t,p)=\varphi _X(t,p)$ for $t\in I_p\cap \{t>0\}$. For the case $(Xh)(p),(Yh)(p)<0$ the definition is the same reversing time;

• (ii) for $p\in \Sigma ^e\cup \Sigma ^s$ such that $Z_s(p)\ne 0$, $\varphi _Z(t,p)=\varphi _{Z_s}(t,p)$ for $t\in I_p\subset \mathbb {R}$.

Here $\varphi _W$ denotes the flow of a vector field W.

For more details about discontinuous differential equation see Filippov’s book [7].