Complex dynamics, hidden attractors and continuous approximation of a fractional-order hyperchaotic PWC system

Abstract

In this paper, a continuous approximation to studying a class of PWC systems of fractional-order is presented. Some known results of set-valued analysis and differential inclusions are utilized. The example of a hyperchaotic PWC system of fractional order is analyzed. It is found that without equilibria, the system has hidden attractors.

Appendices

Basic notions and results

Because the set-valued property of F in (6) is generated by $S_i$, which are real functions, the notions and results presented here are considered in $\mathbb {R}$, for the case of $n=1$, but they are also valid in the general cases of $n>1$.

The graph of a set-valued function F is defined as follows:

$$\begin{aligned} Graph(F):=\{(x,y)\in \mathbb {R}\times \mathbb {R}, ~y\in F(x)\}. \end{aligned}$$

Remark A.1

Due to the symmetric interpretation of a set-valued function as a graph (see, e.g., [43]), a set-valued function satisfies a property if and only if its graph satisfies it. For instance, a set-valued function is closed or convex if and only if its graph is closed or convex.

Definition A.1

A set-valued function F is upper semicontinuous (u.s.c.) at $x^0\in \mathbb {R}$ if, for any open set B containing $F(x^0)$, there exists a neighborhood A of $x^0$ such that $F(A)\in B$.

F is u.s.c. if it is so at every $x^0\in \mathbb {R}$, which means that the graph of F is closed.

Definition A.2

A generalized solution to (4) is an absolutely continuous function $x:[0,T]\rightarrow \mathbb {R}$, satisfying (4) for a.a. $t\in [0,T]$.

Definition A.3

A single-valued function $h:\mathbb {R}\rightarrow \mathbb {R}$ is called an approximation (selection) of the set-valued function F, if

$$\begin{aligned} h(x)\in F(x),~~\forall x\in \mathbb {R}. \end{aligned}$$

Generally, a set-valued function admits (infinitely) many approximations.

Theorem A.1

(Cellina’s Theorem  [43, p. 84] and [44, p. 358]) Let $F:\mathbb {R}\rightrightarrows \mathbb {R}$ have convex values F(x), $x\in X$. Then, for every $\varepsilon >0$, there exists a single-valued continuous $\varepsilon$-approximation of F.

See Fig. 4c for the case of the ${{\mathrm{Sgn}}}$ function.

Theorem A.2

(Weiestrass Approximation Theorem)  Suppose f is a continuous real-valued function defined on the real interval [a, b]. Then, for every $\varepsilon > 0$, there exists a polynomial p such that for all $x\in [a, b]$, $|f(x) - p(x)| < \varepsilon$.

Explicit solutions of IVP

Because the considered system (3) is actually PWL in each of the open half spaces $\varOmega _\pm$, one can find explicit solutions (solutions existence is ensured by Theorem 1).

Indeed, (3) can be written as

$$\begin{aligned} D^q_*x=M_{\pm }x+m,\quad x\in \varOmega _\pm , \end{aligned}$$

(B.1)

where

$$\begin{aligned} M_{\pm }=\begin{pmatrix} -\,1 &{} 1 &{} 0 &{} 0 \\ 0 &{} 0 &{} \mp \,1 &{} 1 \\ \pm \,1 &{} 0 &{} 0 &{} 0 \\ 0 &{} -\,b &{} 0 &{} 0 \end{pmatrix}, \quad m=-\,ae_3,\quad e_3=\begin{pmatrix} 0\\ 0\\ 1\\ 0 \end{pmatrix}, \end{aligned}$$

in which (3) is piecewise affine, so $M_+x+m=g(x)+A(x)s(x)$ for $x\in \varOmega _+$ and $M_-x+m=g(x)+A(x)s(x)$ for $x\in \varOmega _-$. Then, it follows from [45] that the solution of (B.1) with $x(0)=x_0$ on each $\varOmega _\pm$ is given by

$$\begin{aligned} x(t)= & {} E_q(t^qM_{\pm })x_0\nonumber \\&-\,a\int _0^t(t-s)^{q-1}E_{q,q}((t-s)^qM_{\pm })e_3\hbox {d}s,\nonumber \\ \end{aligned}$$

(B.2)

where the Mittag–Leffler matrix functions $E_{\alpha }(M_{\pm })$ and $E_{\alpha ,\beta }(M_{\pm })$ are defined as [46, p. 56]

$$\begin{aligned}&E_{\alpha ,\beta }(M_{\pm })=\sum _{k=0}^\infty \frac{M_{\pm }^k}{\varGamma (\alpha k+\beta )},\\&E_\alpha (M_{\pm }) = E_{\alpha ,1}(M_{\pm }). \end{aligned}$$

Next, using [46, formula (4.4.4)], it follows from (B.2) that

$$\begin{aligned} x(t)= & {} E_q(t^qM_{\pm })x_0\nonumber \\&-\,a\int _0^t(t-s)^{q-1}E_{q,q}((t-s)^qM_{\pm })e_3\hbox {d}s\nonumber \\= & {} E_q(t^qM_{\pm })x_0-a\int _0^ts^{q-1}E_{q,q}(s^qM_{\pm })e_3\hbox {d}s\nonumber \\= & {} E_q(t^qM_{\pm })x_0-at^qE_{q,q+1}(t^qM_{\pm })e_3, \end{aligned}$$

(B.3)

and

$$\begin{aligned} e_3=\begin{pmatrix} 0\\ 0\\ 1\\ 0 \end{pmatrix} . \end{aligned}$$

The last formula of (B.3) gives explicit solutions of (12) on each $\varOmega _\pm$, respectively.

Periodic solutions for Caputo derivative with lower limit at $-\,\infty$

Consider the following simple scalar linear FDE with Caputo derivative with the lower limit at $-\infty$ as

$$\begin{aligned} D^q_{-\infty }x(t)+\beta x(t)=\gamma \cos (\varOmega t+\alpha ), \end{aligned}$$

(C.1)

where $\alpha ,\beta ,\gamma ,\varOmega \in \mathbb {R}$. Note that [2]

$$\begin{aligned} D^q_{-\infty }x(t)=\frac{1}{\varGamma (1-q)}\int _{-\infty }^t(t-s)^{-q}x'(s)\hbox {d}s,\quad t\in \mathbb {R}. \end{aligned}$$

To find a solution of (C.1) in the form of

$$\begin{aligned} x(t)=A\cos \varOmega t+B\sin \varOmega t. \end{aligned}$$

(C.2)

one can use formulas [28, (24), (26)] to derive

$$\begin{aligned}&D^q_{-\infty }x(t)\nonumber \\&\quad =\Big (A\varOmega ^q\cos \frac{\pi q}{2}+B\varOmega ^q\sin \frac{\pi q}{2}\Big )\cos \varOmega t\nonumber \\&\qquad +\Big (B\varOmega ^q\cos \frac{\pi q}{2}-A\varOmega ^q\sin \frac{\pi q}{2}\Big )\sin \varOmega t. \end{aligned}$$

(C.3)

Inserting (C.3) into (C.1) yields

$$\begin{aligned} A\varOmega ^q\cos \frac{\pi q}{2}+B\varOmega ^q\sin \frac{\pi q}{2}+\beta A= & {} \gamma \cos \alpha ,\\ B\varOmega ^q\cos \frac{\pi q}{2}-A\varOmega ^q\sin \frac{\pi q}{2}+\beta B= & {} -\gamma \sin \alpha , \end{aligned}$$

which has a solution

$$\begin{aligned} A= & {} \frac{\gamma \left( \beta \cos \alpha +\varOmega ^q\cos \left( \alpha -\frac{\pi q}{2}\right) \right) }{\beta ^2+2\beta \varOmega ^q\cos \frac{\pi q}{2}+\varOmega ^{2q}},\nonumber \\ B= & {} -\frac{\gamma \left( \beta \sin \alpha +\varOmega ^q\sin \left( \alpha -\frac{\pi q}{2}\right) \right) }{\beta ^2+2\beta \varOmega ^q\cos \frac{\pi q}{2}+\varOmega ^{2 q}}. \end{aligned}$$

(C.4)

Inserting (C.4) into (C.2), one obtains

$$\begin{aligned} x(t)=\frac{\gamma \left( \beta \cos (\varOmega t+\alpha )+\varOmega ^q\cos \left( \varOmega t+\alpha -\frac{\pi q}{2}\right) \right) }{\beta ^2+2\beta \varOmega ^q\cos \frac{\pi q}{2}+\varOmega ^{2 q}}. \end{aligned}$$