Entrainment by Chaos

Abstract

A new phenomenon, the entrainment of limit cycles by chaos, which results from the appearance of cyclic irregular behavior, is discussed. In this study, sensitivity is considered as the main ingredient of chaos to be captured, and the period-doubling cascade is chosen for extension. Theoretical results are supported by simulations and discussions regarding Chua’s oscillators, entrainment of toroidal attractors by chaos, synchronization, and controlling problems. It is demonstrated that the entrainment cannot be considered as generalized synchronization of chaotic systems.

Appendix: Motion Near the Limit Cycle

In this part, we provide the needed information from the proof of the Andronov–Witt theorem (Farkas 2010) and clarify the decay of the solutions regarding the initial value.

Without loss of generality, let us assume that $p(0)=0$ and $p'(0)=\left( \bar{p}_1,0,0,\ldots ,0 \right)$ for some positive number $\bar{p}_1$.

According to our assumption that system (2.7) admits the number $1$ as a simple characteristic multiplier and the remaining $n-1$ characteristic multipliers are smaller than one in modulus, system (2.7) has a real fundamental matrix $\Phi (t)$ of the form ${\small \Phi (t)=P(t) \left( \begin{array}{lll} 1 &{} 0\\ 0 &{} e^{B_1t} \end{array} \right) }$, where $P(t)$ is a regular, continuously differentiable $T$-periodic matrix and $B_1$ is an $(n-1) \times (n-1)$ matrix all of whose eigenvalues have negative real parts.

We emphasize that for an arbitrary solution $u(t)$ of Eq. (2.2), the differential equation satisfied by the function $z(t)=u(t)-p(t)$ is

$$\begin{aligned} z'=A(t)z + \varphi (t,z), \end{aligned}$$

(9.1)

where $A(t)= \frac{\partial f(p(t))}{\partial u}$ and $\varphi (t,z)=f(p(t)+z)-f(p(t))-A(t)z$. It is clear that $\varphi (t+T,z)=\varphi (t,z)$ and $\varphi (t,0)=\varphi _z (t,0)=0$ for all $t \in \mathbb R_+$.

Since $\varphi _z(t,z)=o(1)$ as $z \rightarrow 0$ uniformly in $t \in \mathbb R_+$, there exist numbers $L_{\varphi }>0$ and $\widetilde{\delta }(L_{\varphi })>0$ such that if $\left\| z_1\right\| < \widetilde{\delta }(L_{\varphi }),\,\left\| z_2\right\| < \widetilde{\delta }(L_{\varphi })$, then the inequality $\left\| \varphi (t,z_1)-\varphi (t,z_2)\right\| \le L_{\varphi } \left\| z_1-z_2\right\|$ holds uniformly for $t \in \mathbb R_+$.

Suppose that $a=\left( 0,a_2,a_3,\ldots ,a_n \right)$ is an $n$-dimensional vector that is orthogonal to $p'(0)$. There exist positive numbers $K_1$ and $\alpha$ such that $\left\| \Phi (t)a\right\| \le K_1\left\| a\right\| e^{-\alpha t}$ for all $t\in \mathbb R_+$. Moreover, if $\left\| a \right\| < \widetilde{\delta }(L_{\varphi })/(2K_1)$, then a solution $z(t,a)$ of (9.1) exists on $[0,\infty )$ and satisfies the following inequality:

$$\begin{aligned} \left\| z(t,a)\right\| \le 2K_1\left\| a\right\| e^{-\alpha t /2}, ~~t\ge 0. \end{aligned}$$

(9.2)

A solution $\zeta (t,\zeta _0)$ to (2.2) satisfies the relation $\zeta (t,\zeta _0)=z(t,a)+p(t)$, where $z(t,a)$ is a solution to (9.1) with $z(0,a)=\zeta _0$. Additionally, the equation

$$\begin{aligned} \zeta _0= P(0)a-\widetilde{h}(a) \end{aligned}$$

(9.3)

holds, where $\widetilde{h}(a)=\left( \widetilde{h}_1(a_2,\ldots ,a_n),0,\ldots ,0\right)$, for some continuously differentiable function $\widetilde{h}_1$, and $\widetilde{h}(a)=o(\left\| a\right\| )$.

Suppose that $\zeta _0=(\zeta _1^0,\zeta _2^0,\ldots ,\zeta _n^0)$ and $p_{ij}$ are the coordinates of the matrix $P(0)$, where $i,j=1,2,\ldots ,n$. Equation (9.3) is equivalent to

$$\begin{aligned} \zeta _1^0+\displaystyle \sum _{i=2}^n q_i \zeta _i^0 - h(\eta _2^0,\zeta _3^0,\ldots ,\zeta _n^0)=0, \end{aligned}$$

(9.4)

where $q_i, i=2,\ldots ,n$, are constants and $h$ is a continuously differentiable function such that

$$\begin{aligned} h(\zeta _2^0,\ldots ,\zeta _n^0)=o\left( \left( \, \sum _{i=2}^n (\zeta _i^0)^2 \right) ^{1/2} \right) . \end{aligned}$$

Denote by $S$ the $(n-1)$ dimensional, $C^1$ manifold determined by the equation

$$\begin{aligned} x_1+\displaystyle \sum _{i=2}^{n} q_ix_i-h(x_2,x_3,\ldots ,x_n)=0. \end{aligned}$$

(9.5)

The hypersurface $S$ crosses the orbit $\gamma$, which is defined by Eq. (2.6), transversally, so that for any solution $\zeta (t,\zeta _0)$ starting on this initial manifold we have $\left\| \zeta (t,\zeta _0)-p(t)\right\| \rightarrow 0$ exponentially as $t \rightarrow \infty$.

We will now prove that for each number $l\in (0,1)$ there exists a natural number $n_0=n_0(l)$ such that if $\zeta _0$ belongs to $S$, then

$$\begin{aligned} \left\| \zeta (n_0T,\zeta _0)\right\| \le l\left\| \zeta _0\right\| . \end{aligned}$$

(9.6)

Let $\overline{\epsilon }=1/\left( 2\left\| P^{-1}(0)\right\| \right)$. It is possible to find a number $\overline{\delta } \left( \overline{\epsilon }\right) >0$ such that if

$$\begin{aligned} \left\| a\right\| < \displaystyle \min \left\{ \widetilde{\delta }(L_{\varphi })/(2K_1), \overline{\delta } \left( \overline{\epsilon }\right) \right\} , \end{aligned}$$

then the inequality

$$\begin{aligned} \left\| \widetilde{h}(a)\right\| < \overline{\epsilon } \left\| a\right\| \end{aligned}$$

(9.7)

is valid.

Let us fix a solution $\zeta (t,\zeta _0)$ such that $\zeta _0$ belongs to $S$. In the case $\left\| a\right\| < \displaystyle \min \left\{ \widetilde{\delta }(L_{\varphi })/(2K_1), \overline{\delta } \left( \overline{\epsilon }\right) \right\}$, taking advantage of (9.3) and (9.7) one can find that $\left\| a\right\| \le 2 \left\| P^{-1}(0)\right\| \left\| \zeta _0 \right\|$, and, according to (9.2), we have

$$\begin{aligned} \left\| \zeta (t,\zeta _0)-p(t)\right\| \le 4K_1\left\| P^{-1}(0)\right\| \left\| \zeta _0 \right\| e^{-\alpha t/2}, ~t\ge 0. \end{aligned}$$

(9.8)

Let us fix an arbitrary number $l \in (0,1)$. There exists a natural number $n_0=n_0(l)$ such that $4K_1\left\| P^{-1}(0)\right\| e^{-\alpha T n_0/2}<l$. Making use of (9.8) we obtain that $\left\| \zeta (n_0T,\zeta _0)-p(n_0T)\right\| < l \left\| \zeta _0 \right\|$. Since $p(n_0T)=0$, inequality (9.6) holds.