Let \(u_{j}(n):=x(n+j-k-1)\) for \(1\leq j\leq k+1\), then (3) is changed into a \((k+1)\)-dimensional system on \({\mathbf{R}}^{k+1}\),
(4)
where \(u=(u_{1},u_{2},\ldots,u_{k+1})^{T}\in{\mathbf{R}}^{k+1}\).
System (4) is called the system induced by (3) in \({\mathbf{R}}^{k+1}\). It is clear that a solution \(\{x(n-k),\ldots,x(n)\}_{n=1}^{\infty}\) of (3) corresponds to a solution \(\{u(n)\}_{n=1}^{\infty}\) of system (4), where the initial condition \(\{x(-k),\ldots,x(0)\}\) of (3) corresponds to an initial condition \(u(0)=(u_{1}(0),\ldots,u_{k+1}(0))^{T}\in{\mathbf{R}}^{k+1}\) of system (4). Hence, we can study the dynamical behavior of (3) by studying that of its induced system (4) in \({\mathbf{R}}^{k+1}\). So, we call (3) is chaotic in the sense of Devaney (or Li-Yorke) on \(V\subset{\mathbf{R}}^{k+1}\) if its induced system (4) is chaotic in the sense of Devaney (or Li-Yorke) on \(V\subset {\mathbf{R}}^{k+1}\).
Now, we state the main result of this paper as the following theorem.
Theorem 1
There exists a constant
\(\beta _{0}>0\)
such that for arbitrary
β
satisfying
\(\vert \beta \vert >\beta _{0}\)
and for some
γ
satisfying
$$ \vert \gamma \vert >\frac{1+\vert \alpha +\beta \vert }{\vert \beta \vert }, $$
(5)
system (4), and consequently (3), is chaotic in the sense of both Devaney and Li-Yorke.
Proof
Lemma 1 will be used to prove this theorem. Therefore, we only need to show that the map F of system (4) satisfies all the assumptions in Lemma 1.
It is clear that \(O:=(0,\ldots,0)^{T}\in{\mathbf{R}}^{k+1}\) is always a fixed point of system (4), and other fixed points \(P:=(x_{0},\ldots,x_{0})^{T}\in{\mathbf{R}}^{k+1}\) satisfy
$$\sin\bigl[(1-\gamma )x\bigr]=\frac{(1-\alpha )x}{\beta }. $$
For simplicity, we will only prove that the fixed point O may be a regular and nondegenerate snap-back repeller of the map F when parameters satisfy the conditions in Theorem 1.
For simplifying the proof and convenience, γ is taken as an integer and satisfies condition (5) throughout the proof.
Firstly, it is to show that O is an expanding fixed point of F in \({\mathbf{R}}^{k+1}\) under condition (5). It is obvious that F is continuously differentiable in \({\mathbf{R}}^{k+1}\), and its Jacobian matrix at O is
The characteristic equation of \(\operatorname {DF}(O)\) is
$$\lambda^{k+1}-(\alpha +\beta )\lambda^{k}+\beta \gamma =0, $$
from which we obtain the result that all the eigenvalues of \(\operatorname {DF}(O)\) have absolute values larger than 1 under condition (5). If it is not true, then there will exist at least an eigenvalue \(\lambda_{0}\) of \(\operatorname {DF}(O)\) satisfies \(\vert \lambda_{0}\vert \leq1\). So, we can obtain the following contradiction:
$$\begin{aligned} 1+\vert \alpha +\beta \vert &\geq\bigl\vert \lambda_{0}^{k+1}\bigr\vert +\bigl\vert (\alpha +\beta )\lambda_{0}^{k}\bigr\vert \\ & \geq\bigl\vert \lambda_{0}^{k+1}-(\alpha +\beta )\lambda_{0}^{k}\bigr\vert =\vert -\beta \gamma \vert >1+\vert \alpha +\beta \vert . \end{aligned}$$
Therefore, we find that O is an expanding fixed point of F from the first condition of Lemma 1, that is,
$$\bigl\Vert F(x)-F(y)\bigr\Vert ^{*}\geq \mu \Vert x-y\Vert ^{*}, \quad \forall x,y\in \bar{B}_{r}(O), $$
where \(r>0\) is a constant, \(\Vert \cdot \Vert ^{*}\) is some norm in \({\mathbf{R}}^{k+1}\), and \(\mu >1\) is an expanding coefficient of F in \(\bar{B}_{r}(O)\).
Secondly, one is to prove that O is a snap-back repeller of the map F. Let \(W\subset\bar{B}_{r}(O)\) be an arbitrary neighborhood of O in \({\mathbf{R}}^{k+1}\). Then we can obtain a small interval \(U\subset{\mathbf{R}}\) containing 0 such that \(\underbrace{U\times U\times\cdots\times U}_{k+1}\subset W\). Now, one is to show that there exists a point \(O_{0}\in W\) such that \(O_{0}\neq O\) and
When \(k=1\), we can achieve a positive constant \(\beta _{1}\) such that for arbitrary \(\vert \beta \vert >\beta _{1}\), there exist two points \(x_{1},x_{2}\in U\) satisfying
$$ \begin{cases} \beta \sin(\gamma x_{2})=-\alpha \pi,\\ \beta \sin(x_{2}-\gamma x_{1})=\pi-\alpha x_{2}. \end{cases} $$
(6)
Let \(O_{0}=(x_{1},x_{2})^{T}\in{\mathbf{R}}^{2}\), it follows that \(O_{0}\in U\times U\subset W\) with \(O_{0}\neq O\) for arbitrary \(\vert \beta \vert >\beta _{1}\). From (6), we obtain the result that \(F(O_{0})=(x_{2},\pi)^{T}\), \(F^{2}(O_{0})=(\pi,0)^{T}\), \(F^{3}(O_{0})=O\).
When \(k>1\), we can also achieve a positive constant \(\beta _{2}\) such that for arbitrary \(\vert \beta \vert >\beta _{2}\), there exist two points \(x_{1},x_{2}\in U\) satisfying
$$ \begin{cases} \beta \sin(\gamma x_{1})=-\pi,\\ \beta \sin(\gamma x_{2})=-\alpha \pi. \end{cases} $$
(7)
Let \(O_{0}=(x_{1},x_{2},0,\ldots,0)^{T}\in{\mathbf{R}}^{k+1}\), it also follows that \(O_{0}\in\underbrace{U\times U\times\cdots\times U}_{k+1}\subset W\) with \(O_{0}\neq O\) for arbitrary \(\vert \beta \vert >\beta _{2}\). From (7), we also obtain the result that \(F(O_{0})=(x_{2},0,\ldots,\pi)^{T}\), \(F^{j}(O_{0})=(0,\ldots,0,\underbrace{\pi,0,\ldots,0}_{j})^{T}\) for \(2\leq j\leq k+1\), and \(F^{k+2}(O_{0})=O\).
Set \(\beta _{0}:=\max\{\beta _{1},\beta _{2}\}\). From the above discussion, it follows that for arbitrary β satisfying \(\vert \beta \vert >\beta _{0}\), there exists a point \(O_{0}\in W\) satisfying \(O_{0}\neq O\) and \(F^{k+2}{(O_{0})}=O\). Therefore, O is a snap-back repeller of F for arbitrary β satisfying \(\vert \beta \vert >\beta _{0}\).
Thirdly, one is to prove that for arbitrary \(\vert \beta \vert >\beta _{0}\), the following holds:
$$\det \operatorname {DF}(O_{j})\neq0,\quad 0\leq j\leq k+1, $$
where \(O_{j}:=F(O_{j-1})\) for \(1\leq j\leq k+1\). The existence of the Jacobian matrices of F at \(O_{j}\) (\(0\leq j\leq k+1\)) is because F is continuously differentiable in \({\mathbf{R}}^{k+1}\).
It is easy to conclude that for arbitrary \(u=(u_{1},\ldots,u_{k+1})^{T} \in{\mathbf{R}}^{k+1}\), the following holds:
$$ \det \operatorname {DF}(u)=(-1)^{k+1}\beta \gamma \cos(u_{k+1}-\gamma u_{1}). $$
(8)
When \(k=1\), it follows that \(O_{0}=(x_{1},x_{2})^{T}\), \(O_{1}=(x_{2},\pi)^{T}\), \(O_{2}=(\pi,0)^{T}\in{\mathbf{R}}^{2}\). From (6), we get \(\sin(\gamma x_{2})\neq1\) and \(\sin(x_{2}-\gamma x_{1})\neq1\) for arbitrary \(\vert \beta \vert >\beta _{0}\). Together with (8), we get the following for arbitrary \(\vert \beta \vert >\beta _{0}\):
$$\begin{aligned} &\det \operatorname {DF}(O_{0})=\beta \gamma \cos(x_{2}-\gamma x_{1})\neq0, \\ &\det \operatorname {DF}(O_{1})=-\beta \gamma \cos(\gamma x_{2})\neq0, \\ &\det \operatorname {DF}(O_{2})=\beta \gamma \cos(\gamma \pi)=\pm \beta \gamma \neq0. \end{aligned}$$
When \(k>1\), it follows that \(O_{0}=(x_{1},x_{2},0,\ldots,0)^{T}\), \(O_{1}=(x_{2},0,\ldots,0,\pi)^{T}\), and \(O_{j}=(0,\ldots,0,\underbrace{\pi,0,\ldots,0}_{j})^{T}\in {\mathbf{R}}^{k+1}\) for \(2\leq j\leq k+1\). Similarly, it follows from (7) that \(\sin(\gamma x_{1})\neq1\) and \(\sin(\gamma x_{2})\neq1\) for arbitrary \(\vert \beta \vert >\beta _{0}\). Consequently, the following hold for arbitrary \(\vert \beta \vert >\beta _{0}\):
$$\begin{aligned}& \det \operatorname {DF}(O_{0})=(-1)^{k+1}\beta \gamma \cos(\gamma x_{1})\neq0, \\& \det \operatorname {DF}(O_{1})=(-1)^{k}\beta \gamma \cos(\gamma x_{2})\neq0, \\& \det \operatorname {DF}(O_{j})=(-1)^{k+1}\beta \gamma \neq0, \quad \mbox{for } 2\leq j\leq k, \\& \det \operatorname {DF}(O_{k+1})=(-1)^{k+1}\beta \gamma \cos(\gamma \pi)=\pm \beta \gamma \neq0. \end{aligned}$$
In summary, the map F satisfies all the assumptions in Lemma 1. Consequently, system (4), i.e., (3), is chaotic in the sense of both Devaney and Li-Yorke. This completes the proof. □
Remark 5
For simplifying the proof of Theorem 1, the parameter γ is taken as an integer. It should be pointed out that γ may be taken as other values such that system (4) is chaotic. In addition, it follows from the above proof that there exists a constant \(\beta _{0}>0\) such that for arbitrary \(\vert \beta \vert >\beta _{0}\), system (4) is chaotic in the sense of both Devaney and Li-Yorke. However, there are few methods to determine the concrete expanding area of a fixed point in the literature. So it is not easy to get the particular value \(\beta _{0}\). In practical problems, we can take the parameter \(\vert \beta \vert \) large enough such that (6) or (7) in the proof of Theorem 1 are satisfied.
For illustrating the theoretical result, we present two computer simulations of system (4), from which we can see that system (4), i.e., (3) indeed has complex dynamical behaviors. The parameters are taken as \(\alpha =0.1\), \(\beta =200\), \(\gamma =6\), \(k=1,2\). From the proof of Theorem 1, we see that O is an expanding fixed point of the map F for \(\vert \gamma \vert =6>[1+\vert \alpha +\beta \vert ]/\vert \beta \vert =1.0055\). It is also easy to obtain the result that there exist two pairs of points \(x_{1}\approx-0.1388\), \(x_{2}\approx-0.2618\) satisfying (6) when \(k=1\), and \(x_{1}\approx-0.0026\), \(x_{2}\approx-0.2618\) satisfying (7) when \(k>1\). Therefore, O is a regular and nondegenerate snap-back repeller of the map F. Two simulation results are given in Figures 1 and 2 for \(k=1,2\), which exhibit complex dynamical behaviors of the system.