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 \({\mathbf{R}}^{k+1}\),

\begin{array}{rl}u(n+1)& =\left(\begin{array}{c}{u}_{2}(n)\\ {u}_{3}(n)\\ \vdots \\ {u}_{k+1}(n)\\ \alpha {u}_{k+1}(n)+\beta sin[{u}_{k+1}(n)-\gamma {u}_{1}(n)]\end{array}\right)\\ & :=F(u(n)),\end{array}

(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

DF(O)={\left(\begin{array}{ccccc}0& 1& 0& \cdots & 0\\ 0& 0& 1& \cdots & 0\\ \cdot \cdot & \cdot \cdot & \cdot \cdot & \cdots & \cdot \cdot \\ 0& 0& 0& \cdots & 1\\ -\beta \gamma & 0& 0& \cdots & \alpha +\beta \end{array}\right)}_{(k+1)\times (k+1).}

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.