Bifurcation and chaos in simple discontinuous systems separated by a hypersurface

1.   Introduction

In the last two decades, the fractional difference equations have recently received considerable attention in many fields of science and engineering, see ^[1,2,3,4] and the references therein. On the other hand, the $q$-difference equations have numerous applications in diverse fields in recent years and has gained intensive interest ^[5,6,7,8,9]. It is well know that the $q$-fractional difference equations can be used as a bridge between fractional difference equations and $q$-difference equations, many papers have been published on this research direction, see ^{[10,11,12,13,14,15]} for examples. We recommend the monograph ^[16] and the papers cited therein.

For $0 < q < 1$, we define the time scale $\mathbb{T}_q = \{q^n : n \in \mathbb{Z}\} \cup \{0\}$, where $\mathbb{Z}$ is the set of integers. For $a = q^{n_0}$ and $n_0 \in \mathbb{Z}$, we denote $\mathbb{T}_a = [a, \infty)_q = \{q^{-i} a : i = 0, 1, 2, ...\}$.

In ^[17], Abdeljawad et.al generalized the $q$-fractional Gronwall-type inequality in ^[18], they obtained the following $q$-fractional Gronwall-type inequality.

Theorem 1.1 (^[17]). Let $\alpha > 0$, $u$ and $\nu$ be nonnegative functions and $w(t)$ be nonnegative and nondecreasing function for $t \in [a, \infty)_q$ such that $w(t) \le M$ where $M$ is a constant. If

then

Based on the above result, Abdeljawad et al. investigated the following nonlinear delay $q$-fractional difference system:

where ${\rm{}}_q C_a^{\alpha}$ means the Caputo fractional difference of order $\alpha \in (0, 1)$, $\bar{\mathbb{I}}_{\tau} = \{\tau a, q^{-1} \tau a, q^{-2} \tau a, ..., a\}$, $\tau = q^d \in \mathbb{T}_q$ with $d \in \mathbb{N}_0 = \{0, 1, 2, ... \}$.

Remark 1.1. The domain of $t$ in (1.2) is inaccurate, please see the reference ^[19].

In ^[20], Sheng and Jiang gave the following extended form of the fractional Gronwall inequality :

Theorem 1.2 (^[20]). Suppose $\alpha > 0$, $\beta > 0$, $a(t)$ is a nonnegative function locally integrable on $[0, T)$, $\tilde{g}(t)$, and $\bar{g}(t)$ are nonnegative, nondecreasing, continuous functions defined on $[0, T)$; $\tilde{g}(t) \le \tilde{M}$, $\bar{g}(t) \le \bar{M}$, where $\tilde{M}$ and $\bar{M}$ are constants. Suppose $x(t)$ is a nonnegative and locally integrable on $[0, T)$ with

Then

where $t \in [0, T)$, $g(t) = \tilde{g}(t) + \bar{g}(t)$ and $C_n^k = \frac{n(n-1) \cdots (n-k+1)}{k!}$.

Corollary 1.3 ^[20] Under the hypothesis of Theorem 1.2, let $a(t)$ be a nondecreasing function on $[0, T)$. Then

where $\gamma = \min\{\alpha, \beta\}$, $E_{\gamma}$ is the Mittag-Leffler function defined by $E_{\gamma}(z) = \sum \limits_{k = 0}^{\infty} \frac{z^k}{\Gamma(k \gamma + 1)}$.

Finite-time stability is a more practical method which is much valuable to analyze the transient behavior of nature of a system within a finite interval of time. It has been widely studied of integer differential systems. In recent decades, the finite-time stability analysis of fractional differential systems has received considerable attention, for instance ^{[21,22,23,24,25]} and the references therein. In ^[26], Du and Jia studied the finite-time stability of a class of nonlinear fractional delay difference systems by using a new discrete Gronwall inequality and Jensen inequality. Recently, Du and Jia in ^[27] obtained a criterion on finite time stability of fractional delay difference system with constant coefficients by virtue of a discrete delayed Mittag-Leffler matrix function approach. In ^[28], Ma and Sun investigated the finite-time stability of a class of fractional q-difference equations with time-delay by utilizing the proposed delayed $q$-Mittag-Leffler type matrix and generalized $q$-Gronwall inequality, respectively. Based on the generalized fractional $(q, h)$-Gronwall inequality, Du and Jia in ^[19] derived the finite-time stability criterion of nonlinear fractional delay $(q, h)$-difference systems.

Motivated by the above works, we will extend the $q$-fractional Gronwall-type inequality (Theorem 1.1) to the spreading form of the $q$-fractional Gronwall inequality. As applications, we consider the existence and uniqueness of the solution of the following nonlinear delay $q$-fractional difference damped system :

where $[a, b)_q = [a, b) \cap \mathbb{T}_a$, $b \in \mathbb{T}_a$, $\mathbb{I}_{\tau} = \{q \tau a, \tau a, q^{-1} \tau a, q^{-2} \tau a, ..., a\}$, $\tau = q^d \in \mathbb{T}_q$ with $d \in \mathbb{N}_0 = \{0, 1, 2, ... \}$, ${\rm{}}_q C_a^{\alpha}$ and ${\rm{}}_q C_a^{\beta}$ mean the Caputo fractional difference of order $\alpha \in (1, 2)$ and order $\beta \in (0, 1)$, respectively, and the constant matrices $A_0$, $B_0$ and $B_1$ are of appropriate dimensions. Moreover, a novel criterion of finite-time stability criterion of (1.5) is established. We generalized the main results of ^[17] in this paper.

The organization of this paper is given as follows: In Section 2, we give some notations, definitions and preliminaries. Section 3 is devoted to proving a spreading form of the $q$-fractional Gronwall inequality. In Section 4, the existence and uniqueness of the solution of system (1.5) are given and proved, and the finite-time stability theorem of nonlinear delay $q$-fractional difference damped system is obtained. In Section 5, an example is given to illustrate our theoretical result. Finally, the paper is concluded in Section 6.

2.   Preliminaries

In this section, we provided some basic definitions and lemmas which are used in the sequel.

Let $f : \mathbb{T}_q \to \mathbb{R}$ ($q \in (0, 1)$), the nabla $q$-derivative of $f$ is defined by Thabet et al. as follows:

and $q$-derivatives of higher order by

The nabla $q$-integral of $f$ has the following form

and for $0 \le a \in \mathbb{T}_q$

The definition of the $q$-factorial function for a nonpositive integer $\alpha$ is given by

For a function $f : \mathbb{T}_q \to \mathbb{R}$, the left $q$-fractional integral ${\rm{}}_q \nabla_a^{-\alpha}$ of order $\alpha \not = 0, -1, -2, ...$ and starting at $0 < a \in \mathbb{T}_q$ is defined by

where

The left $q$-fractional derivative ${\rm{}}_q \nabla_a^{\beta}$ of order $\beta > 0$ and starting at $0 < a \in \mathbb{T}_q$ is defined by

where $m$ is the smallest integer greater or equal than $\beta$.

Definition 2.1 (^[11]). Let $0 < \alpha \not\in \mathbb{N}$ and $f : \mathbb{T}_a \to \mathbb{R}$. Then the Caputo left $q$-fractional derivative of order $\alpha$ of a function $f$ is defined by

where $n = [\alpha] + 1$.

Let us now list some properties which are needed to obtain our results.

Lemma 2.1 (^[29]). Let $\alpha, \beta > 0$ and $f$ be a function defined on $(0, b)$. Then the following formulas hold:

Lemma 2.2 (^[11]). Let $\alpha > 0$ and $f$ be defined in a suitable domain. Thus

and if $0 < \alpha \le 1$ we have

The following identity plays a crucial role in solving the linear $q$-fractional equations:

where $\alpha \in \mathbb{R}^+$ and $\mu \in (-1, \infty)$.

Apply ${\rm{}}_q \nabla_a^{\alpha}$ on both sides of (2.10), by virtue of Lemma 2.1, one can obtain

where $\alpha \in \mathbb{R}^+$ and $\mu \in (-1, \infty)$.

By Theorem 7 in ^[11], for any $0 < \beta < 1$, one has

For any $1 < \alpha \le 2$, by (2.8), one has

Apply ${\rm{}}_q \nabla_a^{\alpha}$ on both sides of (2.13), by Lemma 2.1 and (2.11), we get

3.   A generalized $q$-fractional Gronwall inequality

In this section, we give and prove the following spreading form of generalized $q$-fractional Gronwall inequality, which extend a $q$-fractional Gronwall inequality in Theorem 1.1.

Theorem 3.1. Let $\alpha > 0$ and $\beta > 0$. Assume that $u(t)$ and $g(t)$ are nonnegative functions for $t \in [a, T)_q$. Let $w_i(t)$ ($i = 1, 2$) be nonnegative and nondecreasing functions for $t \in [a, T)_q$ with $w_i(t) \le M_i$, where $M_i$ are positive constants ($i = 1, 2$) and

If

then

where $w(t) = \max \left\{\frac{w_1(t)}{\Gamma_q(\alpha)}, \ \frac{w_2(t)}{\Gamma_q(\beta)}\right\}$.

Proof. Define the operator

According to (3.2), one has

By (3.5) and the monotonicity of the operator $A$, we obtain

In the following, we will prove that

and

Obviously, the inequality (3.7) holds for $n = 1$. Assume that (3.7) is true for $n = m$, that is

When $n = m+1$, by using (3.4), (3.9), (2.10) and the nondecreasing of function $w(t)$, we get

${ A^{m+1} u(t) = A(A^m u(t)) }$

${ \le w(t) \int_a^t [(t-qs)_q^{\alpha-1} + (t-qs)_q^{\beta-1}] }$

${\times \left(w(s)^m \sum \limits_{k = 0}^m \frac{C_m^k \Gamma_q(\alpha)^{m-k} \Gamma_q(\beta)^k} {\Gamma_q((m-k)\alpha+k\beta)} \int_a^s (s - qr)_q^{(m-k)\alpha+k\beta-1} u(r) \nabla_q r\right)\nabla_q s }$

${ \le w(t)^{m+1} \int_a^t \sum \limits_{k = 0}^m \frac{C_m^k \Gamma_q(\alpha)^{m-k} \Gamma_q(\beta)^k} {\Gamma_q((m-k)\alpha+k\beta)} [(t-qs)_q^{\alpha-1} + (t-qs)_q^{\beta-1}] }$

${\times \left[\int_a^s (s - qr)_q^{(m-k)\alpha+k\beta-1} u(r) \nabla_q r\right]\nabla_q s }$

${ = w(t)^{m+1} \sum \limits_{k = 0}^m \frac{C_m^k \Gamma_q(\alpha)^{m-k} \Gamma_q(\beta)^k} {\Gamma_q((m-k)\alpha+k\beta)} \left[ \int_a^t (t-qs)_q^{\alpha-1} \int_a^s (s - qr)_q^{(m-k)\alpha+k\beta-1} u(r) \nabla_q r \nabla_q s \right. }$

${\left. + \int_a^t (t-qs)_q^{\beta-1}\int_a^s (s - qr)_q^{(m-k)\alpha+k\beta-1} u(r) \nabla_q r \nabla_q s \right] }$

${ = w(t)^{m+1} \sum \limits_{k = 0}^m \frac{C_m^k \Gamma_q(\alpha)^{m-k} \Gamma_q(\beta)^k} {\Gamma_q((m-k)\alpha+k\beta)} \left[ \int_a^t \int_{qr}^t (t-qs)_q^{\alpha-1} (s - qr)_q^{(m-k)\alpha+k\beta-1} u(r) \nabla_q r \nabla_q s \right. }$

${\left. + \int_a^t \int_{qr}^t (t-qs)_q^{\beta-1} (s - qr)_q^{(m-k)\alpha+k\beta-1} u(r) \nabla_q r \nabla_q s \right] }$

${ = w(t)^{m+1} \sum \limits_{k = 0}^m \frac{C_m^k \Gamma_q(\alpha)^{m-k} \Gamma_q(\beta)^k} {\Gamma_q((m-k)\alpha+k\beta)} }$

${ \times \left(\Gamma_q(\alpha) \int_a^t \left[\frac{1}{\Gamma_q(\alpha)} \int_{qr}^t (t-qs)_q^{\alpha-1} (s - qr)_q^{(m-k)\alpha+k\beta-1} \nabla_q s\right] u(r) \nabla_q r \right. }$

${\left. + \Gamma_q(\beta) \int_a^t \left[\frac{1}{\Gamma_q(\beta)} \int_{qr}^t (t-qs)_q^{\beta-1} (s - qr)_q^{(m-k)\alpha+k\beta-1} \nabla_q s \right] u(r) \nabla_q r \right) }$

${ = w(t)^{m+1} \sum \limits_{k = 0}^m \frac{C_m^k \Gamma_q(\alpha)^{m-k} \Gamma_q(\beta)^k} {\Gamma_q((m-k)\alpha+k\beta)} }$

${ \times \left(\Gamma_q(\alpha) \int_a^t {\rm{}}_q \nabla_{qr}^{-\alpha} (t - qr)_q^{(m-k)\alpha+k\beta-1} u(r) \nabla_q r \right. }$

${\left. + \Gamma_q(\beta) \int_a^t {\rm{}}_q \nabla_{qr}^{-\beta} (t - qr)_q^{(m-k)\alpha+k\beta-1} u(r) \nabla_q r \right) }$

${ = w(t)^{m+1} \sum \limits_{k = 0}^m \frac{C_m^k \Gamma_q(\alpha)^{m-k} \Gamma_q(\beta)^k} {\Gamma_q((m-k)\alpha+k\beta)} }$

${ \times \left(\frac{\Gamma_q(\alpha) \Gamma_q((m-k)\alpha + k\beta)}{\Gamma_q((m-k+1)\alpha + k\beta)} \int_a^t (t-qr)_q^{(m-k+1)\alpha + k\beta - 1} u(r) \nabla_q r \right. }$

${\left. + \frac{\Gamma_q(\beta) \Gamma_q((m-k)\alpha + k\beta)}{\Gamma_q((m-k)\alpha + (k+1)\beta)} \int_a^t (t-qr)_q^{(m-k)\alpha + (k+1)\beta - 1} u(r) \nabla_q r \right) }$

${ = w(t)^{m+1} \sum \limits_{k = 0}^m C_m^k \Gamma_q(\alpha)^{m-k} \Gamma_q(\beta)^k }$

${ \times \left(\Gamma_q(\alpha){\rm{}}_q \nabla_a^{-((m-k+1)\alpha + k\beta)} u(t) + \Gamma_q(\beta) {\rm{}}_q \nabla_a^{-((m-k)\alpha + (k+1)\beta)} u(t) \right) }$

${ = w(t)^{m+1} \sum \limits_{k = 0}^m C_m^k \Gamma_q(\alpha)^{m+1-k} \Gamma_q(\beta)^k {\rm{}}_q \nabla_a^{-((m-k+1)\alpha + k\beta)} u(t) }$

${ + w(t)^{m+1} \sum \limits_{k = 1}^{m+1} C_m^{k-1} \Gamma_q(\alpha)^{m+1-k} \Gamma_q(\beta)^k {\rm{}}_q \nabla_a^{-((m+1-k)\alpha + k\beta)} u(t) }$

${ = w(t)^{m+1} \left[ C_m^0 \Gamma_q(\alpha)^{m+1} {\rm{}}_q \nabla_a^{-((m+1)\alpha)} u(t) \right. }$

${ \left. + \sum \limits_{k = 1}^m (C_m^k + C_m^{k-1}) \Gamma_q(\alpha)^{m+1-k} \Gamma_q(\beta)^k {\rm{}}_q \nabla_a^{-((m-k+1)\alpha + k\beta)} u(t) \right. }$

${ \left. + C_m^m \Gamma_q(\beta)^{m+1} {\rm{}}_q \nabla_a^{-((m+1)\beta)} u(t) \right] }$

${ = w(t)^{m+1} \sum \limits_{k = 0}^{m+1} C_{m+1}^k \Gamma_q(\alpha)^{m+1-k} \Gamma_q(\beta)^k {\rm{}}_q \nabla_a^{-((m+1-k)\alpha + k\beta)} u(t). }$

Thus, (3.7) is proved.

Using Stirling's formula of the $q$-gamma function ^[30], yields that

that is

where $D = [2]_q^{1/2} \Gamma_{q^2}(1/2)$. Moreover, if $t > a > 0$ and $\gamma > 0$ ($\gamma$ is not a positive integer), then $1-\frac{a}{t}q^j < 1-\frac{a}{t}q^{\gamma+j}$ for each $j = 0, 1, ...$, and

By $w_1(t) < M_1$ and $w_2(t) < M_2$, one has that $w(t) < \max \left\{\frac{M_1}{\Gamma_q(\alpha)}, \ \frac{M_2}{\Gamma_q(\beta)}\right\} : = M$. Applying the first mean value theorem for definite integrals ^[31], (3.10) and (3.11), there exists a $\xi \in [a, t]_q$ such that

From (3.1), for each $t \in [a, T)_q$, we have

Thus, $A^n u(t) \to 0$ as $n \to \infty$. Let $n \to \infty$ in (3.6), by (3.8) we get

From (3.7) and (3.12), we obtain (3.3). This completes the proof.

Corollary 3.2. Under the hypothesis of Theorem 3.1, let $g(t)$ be a nondecreasing function on $t \in [a, T)_q$. Then

Proof. By (3.3), (2.10) and the assumption that $g(t)$ is nondecreasing function for $t \in [a, T)_q$, we have

4.   Main results

Throughout this paper, we make the following assumptions:

(H1) $f \in D(\mathbb{T}_q \times \mathbb{R}^n \times \mathbb{R}^n, \mathbb{R}^n)$ is a Lipschitz-type function. That is, for any $x, y : \mathbb{T}_{\tau a} \to \mathbb{R}^n$, there exists a positive constant $L > 0$ such that

for $t \in [a, T)_q$.

(H2)

(H3)

Definition 4.1. The system (1.5) is finite-time stable w.r.t.$\{\delta, \epsilon, T_e\}$, with $\delta < \epsilon$, if and only if $\max \{\|\phi\|, \|\psi\|\} < \delta$ implies $\|x(t)\| < \epsilon$, $\forall t \in [a, T_e]_q = [a, T_e] \cap [a, T)_q$.

Theorem 4.1. Assume that (H1) and (H3) hold. Then the problem (1.5) has a unique solution.

Proof. First we have to prove that $x : \mathbb{T}_{\tau a} \to \mathbb{R}^m$ is a solution of system (1.5) if and only if

For $t \in \mathbb{I}_{\tau}$, it is clear that $x(t) = \phi(t)$ with $\nabla_q x(t) = \psi(t)$ is the solution of (1.5). For $t \in [a, T)_q$, we apply ${\rm{}}_q \nabla_a^{\alpha}$ on both sides of (4.4) to obtain

where $({\rm{}}_q \nabla_a^{\alpha} {\rm{}}_q \nabla_a^{-\alpha} x)(t) = x(t)$ and $({\rm{}}_q \nabla_a^{\alpha} {\rm{}}_q \nabla_a^{-(\alpha-\beta)} x)(t) = {\rm{}}_q \nabla_a^{\beta} x(t)$ (by Lemma 2.1) have been used. By using (2.12) and (2.14), we get

Conversely, from system (1.5), we can see that $x(t) = \phi(t)$ and $\nabla_q x(t) = \psi(t)$ for $t \in \mathbb{I}_{\tau}$. For $t \in [a, T)_q$, we apply ${\rm{}}_q \nabla_a^{-\alpha}$ on both sides of (1.5) to get

According to Lemma 2.2, we obtain

Secondly, we will prove the uniqueness of solution to system (1.5). Let $x$ and $y$ be two solutions of system (1.5). Denote $z$ by $z(t) = x(t) - y(t)$. Obviously, $z(t) = 0$ for $t \in \mathbb{I}_{\tau}$, which implies that system (1.5) has a unique solution for $t \in \mathbb{I}_{\tau}$.

For $t \in [a, T)_q$, one has

If $t \in \mathbb{J}_{\tau} = \{a, q^{-1} a, ..., \tau^{-1} a\}$, then $\tau t \in \mathbb{I}_{\tau}$ and $z(\tau t) = 0$. Hence,

which implies that

By applying Corollary 3.2 and (H3), we get

where $w_1 = \max \left\{\frac{\|A_0\|}{\Gamma(\alpha-\beta)}, \frac{\|B_0\|+L}{\Gamma(\alpha)}\right\}$. This implies $x(t) = y(t)$ for $t \in \mathbb{J}_{\tau}$.

For $t \in [\tau^{-1} a, T)_q$, we obtain

Therefore,

Let $z^*(t) = \max_{\theta \in [a, t]_q } \{\|z(\theta)\|, \|z(\tau \theta)\|\}$ for $t \in [\tau^{-1} a, T)_q$, where $[a, t]_q = [a, t] \cap \mathbb{T}_a$, it is obvious that $z^*(t)$ is a increasing function. From (4.10), we obtain that

By applying Corollary 3.2 and (H3) again, we get

where $w_2 = \max \left\{\frac{\|A_0\|}{\Gamma(\alpha-\beta)}, \frac{\|B_0\|+\|B_1\|+2L}{\Gamma(\alpha)}\right\}$. Thus, we end up with $x(t) = y(t)$ for $t \in [\tau^{-1} a, T)_q$. The proof is completed.

Theorem 4.2. Assume that the conditions (H1), (H2) and (H3) hold. Then the system (1.5) is finite-time stable if the following condition is satisfied:

where $w_2 = \max \left\{\frac{\|B_0\|+\|B_1\|+2L}{\Gamma_q(\alpha)}, \frac{\|A_0\|}{\Gamma_q(\alpha-\beta)}\right\}$.

Proof. Applying left $q$-fractional integral on both sides of (1.5), we obtain

By (4.12) and utilizing Lemma 2.2 we have

Thus, by (H1) and (H2), we get

Let $g(t) = \|\phi\| + \|\psi\| (t-a) + \|A_0\| \|\phi\| \frac{(t-a)_q^{\alpha-\beta}}{\Gamma_q(\alpha-\beta+1)}$, then $g$ is a nondecreasing function.

Set $\bar{x}(t) = \max_{\theta \in [a, t]_q } \{\|x(\theta)\|, \|x(\tau \theta)\|\}$, then by (4.14) we get

Applying the result of Corollary 3.2, we have

Therefore, the system (1.5) is finite-time stable. The proof is completed.

5.   Example

If $x \in \mathbb{R}^n$, then $\|x\| = \sum_{i = 1}^n |x_i|$. If $A \in \mathbb{R}^{n \times n}$, then the induced norm $\|\cdot\|$ is defined as $\|A\| = \max_{1 \le j \le n} \sum_{i = 1}^n |a_{ij}|$.

Example 5.1. Consider the nonlinear delay $q$-fractional differential difference system

where $\alpha = 1.8$, $\beta = 0.8$, $q = 0.6$, $a = q^5 = 0.6^5$, $T = q^{-1} = 0.6^{-1}$, $\tau = q^3 = 0.6^3$, $x(t) = [x_1(t), x_2(t)]^T \in \mathbb{R}^2$,

and

Obviously, $\|\phi\| = \|\psi\| = 0.0085 < 0.1 = \delta$, $\epsilon = 1$. We can see that $f$ satisfies conditions (H1) ($L = \frac{1}{4}$) and (H2). We can calculate $\|A_0\| = 0.62$, $\|B_0\| = 0.109$, $\|B_1\| = 0.15$.

When $T = 0.6^{-1}$, it is easy to check that

that is, (H3) holds. By using Matlab (the pseudo-code to compute different values of $\Gamma_q(\sigma)$, see ^[32]), when $t = 1 \in [a, T)_q$,

Thus, we obtain $T_e = 1$.

6.   Conclusions

In this paper, we introduced and proved new generalizations for $q$-fractional Gronwall inequality. We examined the validity and applicability of our results by considering the existence and uniqueness of solutions of nonlinear delay $q$-fractional difference damped system. Moreover, a novel and easy to verify sufficient conditions have been provided in this paper which are easy to determine the finite-time stability of the solutions for the considered system. Finally, an example is given to illustrate the effectiveness and feasibility of our criterion. Motivated by previous works ^[33,34], the possible applications of fractional $q$-difference in the field of stability theory will be considered in the future.

Acknowledgments

The authors are grateful to the anonymous referees for valuable comments and suggestions that helped to improve the quality of the paper. This work is supported by Natural Science Foundation of China (11571136).

Conflict of interest

The authors declare that there is no conflicts of interest.