The chaotic mechanisms in some jerk systems

Xiaoyan Hu; Bo Sang; Ning Wang; Xiaoyan Hu; Bo Sang; Ning Wang

doi:10.3934/math.2022861

AIMS Mathematics

2022, Volume 7, Issue 9: 15714-15740. doi: 10.3934/math.2022861

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Research article

The chaotic mechanisms in some jerk systems

1.
School of Mathematical Sciences, Liaocheng University, Liaocheng 252059, China
2.
School of Electrical and Information Engineering, Tianjin University, Tianjin 300072, China

Received: 31 December 2021 Revised: 02 June 2022 Accepted: 07 June 2022 Published: 24 June 2022
MSC : 34C05, 34C07, 34C23, 34C28, 34C29

In this work, a five-parameter jerk system with one hyperbolic sine nonlinearity is proposed, in which $\varepsilon$ is a small parameter, and $a$ , $b$ , $c$ , $d$ are some other parameters. For $\varepsilon = 0$ , the system is $Z_{2}$ symmetric. For $\varepsilon \neq {0}$ , the system loses the symmetry. For the symmetrical case, the pitchfork bifurcation and Hopf bifurcation of the origin are studied analytically by Sotomayor's theorem and Hassard's formulas, respectively. These bifurcations can be either supercritical or subcritical depending on the governing parameters. In comparison, it is much more restrictive for the origin of the Lorenz system: Only a supercritical pitchfork bifurcation is available. Thus, the symmetrical system can exhibit very rich local bifurcation structures. The continuation of local bifurcations leads to the main contribution of this work, i.e., the discovery of two basic mechanisms of chaotic motions for the jerk systems. For four typical cases, Cases A–D, by varying the parameter $a$ , the mechanisms are identified by means of bifurcation diagrams. Cases A and B are $Z_{2}$ symmetric, while Cases C and D are asymmetric (caused by constant terms). The forward period-doubling routes to chaos are observed for Cases A and C; meanwhile, the backward period-doubling routes to chaos are observed for Cases B and D. The dynamical behaviors of these cases are studied via phase portraits, two-sided Poincaré sections and Lyapunov exponents. Using Power Simulation (PSIM), a circuit simulation model for a chaotic jerk system is created. The circuit simulations match the results of numerical simulations, which further validate the dynamical behavior of the jerk system.

Keywords:

Citation: Xiaoyan Hu, Bo Sang, Ning Wang. The chaotic mechanisms in some jerk systems[J]. AIMS Mathematics, 2022, 7(9): 15714-15740. doi: 10.3934/math.2022861

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Abstract

1. Introduction

The key to solving the general quadratic congruence equation is to solve the equation of the form $x^2\equiv a\mod p$ , where $a$ and $p$ are integers, $p > 0$ and $p$ is not divisible by $a$ . For relatively large $p$ , it is impractical to use the Euler criterion to distinguish whether the integer $a$ with $(a, p) = 1$ is quadratic residue of modulo $p$ . In order to study this issue, Legendre has proposed a new tool-Legendre's symbol.

Let $p$ be an odd prime, the quadratic character modulo $p$ is called the Legendre's symbol, which is defined as follows:

$\left(\frac{a}{p}\right)= \begin{cases}1, & \text { if } a \text { is a quadratic residue modulo } p; \\ -1, & \text { if } a \text { is a quadratic non-residue modulo } p ;\\ 0, & \text { if } p \mid a.\end{cases}$

The Legendre's symbol makes it easy for us to calculate the level of quadratic residues. The basic properties of Legendre's symbol can be found in any book on elementary number theory, such as ^[1,2,3].

The properties of Legendre's symbol and quadratic residues play an important role in number theory. Many scholars have studied them and achieved some important results. For examples, see the ^{[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21]}.

One of the most representative properties of the Legendre's symbol is the quadratic reciprocal law:

Let $p$ and $q$ be two distinct odd primes. Then, (see Theorem 9.8 in ^[1] or Theorems 4–6 in ^[3])

$\left(\frac{p}{q}\right)\cdot\left(\frac{q}{p}\right) = (-1)^{\frac{(p-1)(q-1)}{4}}.$

For any odd prime $p$ with $p\equiv1\; \rm mod\; 4$ there exist two non-zero integers $\alpha\left(p\right)$ and $\beta\left(p\right)$ such that

$\begin{align*} p = \alpha^2\left(p\right)+\beta^2\left(p\right). \end{align*}$

(1)

In fact, the integers $\alpha\left(p\right)$ and $\beta\left(p\right)$ in the (1) can be expressed in terms of Legendre's symbol modulo $p$ (see Theorems 4–11 in ^[3])

$\alpha\left(p\right) = \frac{1}{2}\sum\limits_{a = 1}^{p-1}\left(\frac{a^3+a}{p}\right)\; \; \; \rm and\; \; \; \it \beta\left(\it p\right) = \frac{\rm1}{\rm 2}\sum\limits_{a = \rm1}^{p-\rm1}\left(\frac {\it a^{\rm3}+\it ra}{\it p}\right),$

where $r$ is any integer, and $\left(r, p\right) = 1$ , $\left(\frac{r}{p}\right) = -1$ , $\left(\frac{\ast}{p}\right) = \chi_2$ denote the Legendre's symbol modulo $p$ .

Noting that Legendre's symbol is a special kind of character. For research on character, Han ^[7] studied the sum of a special character $\chi\left(ma+\bar{a}\right)$ , for any integer $m$ with $\left(m, p\right) = 1$ , then

$\left|\sum\limits_{a = 1}^{p-1}\chi\left(ma+\bar{a}\right)\right|^2 = 2p+\left(\frac{m}{p}\right)\sum\limits_{a = 1}^{p-1}\chi(a)\sum\limits_{b = 1}^{p-1}\left(\frac{b(b-1)(a^2b-1)}{p}\right),$

which is a special case of a general polynomial character sums $\sum_{a = N+1}^{N+M}\chi\left(f\left(a\right)\right)$ , where $M$ and $N$ are any positive integers, and $f\left(x\right)$ is a polynomial.

In ^[8], Du and Li introduced a special character sums $C\left(\chi, m, n, c;p\right)$ in the following form:

$C\left(\chi, m, n, c;p\right) = \sum\limits_{a = 0}^{p-1}\sum\limits_{b = 0}^{p-1}\chi\left(a^2+na-b^2-nb+c\right)\cdot e\left(\frac{mb^2-ma^2}{p}\right),$

and studied the asymptotic properties of it. They obtained

$\begin{equation*} \label{test1} \begin{split} \sum\limits_{c = 1}^{p-1}\left|C\left(\chi, m, n, c;p\right)\right|^{2k} = \left\{ \begin{array}{l} &p^{2k+1}+\frac{k^2-3k-2}{2}\cdot p^{2k}+O\left(p^{2k-1}\right), \; \; \; \; \rm if\; \chi\ \rm is\ \rm the\ \rm Legendre\ \rm symbol\ \rm {modulo} \ \it p \rm; \\ &p^{2k+1}+\frac{k^2-3k-2}{2}\cdot p^{2k}+O\left(p^{2k-1/2}\right), \; \; \rm if\; \chi\ \rm is\ \rm a\ \rm complex\ \rm character\ \rm {modulo} \ \it p.\\ \end{array} \right. \end{split} \end{equation*}$

Recently, Yuan and Zhang ^[12] researched the question about the estimation of the mean value of high-powers for a special character sum modulo a prime, let $p$ be an odd prime with $p\equiv1\; \rm mod\; 6$ , then for any integer $k\geq0$ , they have the identity

$S_k\left(p\right) = \frac{1}{3}\cdot\left[d^k+\left(\frac{-d+9b}{2}\right)^k+\left(\frac{-d-9b}{2}\right)^k\right],$

where

$S_k\left(p\right) = \frac{1}{p-1}\sum\limits_{r = 1}^{p-1}A^k\left(r\right),$

$A\left(r\right) = 1+\sum\limits_{a = 1}^{p-1}\left(\frac{a^2+r\bar{a}}{p}\right),$

and for any integer $r$ with $(r, p) = 1$ .

More relevant research on special character sums will not be repeated. Inspired by these papers, we have the question: If we replace the special character sums with Legendre's symbol, can we get good results on $p\equiv1\; \rm mod\; 4$ ?

We will convert $\beta\left(p\right)$ to another form based on the properties of complete residues

$\beta\left(p\right) = \frac{1}{2}\sum\limits_{a = 1}^{p-1}\left(\frac{a+n\bar{a}}{p}\right),$

where $\bar{a}$ is the inverse of $a$ modulo $p$ . That is, $\bar{a}$ satisfy the equation $x\cdot a\; \equiv1\mod p$ for any integer $a$ with $(a, p) = 1$ .

For any integer $k\geq0$ , $G\left(n\right)$ and $K_k(p)$ are defined as follows:

$G\left(n\right) = 1+\sum\limits_{a = 1}^{p-1}\left(\frac{a^2+n\bar{a}^2}{p}\right)\; \; \; \rm {and} \; \; \; \it K_{\it k}\left(\it p\right) = \frac{\rm 1}{\it p-\rm 1}\sum\limits_{\it n\rm = 1}^{\it p-\rm 1}\it G^{\it k}\left(\it n\right).$

In this paper, we will use the analytic methods and properties of the classical Gauss sums and Dirichlet character sums to study the computational problem of $K_k\left(p\right)$ for any positive integer $k$ , and give a linear recurrence formulas for $K_k\left(p\right)$ . That is, we will prove the following result.

Theorem 1. Let $p$ be an odd prime with $p\equiv1\; \rm mod\; 4$ , then we have

$K_k\left(p\right) = \left(4p+2\right)\cdot K_{k-2}\left(p\right)-8\left(2\alpha^2-p\right)\cdot K_{k-3}\left(p\right)+\left(16\alpha^4-16p\alpha^2+4p-1\right)\cdot K_{k-4}\left(p\right),$

for all integer $k\geq4$ with

$K_0\left(p\right) = 1, \; K_1\left(p\right) = 0, \; K_2\left(p\right) = 2p+1, \; K_3\left(p\right) = -3(4\alpha^2-2p),$

where

$\alpha = \alpha\left(p\right) = \sum\limits_{a = 1}^{\frac{p-1}{2}}\left(\frac{a+\bar{a}}{p}\right).$

Applying the properties of the linear recurrence sequence, we may immediately deduce the following corollaries.

Corollary 1. Let $p$ be an odd prime with $p\equiv1\; \rm mod\; 4$ . Then we have

$\frac{1}{p-1}\sum\limits_{n = 1}^{p-1}\frac{1}{1+\sum _{a = 1}^{p-1}\left(\frac{a^2+n\bar{a}^2}{p}\right)} = \frac{16\alpha^2p-28\alpha^2-8p^2+14p} {16\alpha^4-16\alpha^2p+4p-1}.$

Corollary 2. Let $p$ be an odd prime with $p\equiv1\; \rm mod\; 4$ . Then we have

$\begin{align*} \frac{1}{p-1}\sum\limits_{n = 1}^{p-1}\sum\limits_{m = 0}^{p-1}\left(1+\sum\limits_{a = 1}^{p-1}\left(\frac{a^2+n\bar{a}^2}{p}\right)\right)\cdot e\left(\frac{nm^2}{p}\right) = -\sqrt{p}. \end{align*}$

Corollary 3. Let $p$ be an odd prime with $p\equiv1\; \rm mod\; 4$ . Then we have

$\frac{1}{p-1}\sum\limits_{n = 1}^{p-1}\sum\limits_{m = 0}^{p-1}\left[1+\sum\limits_{a = 1}^{p-1}\left(\frac{a^2+n\bar{a}^2}{p}\right)\right]^2\cdot e\left(\frac{nm^2}{p}\right) = \left(4\alpha^2-2p\right)\cdot\sqrt{p}.$

Corollary 4. Let $p$ be an odd prime with $p\equiv1\; \rm mod\; 8$ . Then we have

$\begin{align*} \sum\limits_{n = 1}^{p-1}\left(1+\sum\limits_{a = 1}^{p-1}\left(\frac{a^2+n\bar{a}^2}{p}\right)\right)\cdot \sum\limits_{m = 0}^{p-1}e\left(\frac{nm^4}{p}\right) = \sqrt{p}\left(-1+B(1)\right)-p, \end{align*}$

where

$B(1) = \sum\limits_{m = 0}^{p-1}e\left(\frac{m^4}{p}\right).$

If we consider such a sequence $F_k\left(p\right)$ as follows: Let $p$ be a prime with $p\equiv1\; \rm mod\; 8$ , $\chi_4$ be any fourth-order character modulo $p$ . For any integer $k\geq0$ , we define the $F_k\left(p\right)$ as

$F_k\left(p\right) = \sum\limits_{n = 1}^{p-1}\frac{1}{G^{k}(n)},$

we have

$\begin{align*} F_{k}\left(p\right) = &\frac{1}{16\alpha^4-16\alpha^2p+4p-1}F_{k-4}\left(p\right)-\frac{(4p+2)}{16\alpha^4-16\alpha^2p+4p-1}F_{k-2}\left(p\right)\\ &+\frac{4(4\alpha^2-2p)}{16\alpha^4-16\alpha^2p+4p-1}F_{k-1}\left(p\right). \end{align*}$

2. Some lemmas

Lemma 1. Let $p$ be an odd prime with $p\equiv1\; \rm mod\; 4$ . Then for any fourth-order character $\chi_4 \; \rm{mod} \; \it p$ , we have the identity

$\tau^2(\chi_4)+\tau^2(\bar{\chi_4}) = 2\sqrt{p}\cdot\alpha,$

where

$\tau(\chi_4) = \sum\limits_{a = 1}^{p-1}\chi_4(a)e\left(\frac{a}{p}\right)$

denotes the classical Gauss sums, $e(y) = e^{2\pi iy}, i^2 = -1$ , and $\alpha$ is the same as in the Theorem 1.

Proof. See Lemma 2.2 in ^[9].

Lemma 2. Let $p$ be an odd prime. Then for any non-principal character $\psi$ modulo $p$ , we have the identity

$\tau(\psi^2) = \frac{\psi^2(2)}{\tau(\chi_2)}\cdot\tau(\psi)\cdot\tau(\psi\chi_2),$

where $\chi_2 = \left(\frac{\ast}{p}\right)$ denotes the Legendre's symbol modulo $p$ .

Proof. See Lemma 2 in ^[12].

Lemma 3. Let $p$ be a prime with $p\equiv 1\; \rm mod\; 4$ , then for any integer $n$ with $\left(n, p\right) = 1$ and fourth-order character $\chi_{4}\; \rm mod \; \it p$ , we have the identity

$\sum\limits_{a = 1}^{p-1}\left(\frac{a^2+n\bar{a}^2}{p}\right) = -1-\chi_2(n)+\frac{1}{\sqrt{p}}\cdot(\chi_{4}(n)\cdot\tau^2(\bar{\chi_{4}})+\bar{\chi_{4}}(n)\cdot\tau^2(\chi_{4})).$

Proof. For any integer $a$ with $(a, p) = 1$ , we have the identity

$1+\chi_{4}(a)+\chi_2(a)+\bar{\chi_{4}}(a) = 4,$

if $a$ satisfies $a\equiv b^4\; \rm mod \; \it p$ for some integer $b$ with $(b, p) = 1$ and

$1+\chi_{4}(a)+\chi_2(a)+\bar{\chi_{4}}(a) = 0,$

otherwise. So from these and the properties of Gauss sums we have

$\begin{align*} \mathop\sum\limits_{a = 1}^{p-1}\left(\frac{a^2+n\bar{a}^2}{p}\right) = &\sum\limits_{a = 1}^{p-1}\left(\frac{a^2}{p}\right)\left(\frac{a^4+n}{p}\right)\nonumber\\ = &\sum\limits_{a = 1}^{p-1}\chi_2\left(a^4\right)\chi_2\left(a^4+n\right)\nonumber\\ = &\sum\limits_{a = 1}^{p-1}(1+\chi_{4}(a)+\chi_2(a)+\bar{\chi_{4}}(a))\cdot\chi_2(a)\cdot\chi_2(a+n)\nonumber\\ = &\sum\limits_{a = 1}^{p-1}(1+\chi_{4}(na)+\chi_2(na)+\bar{\chi_{4}}(na))\cdot\chi_2(na)\cdot\chi_2(na+n)\nonumber\\ = &\sum\limits_{a = 1}^{p-1}\chi_2(a)\chi_2(a+1)+\sum\limits_{a = 1}^{p-1}\chi_{4}(na)\chi_2(a)\chi_2(a+1) \end{align*}$

(2)

$\begin{align*} &+\sum\limits_{a = 1}^{p-1}\chi_2(na)\chi_2(a)\chi_2(a+1)+\sum\limits_{a = 1}^{p-1}\bar{\chi_{4}}(na)\chi_2(a)\chi_2(a+1)\nonumber\\ = &\sum\limits_{a = 1}^{p-1}\chi_2(1+\bar{a})+\sum\limits_{a = 1}^{p-1}\chi_{4}(na)\chi_2(a)\chi_2(a+1)\nonumber\\ &+\sum\limits_{a = 1}^{p-1}\chi_2(n)\chi_2(a+1)+\sum\limits_{a = 1}^{p-1}\bar{\chi_{4}}(na)\chi_2(a)\chi_2(a+1)\nonumber. \end{align*}$

Noting that for any non-principal character $\chi$ ,

$\sum\limits_{a = 1}^{p-1}\chi(a) = 0$

and

$\sum\limits_{a = 1}^{p-1}\chi(a)\chi(a+1) = \frac{1}{\tau(\bar{\chi})}\sum\limits_{b = 1}^{p-1}\sum\limits_{a = 1}^{p-1}\bar{\chi}(b)\chi(a)e\left(\frac{b(a+1)}{p}\right).$

Then we have

$\sum\limits_{a = 1}^{p-1}\chi_2(1+\bar{a}) = -1, \; \; \; \sum\limits_{a = 1}^{p-1}\chi_2(a+1) = -1,$

$\begin{align*} \sum\limits_{a = 1}^{p-1}\chi_{4}(a)\chi_2(a)\chi_2(a+1)& = \frac{1}{\tau(\chi_2)}\sum\limits_{b = 1}^{p-1}\sum\limits_{a = 1}^{p-1}\chi_2(b)\chi_{4}(a)\chi_2(a)e\left(\frac{b(a+1)}{p}\right)\nonumber\\ & = \frac{1}{\tau(\chi_2)}\sum\limits_{b = 1}^{p-1}\bar{\chi_{4}}(b)e\left(\frac{b}{p}\right)\sum\limits_{a = 1}^{p-1}\chi_{4}(ab)\chi_2(ab)e\left(\frac{ab}{p}\right) \end{align*}$

(3)

$\begin{align*} & = \frac{1}{\tau(\chi_2)}\cdot\tau(\bar{\chi_{4}})\cdot\tau(\chi_{4}\chi_2)\nonumber. \end{align*}$

For any non-principal character $\psi$ , from Lemma 2 we have

$\begin{align*} \tau(\psi^2) = \frac{\psi^2(2)}{\tau(\chi_2)}\cdot\tau(\psi)\cdot\tau(\psi\chi_2). \end{align*}$

(4)

Taking $\psi = \chi_{4}$ , note that

$\tau(\chi_2) = \sqrt{p}, \ \ \tau(\chi_{4})\cdot\tau(\bar{\chi_{4}}) = \chi_{4}(-1)\cdot p,$

from (3) and (4), we have

$\begin{align*} \sum\limits_{a = 1}^{p-1}\chi_{4}(a)\chi_2(a)\chi_2(a+1)& = \frac{\bar{\chi_{4}}^2(2)\cdot\tau(\chi_{4}^2)\cdot\tau(\chi_2)\cdot\tau(\bar{\chi_{4}})}{\tau(\chi_2)\cdot\tau(\chi_{4})}\nonumber\\ & = \frac{\chi_2(2)\cdot\tau(\chi_2)\cdot\tau^2(\bar{\chi_{4}})}{\tau(\chi_{4})\cdot\tau(\bar{\chi_{4}})}\nonumber\\ & = \frac{\chi_2(2)\cdot\sqrt{p}\cdot\tau^2(\bar{\chi_{4}})}{\chi_4(-1)\cdot p} \end{align*}$

(5)

$\begin{align*} & = \frac{\chi_2(2)\cdot\tau^2(\bar{\chi_{4}})}{\chi_4(-1)\cdot\sqrt{p}}.\nonumber \end{align*}$

Similarly, we also have

$\begin{align*} \sum\limits_{a = 1}^{p-1}\bar{\chi_{4}}(a)\chi_2(a)\chi_2(a+1) = \frac{\chi_2(2)\cdot\tau^2(\chi_{4})}{\chi_4(-1)\cdot\sqrt{p}}. \end{align*}$

(6)

Consider the quadratic character modulo $p$ , we have

$\begin{align*} \left(\frac{2}{p}\right) = \chi_2(2) = \begin{cases} \; \; 1, &\rm{ if\; \ p\equiv\pm1\; mod\; 8};\\ -1, &\rm{ if\; \ p\equiv\pm3\; mod\; 8}.\\ \end{cases} \end{align*}$

(7)

And when $p\equiv1\; \rm mod\; 8$ , we have $\chi_{4}(-1) = 1$ ; when $p\equiv5\; \rm mod\; 8$ , we have $\chi_{4}(-1) = -1.$ Combining (2) and (5)–(7) we can deduce that

$\sum\limits_{a = 1}^{p-1}\left(\frac{a^2+n\bar{a}^2}{p}\right) = -1-\chi_2(n)+\frac{1}{\sqrt{ p}}\cdot\left(\chi_{4}(n)\cdot\tau^2(\bar{\chi_{4}})+\bar{\chi_{4}}(n)\cdot\tau^2(\chi_{4})\right).$

This prove Lemma 3.

Lemma 4. Let $p$ be an odd prime with $p\equiv1\; \rm mod\; 4$ . Then for any integer $k\geq4$ and $n$ with $(n, p) = 1$ , we have the fourth-order linear recurrence formula

$G^k(n) = (4p+2)\cdot G^{k-2}(n)+8(p-2\alpha^2)\cdot G^{k-3}(n)+[(4\alpha^2-2p)^2-(2p-1)^2]\cdot G^{k-4}(n),$

where

$\alpha = \alpha(p) = \frac{1}{2}\sum\limits_{a = 1}^{p-1}\left(\frac{a^3+a}{p}\right) = \sum\limits_{a = 1}^{\frac{p-1}{2}}\left(\frac{a+\bar{a}}{p}\right),$

$\left(\frac{\ast} {p}\right) = \chi_2$ denotes the Legendre's symbol.

Proof. For $p\equiv1\; \rm mod\; 4$ , any integer $n$ with $\left(n, p\right) = 1$ , and fourth-order character $\chi_{4}$ modulo $p$ , we have the identity

$\chi_{4}^4(n) = \bar{\chi_{4}}^4(n) = \chi_0(n), \ \ \chi_{4}^2(n) = \chi_2(n),$

where $\chi_0$ denotes the principal character modulo $p$ .

According to Lemma 3,

$\sum\limits_{a = 1}^{p-1}\left(\frac{a^2+n\bar{a}^2}{p}\right) = -1-\chi_2(n)+\frac{1}{\sqrt{ p}}\cdot\left(\chi_{4}(n)\cdot\tau^2(\bar{\chi_{4}})+\bar{\chi_{4}}(n)\cdot\tau^2(\chi_{4})\right),$

$G(n) = 1+\sum\limits_{a = 1}^{p-1}\left(\frac{a^2+n\bar{a}^2}{p}\right).\quad\quad\quad\quad\quad\quad\quad\quad\ \$

We have

$\begin{align*} G(n) = -\chi_2(n)+\frac{1}{\sqrt{p}}\cdot\left(\chi_{4}(n)\cdot\tau^2(\bar{\chi_{4}})+\bar{\chi_{4}}(n)\cdot\tau^2(\chi_{4})\right), \end{align*}$

(8)

$\begin{align*} G^2(n) = &[-\chi_2(n)+\frac{1}{\sqrt{p}}\cdot(\chi_{4}(n)\cdot\tau^2(\bar{\chi_{4}})+\bar{\chi_{4}}(n)\cdot\tau^2(\chi_{4}))]^2\\ = &1-2\chi_2(n)\cdot\frac{1}{\sqrt{p}}\cdot\left(\chi_{4}(n)\cdot\tau^2(\bar{\chi_{4}})+\bar{\chi_{4}}(n)\cdot\tau^2(\chi_{4})\right)\\ &+\frac{1}{ p}\cdot\left(\chi_2(n)\cdot\tau^4(\bar{\chi_{4}})+\chi_2(n)\cdot\tau^4(\chi_{4})+2p^2\right)\\ = &1-2\chi_2(n)\cdot\frac{1}{\sqrt{p}}\cdot\left(\chi_{4}(n)\cdot\tau^2(\bar{\chi_{4}})+\bar{\chi_{4}}(n)\cdot\tau^2(\chi_{4})\right)\\ &+\frac{1}{p}\cdot\left(\chi_2(n)\cdot(\tau^4(\bar{\chi_{4}})+\tau^4(\chi_{4}))+2p^2\right). \end{align*}$

According to Lemma 1, we have

$\left(\tau^2(\chi_{4})+\tau^2(\bar{\chi_{4}})\right)^2 = \tau^4(\bar{\chi_{4}})+\tau^4(\chi_{4})+2p^2 = 4p\alpha^2.$

Therefore, we may immediately deduce

$\begin{align*} G^2(n) = &1-2(\chi_2(n)\cdot\left( G(n)+\chi_2(n)\right)\nonumber\\ &+\frac{1}{p}\left(\chi_2(n)\cdot(\tau^4(\bar{\chi_{4}})+\tau^4(\chi_{4}))+2p^2\right)\nonumber\\ = &1-2\chi_2(n)\cdot\left( G(n)+\chi_2(n)\right) \end{align*}$

(9)

$\begin{align*} &+\frac{1}{p}\cdot\left[\chi_2(n)((\tau^2(\bar{\chi_{4}})+\tau^2(\chi_{4}))^2-2p^2)+2p^2\right]\nonumber\\ = &2p-1-2\chi_2(n)\cdot G(n)+\left(4\alpha^2-2p\right)\cdot\chi_2(n)\nonumber, \end{align*}$

$\begin{align*} \; \; \; \; \; \; \; \; \; \; \; \; \; \; G^3(n) = &[-\chi_2(n)+\frac{1}{\sqrt{p}}\cdot(\chi_{4}(n)\cdot\tau^2(\bar{\chi_{4}})+\bar{\chi_{4}}(n)\cdot\tau^2(\chi_{4}))]^3\nonumber\\ = &\left(2p-1-2\chi_2(n)\cdot G(n)+\left(4\alpha^2-2p\right)\cdot\chi_2(n)\right)\cdot G(n) \end{align*}$

(10)

$\begin{align*} = &(4\alpha^2-2p)\chi_2(n)\cdot G(n)+(2p+3)G(n)-(4p-2)\chi_2(n) -2(4\alpha^2-2p)\nonumber \end{align*}$

and

$\begin{align*} \left[G^2(n)-(2p-1)\right]^2 = \left[\chi_2(n)\cdot(4\alpha^2-2p)-2\chi_2(n)\cdot G(n)\right]^2, \end{align*}$

which implies that

$\begin{align*} G^4(n) = (4p+2)\cdot G^{2}(n)+8(p-2\alpha^2)\cdot G(n)+[(4\alpha^2-2p)^2-(2p-1)^2]. \end{align*}$

(11)

So for any integer $k\geq4$ , from (8)–(11), we have the fourth-order linear recurrence formula

$\begin{align*} G^k(n)& = G^{k-4}(n)\cdot G^4(n)\\ & = (4p+2)\cdot G^{k-2}(n)+8(p-2\alpha^2)\cdot G^{k-3}(n)+[(4\alpha^2-2p)^2-(2p-1)^2]\cdot G^{k-4}(n). \end{align*}$

This proves Lemma 4.

3. Proof of the theorem

In this section, we will complete the proof of our theorem.

Let $p$ be any prime with $p\equiv 1\; \rm mod\; 4$ , then we have

$\begin{align*} K_0(p) = \frac{1}{p-1}\sum\limits_{n = 1}^{p-1}G^0(n) = \frac{p-1}{p-1} = 1.\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \end{align*}$

(12)

$\begin{align*} K_1(p)& = \frac{1}{p-1}\sum\limits_{n = 1}^{p-1}G^1(n)\nonumber\\ & = \frac{1}{p-1}\sum\limits_{n = 1}^{p-1}\left(-\chi_2(n)+\frac{1}{\sqrt{p}}\cdot(\chi_{4}(n)\tau^2(\bar{\chi_{4}})+\bar{\chi_{4}}(n)\tau^2(\chi_{4}))\right)\\& = 0, \end{align*}$

(13)

$\begin{align*} \; \; \; \; \; K_2(p)& = \frac{1}{p-1}\sum\limits_{n = 1}^{p-1}G^2(n)\nonumber\\ & = \frac{1}{p-1}\sum\limits_{n = 1}^{p-1}\left(-\chi_2(n)+\frac{1}{\sqrt{p}}\cdot(\chi_{4}(n)\tau^2(\bar{\chi_{4}})+\bar{\chi_{4}}(n)\tau^2(\chi_{4}))\right)^2 \\ & = 2p+1,\end{align*}$

(14)

$\begin{align*} \; \; \; \; \; K_3(p)& = \frac{1}{p-1}\sum\limits_{n = 1}^{p-1}G^3(n)\nonumber\\ & = \frac{1}{p-1}\sum\limits_{n = 1}^{p-1}\left(-\chi_2(n)+\frac{1}{\sqrt{p}}\cdot(\chi_{4}(n)\tau^2(\bar{\chi_{4}})+\bar{\chi_{4}}(n)\tau^2(\chi_{4}))\right)^3 \\ & = -3(4\alpha^2-2p).\end{align*}$

(15)

It is clear that from Lemma 4, if $k\geq4$ , we have

$\begin{align*} \; \; \; \; \; K_k(p) = &\frac{1}{p-1}\sum\limits_{n = 1}^{p-1}G^k(n)\nonumber\\ = &(4p+2)\cdot K_{k-2}(p)-8\left(2\alpha^2-p\right)\cdot K_{k-3}(p) +(16\alpha^4-16p\alpha^2+4p-1)\cdot K_{k-4}(p). \end{align*}$

(16)

Now Theorem 1 follows (12)–(16). Obviously, using Theorem 1 to all negative integers, and that lead to Corollary 1.

This completes the proofs of our all results.

Some notes:

Note 1: In our theorem, know $n$ is an integer, and $(n, p) = 1$ . According to the properties of quadratic residual, $\chi_2(n) = \pm1$ , $\chi_4(n) = \pm1$ .

Note 2: In our theorem, we only discussed the case $p\equiv1\; \rm mod\; 8$ . If $p\equiv3\; \rm mod\; 4$ , then the result is trivial. In fact, in this case, for any integer $n$ with $(n, p) = 1$ , we have the identity

$\begin{align*} G(n)& = 1+\sum\limits_{a = 1}^{p-1}\left(\frac{a^2+n\bar{a}^2}{p}\right) = 1+\sum\limits_{a = 1}^{p-1}\left(\frac{a^4}{p}\right)\cdot\left(\frac{a^4+n}{p}\right)\\ & = 1+\sum\limits_{a = 1}^{p-1}\left(\frac{a}{p}\right)\cdot\left(\frac{a+n}{p}\right) = 1+\sum\limits_{a = 1}^{p-1}\left(\frac{a^2+na}{p}\right)\\ & = 1+\sum\limits_{a = 1}^{p-1}\left(\frac{1+n\bar{a}}{p}\right) = \sum\limits_{a = 0}^{p-1}\left(\frac{1+na}{p}\right) = 0. \end{align*}$

Thus, for all prime $p$ with $p\equiv3\; \rm mod\; 4$ and $k\geq1$ , we have $K_k(p) = 0$ .

4. Conclusions

The main result of this paper is Theorem 1. It gives an interesting computational formula for $K_k(p)$ with $p\equiv1\; \rm mod\; 4$ . That is, for any integer $k$ , we have the identity

$K_k(p) = (4p+2)\cdot K_{k-2}(p)-8\left(2\alpha^2-p\right)\cdot K_{k-3}(p)+(16\alpha^4-16p\alpha^2+4p-1)\cdot K_{k-4}(p).$

Thus, the problems of calculating a linear recurrence formula of one kind special character sums modulo a prime are given.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The authors are grateful to the anonymous referee for very helpful and detailed comments.

This work is supported by the N.S.F. (11971381, 12371007) of China and Shaanxi Fundamental Science Research Project for Mathematics and Physics (22JSY007).

Conflict of interest

The authors declare no conflicts of interest.

References

[1]	A. Jones, N. Strigul, Is spread of COVID-19 a chaotic epidemic? Chaos Solitons Fract., 142 (2021), 110376. https://doi.org/10.1016/j.chaos.2020.110376 doi: 10.1016/j.chaos.2020.110376
[2]	H. Iro, A modern approach to cassical mechanics, Singarpore: World Scientific, 2015. https://doi.org/10.1142/9655
[3]	A. T. Johnson, Biology for engineers, Boca Raton, Florida: CRC Press, 2018. https://doi.org/10.1201/9781351165648
[4]	K. H. Sun, Chaotic secure communication: Principles and technologies, Beijing: Tsinghua University Press, 2016.
[5]	E. N. Lorenz, Deterministic nonperiodic flow, J. Atmos. Sci., 20 (1963), 130–141.
[6]	O. E. Rössler, An equation for continuous chaos, Phys. Lett. A, 57 (1976), 397–398. https://doi.org/10.1016/0375-9601(76)90101-8 doi: 10.1016/0375-9601(76)90101-8
[7]	G. R. Chen, T. Ueta, Yet another chaotic attractor, Int. J. Bifurcat. Chaos, 9 (1999), 1465–1466. https://doi.org/10.1142/S0218127499001024 doi: 10.1142/S0218127499001024
[8]	Q. G. Yang, Z. C. Wei, G. R. Chen, An unusual 3D autonomous quadratic chaotic system with two stable node-foci, Int. J. Bifurcat. Chaos, 20 (2010), 1061–1083. https://doi.org/10.1142/S0218127410026320 doi: 10.1142/S0218127410026320
[9]	J. C. Sprott, Elegant chaos: Algebraically simple chaotic flows, Singapore: World Scientific, 2010.
[10]	J. C. Sprott, Strange attractors with various equilibrium types, Eur. Phys. J. Spec. Top., 224 (2015), 1409–1419. https://doi.org/10.1140/epjst/e2015-02469-8 doi: 10.1140/epjst/e2015-02469-8
[11]	Z. Wang, Z. C. Wei, K. H. Sun, S. B. He, H. H. Wang, Q. Xu, et al., Chaotic flows with special equilibria, Eur. Phys. J. Spec. Top., 229 (2020), 905–919. https://doi.org/10.1140/epjst/e2020-900239-2 doi: 10.1140/epjst/e2020-900239-2
[12]	G. A. Leonov, N. V. Kuznetsov, Hidden attractors in dynamical systems. From hidden oscillations in Hilbert-Kolmogorov, Aizerman, and Kalman problems to hidden chaotic attractor in Chua circuits, Int. J. Bifurcat. Chaos, 23 (2013), 1330002. https://doi.org/10.1142/S0218127413300024 doi: 10.1142/S0218127413300024
[13]	S. N. Chowdhurry, D. Ghosh, Hidden attractors: A new chaotic system without equilibria, Eur. Phys. J. Spec. Top., 229 (2020), 1299–1308. https://doi.org/10.1140/epjst/e2020-900166-7 doi: 10.1140/epjst/e2020-900166-7
[14]	X. Wang, A. Akgul, S. Cicek, V. T. Pham, D. V. Hoang, A chaotic system with two stable equilibrium points: Dynamics, circuit realization and communication application, Int. J. Bifurcat. Chaos, 27 (2017), 1750130. https://doi.org/10.1142/S0218127417501309 doi: 10.1142/S0218127417501309
[15]	S. Jafari, J. C. Sprott, V. T. Pham, C. Volos, C. B. Li, Simple chaotic 3D flows with surfaces of equilibria, Nonlinear Dyn., 86 (2016), 1349–1358. https://doi.org/10.1007/s11071-016-2968-x doi: 10.1007/s11071-016-2968-x
[16]	S. T. Kingni, V. T. Pham, S. Jafari, P. Woafo, A chaotic system with an infinite number of equilibrium points located on a line and on a hyperbola and its fractional-order form, Chaos Solitons Fract., 99 (2017), 209–218. https://doi.org/10.1016/j.chaos.2017.04.011 doi: 10.1016/j.chaos.2017.04.011
[17]	Y. J. Dong, G. Y. Wang, H. H. Iu, G. R. Chen, L. Chen, Coexisting hidden and self-excited attractors in a locally active memristor-based circuit, Chaos, 30 (2020), 103123. https://doi.org/10.1063/5.0002061 doi: 10.1063/5.0002061
[18]	T. Kapitaniak, G. A. Leonov, Multistability: Uncovering hidden attractors, Eur. Phys. J. Spec. Top., 224 (2015), 1405–1408. https://doi.org/10.1140/epjst/e2015-02468-9 doi: 10.1140/epjst/e2015-02468-9
[19]	N. Wang, G. S. Zhang, N. V. Kuznetsov, H. Bao, Hidden attractors and multistability in a modified Chua's circuit, Commun. Nonlinear Sci. Numer. Simul., 92 (2021), 105494. https://doi.org/10.1016/j.cnsns.2020.105494 doi: 10.1016/j.cnsns.2020.105494
[20]	X. Wang, N. V. Kuznetsov, G. R. Chen, Chaotic systems with multistability and hidden attractors, New York: Springer, 2021. https://doi.org/10.1007/978-3-030-75821-9
[21]	M. N. Zavareh, F. Nazarimehr, K. Rajagopal, S. Jafari, Hidden attractor in a passive motion model of compass-gait robot, Int. J. Bifurcat. Chaos, 28 (2018), 1850171. https://doi.org/10.1142/S0218127418501717 doi: 10.1142/S0218127418501717
[22]	A. Prasad, Existence of perpetual points in nonlinear dynamical systems and its applications, Int. J. Bifurcat. Chaos, 25 (2015), 1530005. https://doi.org/10.1142/S0218127415300050 doi: 10.1142/S0218127415300050
[23]	D. Dudkowski, A. Prasad, T. Kapitaniak, Perpetual points and hidden attractors in dynamical systems, Phys. Lett. A, 379 (2015), 2591–2596. https://doi.org/10.1016/j.physleta.2015.06.002 doi: 10.1016/j.physleta.2015.06.002
[24]	F. Nazarimehr, B. Saedi, S. Jafari, J. C. Sprott, Are perpetual points sufficient for locating hidden attractors? Int. J. Bifurcat. Chaos, 27(2017), 1750037. https://doi.org/10.1142/S0218127417500377 doi: 10.1142/S0218127417500377
[25]	D. Dudkowski, A. Prasad, T. Kapitaniak, Describing chaotic attractors: Regular and perpetual points, Chaos, 28 (2018), 033604. https://doi.org/10.1063/1.4991801 doi: 10.1063/1.4991801
[26]	A. K. Farhan, N. M. G. Al-Saidi, A. T. Maolood, F. Nazarimehr, I. Hussain, Entropy analysis and image encryption application based on a new chaotic system crossing a cylinder, Entropy, 21 (2019), 1–14. https://doi.org/10.3390/e21100958 doi: 10.3390/e21100958
[27]	U. Çavuçoğlu, S. Panahi, A. Akgül, S. Jafari, S. Kaçar, A new chaotic system with hidden attractor and its engineering applications: Analog circuit realization and image encryption, Analog Integr. Circ. Sig. Process, 98 (2019), 85–99. https://doi.org/10.1007/s10470-018-1252-z doi: 10.1007/s10470-018-1252-z
[28]	A. N. Pisarchik, U. Feudel, Control of multistability, Phys. Rep., 540 (2014), 167–218. https://doi.org/10.1016/j.physrep.2014.02.007 doi: 10.1016/j.physrep.2014.02.007
[29]	S. Morfu, B. Nofiele, P. Marquié, On the use of multistability for image processing, Phys. Lett. A, 367 (2007), 192–198. https://doi.org/10.1016/j.physleta.2007.02.086 doi: 10.1016/j.physleta.2007.02.086
[30]	Z. T. Njitacke, S. D. Isaac, T. Nestor, J. Kengne, Window of multistability and its control in a simple 3D Hopfield neural network: Application to biomedical image encryption, Neural Comput. Appl., 33 (2021), 6733–6752. https://doi.org/10.1007/s00521-020-05451-z doi: 10.1007/s00521-020-05451-z
[31]	M. Lines, Nonlinear dynamical systems in economics, CISM, Vol. 476, Vienna: Springer, 2005. https://doi.org/10.1007/3-211-38043-4
[32]	B. Chen, X. X. Cheng, H. Bao, M. Chen, Q. Xu, Extreme multistability and its incremental integral reconstruction in a non-autonomous memcapacitive oscillator, Mathematics, 10 (2022), 1–13. https://doi.org/10.3390/math10050754 doi: 10.3390/math10050754
[33]	J. C. Sprott, S. Jafari, A. J. M. Khalaf, T. Kapitaniak, Megastability: Coexistence of a countable infinity of nested attractors in a periodically-forced oscillator with spatially-periodic damping, Eur. Phys. J. Spec. Top., 226 (2017), 1979–1985. https://doi.org/10.1140/epjst/e2017-70037-1 doi: 10.1140/epjst/e2017-70037-1
[34]	V. Patidar, K. K. Sud, Bifurcation and chaos in simple jerk dynamical systems, Pramana, 64 (2005), 75–93. https://doi.org/10.1007/BF02704532 doi: 10.1007/BF02704532
[35]	G. Innocenti, A. Tesi, R. Genesio, Complex behavior analysis in quadratic jerk systems via frequency domain Hopf bifurcation, Int. J. Bifurcat. Chaos, 20 (2010), 657–667. https://doi.org/10.1142/S0218127410025946 doi: 10.1142/S0218127410025946
[36]	B. Sang, B. Huang, Zero-Hopf bifurcations of 3D quadratic jerk system, Mathematics, 8 (2020), 1–19. https://doi.org/10.3390/math8091454 doi: 10.3390/math8091454
[37]	Z. C. Wei, J. C. Sprott, H. Chen, Elementary quadratic chaotic flows with a single non-hyperbolic equilibrium, Phys. Lett. A, 379 (2015), 2184–2187. https://doi.org/10.1016/j.physleta.2015.06.040 doi: 10.1016/j.physleta.2015.06.040
[38]	K. E. Chlouverakis, J. C. Sprott, Chaotic hyperjerk systems, Chaos Solitons Fract., 28 (2006), 739–746. https://doi.org/10.1016/j.chaos.2005.08.019 doi: 10.1016/j.chaos.2005.08.019
[39]	F. Y. Dalkiran, J. C. Sprott, Simple chaotic hyperjerk system, Int. J. Bifurcat. Chaos, 26 (2016), 1650189. https://doi.org/10.1142/S0218127416501893 doi: 10.1142/S0218127416501893
[40]	J. P. Singh, V. T. Pham, T. Hayat, S. Jafari, F. E. Alsaadi, B. K. Roy, A new four-dimensional hyperjerk system with stable equilibrium point, circuit implementation, and its synchronization by using an adaptive integrator backstepping control, Chinese Phys. B, 27 (2018), 100501. https://doi.org/10.1088/1674-1056/27/10/100501 doi: 10.1088/1674-1056/27/10/100501
[41]	G. D. Leutcho, J. Kengne, L. K. Kengne, Dynamical analysis of a novel autonomous 4-D hyperjerk circuit with hyperbolic sine nonlinearity: Chaos, antimonotonicity and a plethora of coexisting attractors, Chaos Solitons Fract., 107 (2018), 67–87. https://doi.org/10.1016/j.chaos.2017.12.008 doi: 10.1016/j.chaos.2017.12.008
[42]	I. Ahmad, B. Srisuchinwong, W. San-Um, On the first hyperchaotic hyperjerk system with no equilibria: A simple circuit for hidden attractors, IEEE Access, 6 (2018), 35449–35456. https://doi.org/10.1109/ACCESS.2018.2850371 doi: 10.1109/ACCESS.2018.2850371
[43]	P. Ketthong, B. Srisuchinwong, A damping-tunable snap system: From dissipative hyperchaos to conservative chaos, Entropy, 24 (2022), 1–14. https://doi.org/10.3390/e24010121 doi: 10.3390/e24010121
[44]	M. Joshi, A. Ranjan, An autonomous simple chaotic jerk system with stable and unstable equilibria using reverse sine hyperbolic functions, Int. J. Bifurcat. Chaos, 30 (2020), 2050070. https://doi.org/10.1142/S0218127420500704 doi: 10.1142/S0218127420500704
[45]	K. Rajagopal, S. T. Kingni, G. F. Kuiate, V. K. Tamba, V. T. Pham, Autonomous jerk oscillator with cosine hyperbolic nonlinearity: Analysis, FPGA implementation, and synchronization, Adv. Math. Phys., 2018 (2018), 1–12. https://doi.org/10.1155/2018/7273531 doi: 10.1155/2018/7273531
[46]	C. Volos, A. Akgul, V. T. Pham, I. Stouboulos, I. Kyprianidis, A simple chaotic circuit with a hyperbolic sine function and its use in a sound encryption scheme, Nonlinear Dyn., 89 (2017), 1047–1061. https://doi.org/10.1007/s11071-017-3499-9 doi: 10.1007/s11071-017-3499-9
[47]	J. Kengne, Z. T. Njitacke, A. N. Negou, M. F. Tsostop, H. B. Fotsin, Coexistence of multiple attractors and crisis route to chaos in a novel chaotic jerk circuit, Int. J. Bifurcat. Chaos, 26 (2016), 1650081. https://doi.org/10.1142/S0218127416500814 doi: 10.1142/S0218127416500814
[48]	L. K. Kengne, J. Kengne, J. R. M. Pone, H. T. K. Tagne, Symmetry breaking, coexisting bubbles, multistability, and its control for a simple jerk system with hyperbolic tangent nonlinearity, Complexity, 2020 (2020), 1–24. https://doi.org/10.1155/2020/2340934 doi: 10.1155/2020/2340934
[49]	Y. Li, Y. C. Zeng, J. F. Zeng, A unique jerk system with abundant dynamics: Symmetric or asymmetric bistability, tristability, and coexisting bubbles, Braz. J. Phys., 50 (2020), 153–163. https://doi.org/10.1007/s13538-019-00731-z doi: 10.1007/s13538-019-00731-z
[50]	M. Molaie, S. Jafari, J. C. Sprott, S. M. R. H. Golpayegani, Simple chaotic flows with one stable equilibrium, Int. J. Bifurcat. Chaos, 23 (2013), 1350188. https://doi.org/10.1142/S0218127413501885 doi: 10.1142/S0218127413501885
[51]	M. Liu, B. Sang, N. Wang, I. Ahmad, Chaotic dynamics by some quadratic jerk systems, Axioms, 10 (2021), 1–18. https://doi.org/10.3390/axioms10030227 doi: 10.3390/axioms10030227
[52]	C. B. Li, J. C. Sprott, W. J. C. Thio, Z. Y. Gu, A simple memristive jerk system, IET Circ. Device. Syst., 15 (2021), 388–392. https://doi.org/10.1049/CDS2.12035 doi: 10.1049/CDS2.12035
[53]	H. G. Tian, Z. Wang, P. J. Zhang, M. S. Chen, Y. Wang, Dynamic analysis and robust control of a chaotic system with hidden attractor, Complexity, 2021 (2021), 1–11. https://doi.org/10.1155/2021/8865522 doi: 10.1155/2021/8865522
[54]	S. Jafari, J. C. Sprott, S. M. R. H. Golpayegani, Elementary quadratic chaotic flows with no equilibria, Phys. Lett. A, 377 (2013), 699–702. https://doi.org/10.1016/j.physleta.2013.01.009 doi: 10.1016/j.physleta.2013.01.009
[55]	S. Zhang, Y. C. Zeng, A simple jerk-like system without equilibrium: Asymmetric coexisting hidden attractors, bursting oscillation and double full feigenbaum remerging trees, Chaos Solitons Fract., 120 (2019), 25–40. https://doi.org/10.1016/j.chaos.2018.12.036 doi: 10.1016/j.chaos.2018.12.036
[56]	K. Rajagopal, S. T. Kingni, G. H. Kom, V. T. Pham, A. Karthikeyan, S. Jafari, Self-excited and hidden attractors in a simple chaotic jerk system and in its time-delayed form: Analysis, electronic implementation, and synchronization, J. Korean Phys. Soc., 77 (2020), 145–152. https://doi.org/10.3938/jkps.77.145 doi: 10.3938/jkps.77.145
[57]	J. Guckenheimer, P. Holmes, Nonlinear oscillation, dynamical systems, and bifurcations of vector fields, New York: Springer, 1983. https://doi.org/10.1007/978-1-4612-1140-2
[58]	L. Perko, Differential equations and dynamical systems, New York: Springer, 2001. https://doi.org/10.1007/978-1-4613-0003-8
[59]	B. D. Hassard, N. D. Kazarinoff, Y. H. Wan, Theory and applications of Hopf bifurcation, Cambridge: Cambridge University Press, 1981.
[60]	Y. A. Kuznetsov, Elements of applied bifurcation theory, New York: Springer, 1998.
[61]	B. Sang, B. Huang, Bautin bifurcations of a financial system, Electron. J. Qual. Theory Differ. Equ., 2017 (2017), 1–22. https://doi.org/10.14232/ejqtde.2017.1.95 doi: 10.14232/ejqtde.2017.1.95
[62]	B. Sang, Focus quantities with applications to some finite-dimensional systems, Math. Methods Appl. Sci., 44 (2021), 464–475. https://doi.org/10.1002/mma.6750 doi: 10.1002/mma.6750
[63]	T. Asada, W. Semmler, Growth and finance: An intertemporal model, J. Macroeconom., 17 (1995), 623–649. https://doi.org/10.1016/0164-0704(95)80086-7 doi: 10.1016/0164-0704(95)80086-7

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