Finnish perspective on using synthetic health data to protect privacy: the PRIVASA project

Tinja Pitkämäki; Tapio Pahikkala; Ileana Montoya Perez; Parisa Movahedi; Valtteri Nieminen; Tom Southerington; Juho Vaiste; Mojtaba Jafaritadi; Muhammad Irfan Khan; Elina Kontio; Pertti Ranttila; Juha Pajula; Harri Pölönen; Aysen Degerli; Johan Plomp; Antti Airola; Tinja Pitkämäki; Tapio Pahikkala; Ileana Montoya Perez; Parisa Movahedi; Valtteri Nieminen; Tom Southerington; Juho Vaiste; Mojtaba Jafaritadi; Muhammad Irfan Khan; Elina Kontio; Pertti Ranttila; Juha Pajula; Harri Pölönen; Aysen Degerli; Johan Plomp; Antti Airola

doi:10.3934/aci.2024009

Applied Computing and Intelligence

2024, Volume 4, Issue 2: 138-163. doi: 10.3934/aci.2024009

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Overview

Finnish perspective on using synthetic health data to protect privacy: the PRIVASA project

1.
Department of Computing, University of Turku, Turku, Finland
2.
Faculty of Law, University of Turku, Turku, Finland
3.
Finnish Biobank Cooperative (FINBB), Turku, Finland
4.
Turku School of Economics, University of Turku, Turku, Finland
5.
Faculty of Engineering and Business, Turku University of Applied Sciences, Turku, Finland
6.
VTT Technical Research Centre of Finland Ltd., Tampere, Finland

Academic Editor: Pasi Fränti

Received: 10 September 2024 Revised: 30 September 2024 Accepted: 01 October 2024 Published: 28 October 2024

The use of synthetic data could facilitate data-driven innovation across industries and applications. Synthetic data can be generated using a range of methods, from statistical modeling to machine learning and generative AI, resulting in datasets of different formats and utility. In the health sector, the use of synthetic data is often motivated by privacy concerns. As generative AI is becoming an everyday tool, there is a need for practice-oriented insights into the prospects and limitations of synthetic data, especially in the privacy sensitive domains. We present an interdisciplinary outlook on the topic, focusing on, but not limited to, the Finnish regulatory context. First, we emphasize the need for working definitions to avoid misplaced assumptions. Second, we consider use cases for synthetic data, viewing it as a helpful tool for experimentation, decision-making, and building data literacy. Yet the complementary uses of synthetic datasets should not diminish the continued efforts to collect and share high-quality real-world data. Third, we discuss how privacy-preserving synthetic datasets fall into the existing data protection frameworks. Neither the process of synthetic data generation nor synthetic datasets are automatically exempt from the regulatory obligations concerning personal data. Finally, we explore the future research directions for generating synthetic data and conclude by discussing potential future developments at the societal level.

Keywords:

Citation: Tinja Pitkämäki, Tapio Pahikkala, Ileana Montoya Perez, Parisa Movahedi, Valtteri Nieminen, Tom Southerington, Juho Vaiste, Mojtaba Jafaritadi, Muhammad Irfan Khan, Elina Kontio, Pertti Ranttila, Juha Pajula, Harri Pölönen, Aysen Degerli, Johan Plomp, Antti Airola. Finnish perspective on using synthetic health data to protect privacy: the PRIVASA project[J]. Applied Computing and Intelligence, 2024, 4(2): 138-163. doi: 10.3934/aci.2024009

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Abstract

1. Introduction

Hyperchaotic system is characterized as a chaotic system with at least two Lyapunov exponents ^[1] and the minimal dimension of the phase space that embeds the hyperchaotic attractor should be more than three. These imply that hyperchaos has more complex, dynamical phenomena than chaos. Compared to chaos, hyperchaos has greater potential applications due to its higher dimensions, stronger randomness and unpredictability such as secure communications ^[2,3], nonlinear circuits ^[4,5], lasers ^[6,7], et al. In addition, as far as we know, the complexity of the hyperchaotic system dynamics has not been completely mastered by researchers until now. There are few studies on rich dynamical behaviors of hyperchaotic systems. Furthermore, it is difficult to find effective ways to analysis and study the complex dynamical phenomena of the high-dimensional hyperchaotic systems. Thus, it is necessary to formulate new high-dimensional hyperchaotic systems and to further investigate the properties of hyperchaos. Meanwhile, there have been many hyperchaotic systems presented. By adding a controlled variable to the Lorenz system, a hyperchaotic system was obtained in ^[8,9]. ^[10] presented a 4D generalised Lorenz hyperchaotic system by introducing a linear state feedback control to the first state equation of a generalised Lorenz system. ^[11] constructed a memristive hyperchaotic system by adding the flux-controlled memristor to an extended jerk system. Based on Chua's electrical circuit, ^[12] constructed a hyperchaotic system by introducing an additional inductor and ^[13] designed a hyperchaotic circuit system by replacing the non-linear element with an experimentally realizable memristor. Furthermore, there are also some scholars who constructed their own hyperchaotic systems ^[14,15].

In ^[16], a chaotic system was introduced

$\begin{equation} \left \{ \begin{array}{l} \dot u = a(v-u)+evw, \\ \dot v = cu+dv-uw, \\ \dot w = -bw+uv, \end{array} \right. \end{equation}$

(1.1)

where $(u, v, w)^{T}\in { \mathbb R^3}$ is the state vector. $a, \, b, \, c$ are positive real parameters and $c\in R$ . When $(a, b, c, d, e) = (14, 43, -1, 16, 4)$ , system (1.1) has four-wing chaotic attractor. The chaotic system was different from the Lorenz system family and some scholars have investigated the system including mechanical analysis, chaos control, energy cycle, bound and constructing 4D hyperchaotic systems ^{[17,18,19,20,21,22]}. In this work, we introduce a new 4D hyperchaotic system by adding a linear controller to system (1.1) as:

$\begin{equation} \left \{ \begin{array}{l} \dot u = a(v-u)+evw, \\ \dot v = cu+dv-uw+mp, \\ \dot w = -bw+uv, \\ \dot p = -ku-kv, \end{array} \right. \end{equation}$

(1.2)

where $a, \, b, \, c, \, d, \, e, \, m, \, k\,$ are positive real parameters and $c\in R$ .

The rest of this paper is organized as follows: in the second section, dissipativity and invariance, equilibria and their stability of system (1.2) are discussed. In addition, the complex dynamical behaviors such as quasi-periodicity, chaos and hyperchaos are numerically verified by Lyapunov exponents, bifurcation and Poincaré maps. In the third section, the Hopf bifurcation at the zero equilibrium point of system (1.2) is investigated. In addition, two examples are given to test and verify the theoretical results. The new system always has two unstable nonzero equilibrium points and has rich dynamical behaviors under different system parameters. Thus, the new 4D system may be more useful in some fields such as image encryption, secure communication. In the fourth section, the hyperchaos control is studied. The results show that the linear feedback control method can achieve a good control effect by selecting appropriate feedback coefficients. In the last section, the conclusions are summarized.

2. Dynamical analysis

2.1. Dissipativity and invariance

We can see that system (1.2) is invariant for the coordinate transformation

$(u, v, w, p)\rightarrow (-u, -v, w, -p).$

Then, the nonzero equilibria of (1.2) is symmetric with respect to $w$ axis. The divergence of (1.2) is

$\nabla W = \frac{\partial \dot u}{\partial u}+\frac{\partial \dot v}{\partial v}+\frac{\partial \dot w}{\partial w}+\frac{\partial \dot p}{\partial p} = -(a+b-d).$

By Liouville's theorem, we get

$\frac{dV(t)}{dt} = \int_{\Sigma(t)} \, (d-a-b)\, du\, dv\, dw\, dp = -(a+b-d)V(t),$

where $\Sigma(t)$ is any region in $R^{4}$ with smooth boundary and $\Sigma(t) = \Sigma_{0}(t)$ , $\Sigma_{0}(t)$ denotes the flow of $W$ and the hypervolume of $\Sigma(t)$ is $V(t)$ . For initial volume $V(0)$ , by integrating the equation, we have

$V(t) = exp(-(a+b-d) t)V(0), \, (\forall t\geq 0).$

Hence, system (1.2) is dissipative if and only if $a+b-d > 0$ . It shows that each volume containing the system trajectories shrinks to zero as $t\to \infty$ at an exponential rate $-(a+b-d)$ . There exists an attractor in system (1.2).

2.2. Equilibria

By computations, system (1.2) has three equilibrium points

$E_{0} = (0, 0, 0, 0)$

and

$E_{1} = (\sqrt{\frac{2ab}{e}}, -\sqrt{\frac{2ab}{e}}, -\frac{2a}{e}, \, \frac{\sqrt{2 eab}(de-ce-2 ad)}{e^{2}m}),$

$E_{2} = (-\sqrt{\frac{2ab}{e}}, \sqrt{\frac{2ab}{e}}, -\frac{2a}{e}, -\frac{\sqrt{2 eab}(de-ce-2 ad)}{e^{2}m}).$

The characteristic equation of Jacobian matrix at $E_{0}$ is

$\begin{equation} (\lambda+b)[{\lambda}^{3}+ ( a-d ) {\lambda}^{2}+ (km-ac-da ) \lambda+2\, akm]. \end{equation}$

(2.1)

According to Routh-Hurwitz criterion ^[23], the real parts of eigenvalues are negative if and only if

$a > d, a{d}^{2}+ ( ac-{a}^{2}-km ) d > {a}^{2}c+akm.$

The Jacobian matrices at $E_{1}$ and $E_{2}$ have the same characteristic equation, i.e.,

$\lambda^4+e(a+b-d)\lambda^3+a_{2}\lambda^2+a_{1}\lambda-4abkm,$

where

$a_{2} = km-ab+ac-ad-bd+\frac{2\, {a}^{2}d+2\, abd}{e}, \; a_{1} = (3 ac+ad+km)b+\frac{10 dba^{2}}{e}.$

Note $4abkm > 0$ , then the two nonzero equilibrium points are unstable.

2.3. Lyapunov exponents and Poincaré maps

In this subsection, some properties of the system (1.2) are discussed and the simulation results are further obtained by using numerical methods. Firstly, fix $a = 15, \, b = 43, \, c = 1, \, d = 16, \, e = 5, \, m = 5$ and varies $k$ . indicates the Lyapunov exponent spectrum of system (1.2) with respect to $k\in[1.5, \, 5.5]$ and the corresponding bifurcation diagram is given in Figure 2. From Figures 1 and , the complex dynamical behaviors of system (1.2) can be clearly observed. indicates that system (1.2) has hyperchaotic attractors with two positive Lyapunov exponents when $k$ varies in $[1.5, \, 5.5]$ . When $(a, b, c, d, e, m, k) = (15, 43, 1, 16, 5, 5, 2)$ , system (1.2) has three unstable equilibrium points

$E_{0} = (0, 0, 0, 0), E_{1} = (-16.062, 16.062, -6,260.210), E_{2} = (16.062, -16.062, -6, -260.210).$

Figure 1. Lyapunov exponent spectrum of (1.2) with

$k\in[1.5, \, 5.5]$ .

DownLoad: Full-Size Img PowerPoint

Figure 2. Bifurcation diagram of

$x$ .

DownLoad: Full-Size Img PowerPoint

The first and the second Lyapunov exponents are $4.273$ and $0.092$ , these imply that system (1.2) is hyperchaotic. shows the $(u, v, w, p)$ 4D surface of section and the location of the consequents is given in the $(u, v, w)$ subspace and are colored according to their $p$ value. shows the Poincaré maps on $u-p$ plane and $w-p$ plane, respectively.

Figure 3. Projection on

$(u, v, w, p)$ .

DownLoad: Full-Size Img PowerPoint

Figure 4. Poincaré maps on the

$u-p$ plane and

$w-p$ plane.

DownLoad: Full-Size Img PowerPoint

Assume $b = 43, \, a = 15, \, c = 1, \, e = 5, \, k = 5, \, d = 16$ , the different Lyapunov exponents and dynamical properties with different values of parameter $m$ are given in . As we can see in , system (1.2) is hyperchaotic with different $m$ values. When $b = 43,$ $a = 15,$ $c = 1,$ $e = 5,$ $k = 5,$ $d = 16,$ $m = 0.5$ , the hyperchaotic attractor on $(v, u, w, p)$ space is depicted in , the location of the consequents is given in the $(v, u, w)$ subspace and are colored according to their $p$ value. shows the Poincaré map on $v-w$ plane. It shows that system (1.2) has different hyperchaotic phenomena under different parameter values.

Table 1. Lyapunov exponents of (1.2) with

$(b, a, c, e, k, d) = (43, 15, 1, 5, 5, 16)$ .

$m$	$LE_{1}$	$LE_{2}$	$LE_{3}$	$LE_{4}$	Dynamics
$0.5$	$3.827$	$0.036$	$-0.000$	$-45.862$	Hyperchaos
$1$	$3.901$	$0.052$	$0.000$	$-45.953$	Hyperchaos
$1.5$	$3.943$	$0.081$	$-0.000$	$-46.022$	Hyperchaos
$2$	$4.277$	$0.091$	$-0.000$	$-46.364$	Hyperchaos
$2.5$	$4.221$	$0.106$	$-0.000$	$-46.325$	Hyperchaos
$3$	$3.976$	$0.075$	$-0.000$	$-46.040$	Hyperchaos

| Show Table

DownLoad: CSV

Figure 5. Projection on

$(v, u, w, p)$ .

DownLoad: Full-Size Img PowerPoint

Figure 6. Poincaré map on

$v-w$ plane.

DownLoad: Full-Size Img PowerPoint

When $(b, a, c, d, e, m, k) = (45, 15, 2, 15.9, 5, 0.1, 0.1)$ , (1.2) has three unstable equilibrium points

$E_{0} = (0, 0, 0, 0),$

$E_{1} = (-16.431, 16.431, -6, 13391.816),$

$E_{2} = (16.431, -16.431, -6, -13391.816).$

The Lyapunov exponents are $LE_{1} = 1.979$ , $LE_{2} = 0.000$ , $LE_{3} = -0.002$ , $LE_{4} = -46.077$ , system (1.2) is chaotic. shows the $(v, u, w, p)$ 4D surface of section and the location of the consequents is given in the $(v, u, w)$ subspace and are colored according to their $p$ value. When $(b, a, c, d, e, m, k) = (45, 15, 2, 15.01, 5, 0.1, 0.1)$ , system (1.2) has three unstable equilibrium points

$O = (0, 0, 0, 0),$

$E_{1} = (-16.431, 16.431, -6, 12660.606),$

$E_{2} = (16.431, -16.431, -6, -12660.606).$

Figure 7. Projection on

$(v, u, w, p)$ .

DownLoad: Full-Size Img PowerPoint

The Lyapunov exponents are $LE_{1} = -0.000$ , $LE_{2} = -0.000$ , $LE_{3} = -0.020$ , $LE_{4} = -44.963$ , system (1.2) has quasi-periodic orbit. The $(v, u, w, p)$ 4D surface of section is depicted in , the location of the consequents is given in the $(v, u, w)$ subspace and are colored according to their $p$ value.

Figure 8. Projection on

$(v, u, w, p)$ .

DownLoad: Full-Size Img PowerPoint

When $(b, a, c, d, e, m, k) = (45, 15, 2, 15.1, 5, 0.1, 0.1)$ , system (1.2) has three unstable equilibrium points

$E_{0} = (0, 0, 0, 0),$

$E_{1} = (-16.431, 16.431, -6, 12734.549),$

$E_{2} = (16.431, -16.431, -6, -12734.549).$

The Lyapunov exponents are $LE_{1} = 0.066$ , $LE_{2} = 0.000$ , $LE_{3} = -0.005$ , $LE_{4} = -44.961$ , system (1.2) is chaotic. When $(b, a, c, d, e, m, k) = (45, 15, 2, 15.04, 5, 0.1, 0.1)$ , system (1.2) has three unstable equilibrium points

$E_{0} = (0, 0, 0, 0),$

$E_{1} = (-16.431, 16.431, -6, 12685.254),$

$E_{2} = (16.431, -16.431, -6, -12685.254).$

The Lyapunov exponents are $LE_{1} = 0.000$ , $LE_{2} = -0.000$ , $LE_{3} = -0.010$ , $LE_{4} = -44.948$ , system (1.2) is quasi-periodic. Similar to and , the corresponding chaotic attractor and quasi-periodic orbit on $(v, u, w, p)$ space are given in Figures 9 and 10, respectively. Overall, the results indicate that system (1.2) has rich and complex dynamical behaviors including hyperchaos, chaos and quasi-periodicity with different parameters.

Figure 9. Projection on

$(v, u, w, p)$ .

DownLoad: Full-Size Img PowerPoint

Figure 10. Projection on

$(v, u, w, p)$ .

DownLoad: Full-Size Img PowerPoint

3. Hopf bifurcation

Theorem 3.1. Suppose that $a-d > 0$ and $c+d < 0$ are satisfied. Then, as $k$ varies and passes through the critical value $k = \frac{a(c+d)(d-a)}{m(a+d)}$ , system $(1.2)$ undergoes a Hopf bifurcation at $O(0, 0, 0, 0)$ .

Proof. Assume that system (1.2) has a pure imaginary root $\lambda = { i}\omega, \, (\omega \in {\mathbb {R}}^{+})$ . From (2.1), we get

$(a-d)\omega^2-2akm = 0, \omega^3-(km-ac-ad)\omega = 0,$

then

$\omega = \omega_{0} = \sqrt{km-ac-ad}, k = k_{0} = \frac{a(c+d)(d-a)}{m(a+d)}.$

Substituting $k = k_{0}$ into (2.1), we have

$\lambda_{1} = { i}\omega_{0}, \lambda_{2} = -{ i}\omega_{0}, \lambda_{3} = d-a, \lambda_{4} = -b.$

Therefore, when $a-d > 0$ , $c-d < 0$ and $k = k_{0}$ , the first condition for Hopf bifurcation ^[24] is satisfied. From (2.1), we have

${\rm Re}(\lambda'(k_{0}))|_{\lambda = { i}\omega_{0}} = \frac{m(a+d)^2}{2(a+d)(a-d)^2-4a^2(c+d)} > 0.$

Thus, the second condition for a Hopf bifurcation ^[24] is also met. Hence, Hopf bifurcation exists.

Remark 3.1. When $km-ac-ad\leq 0$ , system $(1.2)$ has no Hopf bifurcation at the zero equilibrium point.

Theorem 3.2. When $a > d$ and $c+d < 0$ , the periodic solutions of $(1.2)$ from Hopf bifurcation at $O(0, 0, 0, 0)$ exist for sufficiently small $0 < |k-k_{0}| = |k-\frac{a(c+d)(d-a)}{m(a+d)}|$ . And the periodic solutions have the following properties:

(I) if $\frac{\delta_{2}}{\delta_{1}} > 0$ (resp., $\frac{\delta_{2}}{\delta_{1}} < 0$ ), then the Hopf bifurcation of system $(1.2)$ at $(0, 0, 0, 0)$ is non-degenerate and supercritical (resp. subcritical), and the bifurcating periodic solution exists for $k < k_{0}$ (resp., $k > k_{0}$ ) and is stable (resp., unstable), where

$\begin{equation} \delta_{1} = -b\sqrt {-{\frac {{a}^{2} ( c+d ) }{a+d}}} ( {a}^{3}-{ a}^{2}d+2\, acd+a{d}^{2}+{d}^{3} ) ( a+d ) ( 8 \, {a}^{2}c+8\, {a}^{2}d-{b}^{2}a-{b}^{2}d), \nonumber \end{equation}$

$\begin{array}{l} \delta_{2} = ( 16\, c+16\, d ) {a}^{5}+ ( 2\, bce+2\, bde+16\, cde+16\, {d}^{2}e-3\, {b}^{2}+4\, bc+4\, bd ) {a}^{4}+ ( {b}^{2}ce-2\, { b}^{2}de\\ -4\, b{c}^{2}e-6\, bcde-2\, b{d}^{2}e-16\, {c}^{2}de-32\, c{d}^{2}e -16\, {d}^{3}e-3\, {b}^{2}d+4\, bcd+\\4\, b{d}^{2}-16\, c{d}^{2}-16\, {d}^{3} ) {a}^{3}\\ + ( -2\, {b}^{2}{c}^{2}e+2\, {b}^{2}cde+{b}^{2}{d}^ {2}e+2\, bc{d}^{2}e+2\, b{d}^{3}e+16\, {c}^{2}{d}^{2}e+16\, c{d}^{3}e-2\, { b}^{2}cd+{b}^{2}{d}^{2} ) {a}^{2}\\ + ( -2\, {b}^{2}{c}^{2}de-3 \, {b}^{2}c{d}^{2}e+2\, {b}^{2}{d}^{3}e-4\, b{c}^{2}{d}^{2}e-6\, bc{d}^{3} e-2\, b{d}^{4}e-2\, {b}^{2}c{d}^{2}+{b}^{2}{d}^{3} ) a- \\4\, {b}^{2}c {d}^{3}e-{b}^{2}{d}^{4}e; \end{array}$

(II) the period and characteristic exponent of the bifurcating periodic solution are

$T = \frac{2\pi}{\omega_{0}}(1+\tau_{2}\varepsilon^2+O(\varepsilon^4)), \; \; \beta = \beta_{2}\varepsilon^2+O(\varepsilon^4),$

where

$\varepsilon = \frac{k-k_{0}}{\mu_{2}}+O[(k-k_{0})^2], \; \; \beta_{2} = \frac{a^4}{4\delta_{1}}[-2\, \sqrt {-{\frac {{a}^{2} ( c+d ) }{a+d}}}( c+d ) \delta_{2}],$

$\tau_{2} = {\frac {\delta_{{2}}{a}^{3}}{4\delta_{{1}} ( a+d ) } [ \sqrt {2}\sqrt {-{\frac {{a}^{2}( c+d ) }{a+d}}} ( {a}^{2}-ac-ad-{d}^{2} ) +{a}^{2}c+{a}^{2}d+acd+a{d}^{2} ] },$

$\mu_{2} = \frac{a^4}{4m \delta_{1}(a+d)^2}\sqrt {-{\frac {{a}^{2} ( c+d ) }{a+d}}}( c+d ) \delta_{2}[2(a+d)(a-d)^2-4a^2(c+d)];$

(III) the expression of the bifurcating periodic solution is

$\left[ \begin{array}{c} u\\ v \\ w \\ p \end{array} \right] = \left[ \begin{array}{c} \varepsilon a\sqrt {-{\frac {2{a}^{2} \left( c+d \right) }{a+d}}} \cos \left({\frac {2\pi t}{T}} \right) \\ \varepsilon[ a\sqrt {-{\frac {2{a}^{2} \left( c+d \right) }{a+d}}} \cos \left({\frac {2\pi t}{T}} \right) +{\frac {2{a}^{2} \left( c+d \right) }{a+d}\sin \left({\frac {2\pi t}{T}} \right)] }\\ { \varepsilon }^{2}L\\ \varepsilon{\frac {a ( c+d ) ( -d+a ) }{m ( a+d ) } [ \sqrt {-{\frac {2{a}^{2}( c+d ) }{a+d}}}\cos \left( {\frac {2\pi t}{T}} \right) +2 a\sin \left( {\frac {2\pi t}{T}} \right) ] } \end{array} \right] +O(\varepsilon^{3}),$

where

$L = -\frac{a^4(c+d)}{ab+bd}+\delta_{4}[2{\rm cos}^{2}(\frac{2\pi t}{T})-1]-\delta_{5}{\rm sin}(\frac{4\pi t}{T}),$

$\delta_{4} = \frac{a^3(c+d)(4\, {a}^{2}c+4\, {a}^{2}d-{a}^{2}b-adb)}{(a+d)({b}^{2}a+{b}^{2}d-8\, {a}^{2}c-8\, {a}^{2}d)}, \; \; \delta_{5} = \frac{a^3(c+d) ( 2 \, a-b ) ( a+d ) \sqrt {-{\frac {2{a}^{2} ( c+d ) }{a+d}}}}{(a+d)({b}^{2}a+{b}^{2}d-8\, {a}^{2}c-8\, {a}^{2}d)}.$

Proof. Let $k = k_{0}$ , by straightforward computations, we can obtain

$t_{1} = \left[ \begin{array}{c} a\omega_{0} \\ a\omega_{0}+(k_{0}m-ac-ad){ i}\\ 0 \\ 2ak_{0}{ i}-k_{0}\omega_{0} \end{array} \right], \; \; t_{3} = \left[ \begin{array}{c} 0 \\ 0\\ 1 \\ 0 \end{array} \right], \; \; t_{4} = \left[ \begin{array}{c} ad-a^2 \\ d^2-ad\\ 0 \\ k_{0}(a+d) \end{array} \right],$

which satisfy

$Jt_{1} = { i}\omega_{0}t_{1}, \; \; Jt_{3} = -bt_{3}, \; \; Jt_{4} = (d-a)t_{4},$

where

$J = \left[ \begin{array}{cccc} -a&a&0&0\\ c&d&0&m \\ 0&0&-b&0\\ -k&-k&0&0 \end{array} \right].$

Now, we use transformation $X = QX_{1}$ , where

$X = (u, v, w, p)^{T}, \; \; X_{1} = (u_{1}, v_{1}, w_{1}, p_{1})^{T},$

and

$Q = \left[ \begin{array}{cccc} aw&0&0&-{a}^{2}+ad\\ aw &ac+ad-k_{{0}}m&0&-ad+{d}^{2}\\ 0&0&1&0 \\ -k_{{0}}w&-2\, ak_{{0}}&0&k_{{0}} \left( a+d \right) \end{array} \right],$

then, system (1.2) is transformed into

$\begin{equation} \left \{ \begin{array}{l} \dot u_{1} = -\omega_{0}v_{1}+F_{1}(u_{1}, v_{1}, w_{1}, p_{1}), \\ \dot v_{1} = \omega_{0}u_{1}+F_{2}(u_{1}, v_{1}, w_{1}, p_{1}), \\ \dot w_{1} = -bw_{1}+F_{3}(u_{1}, v_{1}, w_{1}, p_{1}), \\ \dot p_{1} = -(a-d)p_{1}+F_{4}(u_{1}, v_{1}, w_{1}, p_{1}), \end{array} \right. \end{equation}$

(3.1)

where

$\begin{array}{l} F_{1}(u_{1}, v_{1}, w_{1}, p_{1}) = -\frac{w_{1}}{a\sqrt {-\, {\frac {2{a}^{2} ( c+d ) }{a+d}}} ( {a}^{3 }-{a}^{2}d+2\, acd+a{d}^{2}+{d}^{3} ) }\\ [\sqrt {-{\frac {2{a}^{2} ( c+d ) }{a+d}}}a ( a +d ) ( {a}^{2}-ace-{d}^{2}e\\ -ad ) u_{1}-2\, {a}^{2}e ( c+d ) ( ac+{d}^{2} ) v_{1}-p_{{1}} ( a-d ) ( a+d ) \\({a}^{3} -acde-{d}^{3}e -{a}^{2}d ) ], \end{array}$

$\begin{equation*} \begin{split} F_{2}(u_{1}, v_{1}, w_{1}, p_{1}) = &-\frac{w_{1}}{2a^2( {a}^{3 }-{a}^{2}d+2\, acd+a{d}^{2}+{d}^{3} ) }[\sqrt {-\, {\frac {2{a}^{2} ( c+d ) }{a+d}}}a ( a+d ) ( {a}^{2}e+{d}^{2}e\\&+2\, ad ) u_{{1}} +2\, e{a}^{2}( {a}^{2} +{d}^{2} ) ( c+d ) v_{{1}}-dp_{{1}} ( a-d ) ( a+d ) ( {a}^{2}e+{d}^ {2}e+2\, {a}^{2}) ], \end{split} \end{equation*}$

$\begin{equation*} \begin{split} F_{3}(u_{1}, v_{1}, w_{1}, p_{1}) = &\frac{a}{a+d}(\sqrt {-{\frac {2{a}^{2} ( c+d ) }{a+d}}}u_{{1}}-p_ {{1}}a+dp_{{1}} )[au_{{1}} ( a+d ) \sqrt {-{\frac {2{a}^{2} ( c+ d ) }{a+d}}} \\&+2\, {a}^{2}v_{{1}}c-{a}^{2}dp_{{1}}+2\, {a}^{2}v_{{1}}d+{d}^{3}p_{{1}} ], \end{split} \end{equation*}$

$\begin{equation*} \begin{split} F_{4}(u_{1}, v_{1}, w_{1}, p_{1}) = &-\frac{w_{1}}{(a+d)[a^3+d^3-ad(a-d-2 c)]}[\sqrt {-{\frac {2{a}^{2}( c+d ) }{a+d}}}a ( a +d ) ( ae-ce\\&+a+d ) u_{{1}}+2\, {a}^{2}e ( c+d ) ( a-c ) v_{{1}}-p_{{1}} ( a-d ) ( a+d ) ( ade-cde+{a}^{2 }+ad) ]. \end{split} \end{equation*}$

Furthermore,

$g_{11} = \frac{1}{4}[\frac{\partial^2F_{1}}{\partial u_{1}^2}+\frac{\partial^2F_{1}}{\partial v_{1}^2}+{ i}(\frac{\partial^2F_{2}}{\partial u_{1}^2}+\frac{\partial^2F_{2}}{\partial v_{1}^2})] = 0,$

$g_{02} = \frac{1}{4}[\frac{\partial^2F_{1}}{\partial u_{1}^2}-\frac{\partial^2F_{1}}{\partial v_{1}^2}-\frac{2\partial^2 F_{2}}{\partial u_{1} \partial v_{1}}+{ i}(\frac{\partial^2F_{2}}{\partial u_{1}^2}-\frac{\partial^2F_{2}}{\partial v_{1}^2}+\frac{2\partial^2 F_{1}}{\partial u_{1} \partial v_{1}})] = 0,$

$g_{20} = \frac{1}{4}[\frac{\partial^2F_{1}}{\partial u_{1}^2}-\frac{\partial^2F_{1}}{\partial v_{1}^2}+\frac{2\partial^2 F_{2}}{\partial u_{1} \partial v_{1}}+{ i}(\frac{\partial^2F_{2}}{\partial u_{1}^2}-\frac{\partial^2F_{2}}{\partial v_{1}^2}-\frac{2\partial^2 F_{1}}{\partial u_{1} \partial v_{1}})] = 0,$

$G_{21} = \frac{1}{8}[\frac{\partial^3F_{1}}{\partial u_{1}^3}+\frac{\partial^3F_{2}}{\partial v_{1}^3}+\frac{\partial^3 F_{1}}{\partial u_{1} \partial v_{1}^2}+ \frac{\partial^3 F_{2}}{\partial u_{1}^2 \partial v_{1}}+{ i}(\frac{\partial^3F_{2}}{\partial u_{1}^3}-\frac{\partial^3F_{2}}{\partial v_{1}^3}+\frac{\partial^3 F_{2}}{\partial u_{1} \partial v_{1}^2} -\frac{\partial^3 F_{1}}{\partial u_{1}^2 \partial v_{1}})] = 0.$

By solving the following equations

$\left[\begin{array}{cc} -b&0\\ 0&-(a-d) \end{array} \right]\left[ \begin{array}{c} \omega_{{11}}^1\\ \omega_{{ 11}}^{2}\end{array} \right] = -\left[ \begin{array}{c} h_{{11}}^{1}\\ h_{{11}}^{2} \end{array} \right],$

$\left[ \begin{array}{cc} -b-2{ i}\omega_{{0}}&0 \\ 0&-(a-d)-2{ i}\omega_{{0}} \end{array} \right]\left[ \begin{array}{c} \omega_{{20}}^1\\ \omega_{{20}}^{2}\end{array} \right] = -\left[ \begin{array}{c} h_{{20}}^{1}\\ h_{{20}}^{2}\end{array} \right],$

where

$h_{11}^{1} = -{\frac {{a}^{4} \left( c+d \right) }{a+d}},$

$h_{11}^{2} = \frac{1}{4}(\frac{\partial^2F_{4}}{\partial x_{1}^2}+\frac{\partial^2F_{4}}{\partial y_{1}^2}) = 0,$

$h_{20}^{1} = -{a}^{4} ( c+d ) -{ i}{a}^{3}( c+d )\sqrt {-{\frac {2{a}^{2} ( c+d ) }{a+d}}},$

$h_{20}^{2} = \frac{1}{4}(\frac{\partial^2F_{4}}{\partial x_{1}^2}-\frac{\partial^2F_{4}}{\partial y_{1}^2}-2{ i}\frac{\partial^2F_{4}}{\partial x_{1}\partial y_{1}}) = 0,$

one obtains

$\omega_{11}^{1} = -\frac{a^4(c+d)}{ab+bd}, \; \; \omega_{11}^{2} = 0, \; \; \omega_{20}^{2} = 0,$

$\begin{equation*} \begin{split} \omega_{20}^{1} = &\frac{a^3(c+d)}{(a+d)({b}^{2}a+{b}^{2}d-8\, {a}^{2}c-8\, {a}^{2}d)}[4\, {a}^{2}c+4\, {a}^{2}d-{a}^{2}b-adb\\&+{ i}\sqrt {-{\frac {2{a}^{2} ( c+d ) }{a+d}}} ( 2 \, a-b ) ( a+d ) ], \end{split} \end{equation*}$

$\begin{equation*} \begin{split} G_{110}^{1} = &\frac{1}{2}[(\frac{\partial^2F_{1}}{\partial u_{1}\partial w_{1}}+\frac{\partial^2F_{2}}{\partial v_{1}\partial w_{1}})+{ i}(\frac{\partial^2F_{2}}{\partial u_{1}\partial w_{1}}-\frac{\partial^2F_{1}}{\partial v_{1}\partial w_{1}})]\\ = &-\frac{1}{4\sqrt {-{\frac {{a}^{2} ( c+d ) }{a+d}}}( {a}^{3}-{a} ^{2}d+2\, acd+a{d}^{2}+{d}^{3} ) }[2\, \sqrt {-{\frac {{a}^{2} ( c+d ) }{a+d}}}\\&( a-d ) ( ade-cde+{a}^{2}+ad)-{ i}\sqrt {2} ( {a}^{2}e-2\, ace-{d}^{2}e+2\, ad ) a ( c+d ) ], \end{split} \end{equation*}$

$G_{110}^{2} = \frac{1}{2}[(\frac{\partial^2F_{1}}{\partial u_{1}\partial p_{1}}+\frac{\partial^2F_{2}}{\partial v_{1}\partial p_{1}})+{ i}(\frac{\partial^2F_{2}}{\partial u_{1}\partial p_{1}}-\frac{\partial^2F_{1}}{\partial v_{1}\partial p_{1}})] = 0,$

$\begin{equation*} \begin{split} G_{101}^{1} = &-\frac{1}{4\sqrt {-{\frac {{a}^{2} ( c+d ) }{a+d}}}( {a}^{3}-{a} ^{2}d+2\, acd+a{d}^{2}+{d}^{3} ) }[2\, \sqrt {-{\frac {{a}^{2} ( c+d ) }{a+d}}} ( -2\, {a}^ {2}ce-{a}^{2}de-acde\\&-a{d}^{2}e-c{d}^{2}e-2\, {d}^{3}e+{a}^{3}-a{d}^{2} ) -{ i}\sqrt {2} ( {a}^{2}e+2\, ace+3\, {d}^{2}e+2\, ad ) a ( c+d ) ], \end{split} \end{equation*}$

$G_{101}^{2} = \frac{1}{2}[(\frac{\partial^2F_{1}}{\partial u_{1}\partial p_{1}}-\frac{\partial^2F_{2}}{\partial v_{1}\partial p_{1}})+{ i}(\frac{\partial^2F_{2}}{\partial u_{1}\partial p_{1}}+\frac{\partial^2F_{1}}{\partial v_{1}\partial p_{1}})] = 0,$

$g_{21} = G_{21}+\sum\limits_{j = 1}^2 (2G_{110}^{j}\omega_{11}^{j}+G_{101}^{j}\omega_{20}^{j}) = \frac{a^4}{4\delta_{1}}[-2\, \sqrt {-{\frac {{a}^{2} ( c+d ) }{a+d}}}( c+d ) \delta_{2}+{ i}\sqrt{2}a(c+d)^2\delta_{3}],$

where

$\delta_{1} = -b\sqrt {-{\frac {{a}^{2} ( c+d ) }{a+d}}} ( {a}^{3}-{ a}^{2}d+2\, acd+a{d}^{2}+{d}^{3} ) ( a+d ) ( 8 \, {a}^{2}c+8\, {a}^{2}d-{b}^{2}a-{b}^{2}d) ,$

$\begin{array}{l} \delta_{2} = ( 16\, c+16\, d ) {a}^{5}+ ( 2\, bce+2\, bde+16\, cde+16\, {d}^{2}e-3\, {b}^{2}+4\, bc+4\, bd ) {a}^{4}+ ( {b}^{2}ce-2\, { b}^{2}de\\ -4\, b{c}^{2}e-6\, bcde-2\, b{d}^{2}e-16\, {c}^{2}de-32\, c{d}^{2}e -16\, {d}^{3}e-3\, {b}^{2}d+ \\4\, bcd+4\, b{d}^{2}-16\, c{d}^{2}-16\, {d}^{3} ) {a}^{3}\\ + ( -2\, {b}^{2}{c}^{2}e+2\, {b}^{2}cde+{b}^{2}{d}^ {2}e+2\, bc{d}^{2}e+2\, b{d}^{3}e+16\, {c}^{2}{d}^{2}e+16\, c{d}^{3}e-2\, { b}^{2}cd+{b}^{2}{d}^{2} ) {a}^{2}\\ + ( -2\, {b}^{2}{c}^{2}de-3 \, {b}^{2}c{d}^{2}e+2\, {b}^{2}{d}^{3}e-4\, b{c}^{2}{d}^{2}e-6\, bc{d}^{3} e-2\, b{d}^{4}e- \\2\, {b}^{2}c{d}^{2}+{b}^{2}{d}^{3} ) a-4\, {b}^{2}c {d}^{3}e-{b}^{2}{d}^{4}e, \end{array}$

$\begin{array}{l} \delta_{3} = ( 16\, ce+16\, de+4\, b ) {a}^{4}+ ( -3\, {b}^{2}e-4\, bce -32\, {c}^{2}e-32\, cde-2\, {b}^{2}+32\, cd+32\, {d}^{2} ) {a}^{3}+ ( 6\, {b}^{2}ce\\ -{b}^{2}de+8\, b{c}^{2}e+4\, bcde-4\, b{d}^{2}e-16\, c {d}^{2}e-16\, {d}^{3}e-6\, {b}^{2}d+8\, bcd+4\, b{d}^{2} ) {a}^{2}+ \\( 4\, {b}^{2}cde+{b}^{2}{d}^{2}e\\ +8\, bc{d}^{2}e+4\, b{d}^{3}e-4\, {b }^{2}{d}^{2} ) a+2\, {b}^{2}c{d}^{2}e+3\, {b}^{2}{d}^{3}e. \end{array}$

Based on above calculation and analysis, we get

$C_{1}(0) = \frac{ i}{2\omega_{0}}(g_{20}g_{11}-2|g_{11}|^2-\frac{1}{3}|g_{02}|^2)+\frac{1}{2}g_{21} = \frac{1}{2}g_{21},$

$\mu_{2} = -\frac{{\rm Re}C_{1}(0)}{\alpha'(0)} = \frac{a^4}{4m \delta_{1}(a+d)^2}\sqrt {-{\frac {{a}^{2} ( c+d ) }{a+d}}}( c+d ) \delta_{2}[2(a+d)(a-d)^2-4a^2(c+d)],$

$\tau_{2} = {\frac {\delta_{{2}}{a}^{3}}{4\delta_{{1}} ( a+d ) } [ \sqrt {2}\sqrt {-{\frac {{a}^{2}( c+d ) }{a+d}}} ( {a}^{2}-ac-ad-{d}^{2} ) +{a}^{2}c+{a}^{2}d+acd+a{d}^{2} ] },$

where

$\omega'(0) = -\frac{1}{2}\sqrt {-{\frac {{a}^{2} ( c+d ) }{a+d}}}{\frac { ( {a}^{2}-ac-ad-{d}^{2} ) ( a+d ) \sqrt {2}m}{a ( {a}^{3}c+{a}^{3}d-2\, {a}^{2}{c}^{2}-5\, { a}^{2}cd-3\, {a}^{2}{d}^{2}-ac{d}^{2}-a{d}^{3}+c{d}^{3}+{d}^{4} ) }} ,$

$\alpha'(0) = \frac{m(a+d)^2}{2(a+d)(a-d)^2-4a^2(c+d)},$

$\beta_{2} = 2{\rm Re}C_{1}(0) = \frac{a^4}{4\delta_{1}}[-2\, \sqrt {-{\frac {{a}^{2} ( c+d ) }{a+d}}}( c+d ) \delta_{2}].$

From $a-d > 0$ and $c+d < 0$ , we have if $\frac{\delta_{2}}{\delta_{1}} > 0$ (resp., $\frac{\delta_{2}}{\delta_{1}} < 0$ ), then $\mu_{2} < 0$ (resp., $\mu_{2} > 0$ ) and $\beta_{2} > 0$ (resp., $\beta_{2} < 0$ ), the Hopf bifurcation of system (1.2) at $(0, 0, 0, 0)$ is non-degenerate and supercritical (resp. subcritical), and the bifurcating periodic solution exists for $k < k_{0}$ (resp., $k > k_{0}$ ) and is stable (resp., unstable).

Furthermore, the period and characteristic exponent are

$T = \frac{2\pi}{\omega_{0}}(1+\tau_{2}\varepsilon^2+O(\varepsilon^4)), \; \; \beta = \beta_{2}\varepsilon^2+O(\varepsilon^4),$

where $\varepsilon = \frac{k-k_{0}}{\mu_{2}}+O[(k-k_{0})^2]$ .

And the expression of the bifurcating periodic solution is (except for an arbitrary phase angle)

$X = (u, v, w, p)^{T} = Q( \overline{y}_{1}, \overline{y}_{2}, \overline{y}_{3}, \overline{y}_{4})^{T} = QY,$

where

$\overline{y}_{1} = {\rm Re}\mu, \; \; \overline{y}_{2} = {\rm Im}\mu, \; \; (\overline{y}_{3}, \overline{y}_{4})^{T} = \omega_{11}|\mu|^2+{\rm Re}(\omega_{20}\mu^2)+O(|\mu|^2),$

$\omega_{11} = (\omega_{11}^{1}, 0)^{T}, \; \; \omega_{20} = (\omega_{20}^{1}, 0)^{T},$

and

$\mu = \varepsilon e^{\frac{2{ i}t\pi}{T}}+\frac{{ i}\varepsilon^2}{6\omega_{0}}[g_{02}e^{-\frac{4{ i}t \pi}{T}} -3g_{20}e^{\frac{4{ i}t\pi}{T}}+6g_{11}]+O(\varepsilon^3) = \varepsilon {e^{\frac{2{ i}t\pi}{T}}}+O(\varepsilon^3).$

By computations, we can obtain

$\left[ \begin{array}{c} u\\ v \\ w \\ p \end{array} \right] = \left[ \begin{array}{c} \varepsilon a\sqrt {-{\frac {2{a}^{2} \left( c+d \right) }{a+d}}} \cos \left({\frac {2\pi t}{T}} \right) \\ \varepsilon [a\sqrt {-{\frac {2{a}^{2} \left( c+d \right) }{a+d}}} \cos \left({\frac {2\pi t}{T}} \right) +{\frac {2{a}^{2} \left( c+d \right) }{a+d}\sin \left({\frac {2\pi t}{T}} \right)] }\\ { \varepsilon }^{2}L\\ \varepsilon{\frac {a ( c+d ) ( -d+a ) }{m ( a+d ) } [ \sqrt {-{\frac {2{a}^{2}( c+d ) }{a+d}}}\cos \left( {\frac {2\pi t}{T}} \right) +2 a\sin \left( {\frac {2\pi t}{T}} \right) ] } \end{array} \right] +O(\varepsilon^{3}) ,$

where

$L = -\frac{a^4(c+d)}{ab+bd}+\delta_{4}[2{\rm cos}^{2}(\frac{2\pi t}{T})-1]-\delta_{5}{\rm sin}(\frac{4\pi t}{T}),$

$\delta_{4} = \frac{a^3(c+d)(4\, {a}^{2}c+4\, {a}^{2}d-{a}^{2}b-adb)}{(a+d)({b}^{2}a+{b}^{2}d-8\, {a}^{2}c-8\, {a}^{2}d)},$

$\delta_{5} = \frac{a^3(c+d) ( 2 \, a-b ) ( a+d ) \sqrt {-{\frac {2{a}^{2} ( c+d ) }{a+d}}}}{(a+d)({b}^{2}a+{b}^{2}d-8\, {a}^{2}c-8\, {a}^{2}d)}.$

Based on the above discussion, the conclusions of Theorem 3.2 are proved.

In order to verify the above theoretical analysis, we assume

$a = 1, \; \; b = 0.5, \; \; c = -2, \; \; d = 0.5, \; \; m = 1, \, e = 1.\$

According to Theorem 3.1, we get $k_{0} = 1$ . Then from Theorem 3.2, $\mu_{2} = -51.111$ and $\beta_{2} = 17.037$ , which imply that the Hopf bifurcation of system (1.2) at $(0, 0, 0, 0)$ is nondegenerate and supercritical, a bifurcation periodic solution exists for $k < k_{0} = 1$ and the bifurcating periodic solution is stable. shows the Hopf periodic solution occurs when $k = 0.9999 < k_{0} = 1$ . Based on above conclusions, in this case, it can be seen that the critical value $k_{0}$ and the properties of Hopf bifurcation of system (1.2) at $(0, 0, 0, 0)$ still remain unchanged with the parameters $(a, b, c, d, m, e) = (1, 10, -2, 0.5, 1, 10)$ . By computations, we get $\mu_{2} = -13.292$ and $\beta_{2} = 4.430$ . The corresponding Hopf periodic orbit with $k = 0.9999$ is given in Figure 11(Ⅱ).

Figure 11. Phase portraits of (1.2) with (Ⅰ)

$(b, e) = (0.5, 1)$ , (Ⅱ)

$(b, e) = (10, 10)$ .

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4. Hyperchaos control

In some cases, chaos usually is harmful and need to be suppressed, such as in pendulum system ^[25], wind power system ^[26] and spiral waves chaos ^[27], etc.. Therefore, chaos control has been widely concerned by scholars, and many valuable hyperchaos control methods have emerged, such as universal adaptive feedback control ^[28], linear state-feedback control and fuzzy disturbance-observer-based terminal sliding mode control scheme ^[29,30], etc. In addition, the chaos control can be achieved based on the ultimate boundedness of a chaotic system ^[31,32,33]. In this section, we will control hyperchaotic system (1.2) to stable equilibrium $(0, 0, 0, 0)$ by using the linear feedback control method. Suppose that the controlled hyperchaotic system is given by

$\begin{equation} \left \{ \begin{array}{l} \dot u = a(v-u)+evw+c_{1}u, \\ \dot v = cu+dv-uw+mp+c_{2}v, \\ \dot w = -bw+uv+c_{3}w, \\ \dot p = -ku-kv+c_{4}p, \end{array} \right. \end{equation}$

(4.1)

where $c_{j} \, (j = 1, 2, 3, 4)$ are feedback coefficients. The Jacobian matrix of system (4.1) at zero equilibrium point is

$J_{c} = \left[ \begin{array}{cccc} -a+c_{{1}}&a&0&0\\ c&d+ c_{{2}}&0&m\\ 0&0&-b+c_{{3}}&0\\ - k&-k&0&c_{{4}}\end{array} \right].$

The characteristic equation of $J_{c}$ at zero equilibrium point is

$f(\lambda) = \lambda^4+t_{3}\lambda^3+t_{2}\lambda^2+t_{1}\lambda+t_{0},$

where

$\begin{array}{l} t_{0} = abcc_{{4}}+abdc_{{4}}+2abkm+abc_{{2}}c_{{4}}-acc_{{3}}c_{{4}}-adc_{{ 3}}c_{{4}}-2akmc_{{3}}-ac_{{2}}c_{{3}}c_{{4}}-bdc_{{1}}c_{{4}}-bkmc_ {{1}} \\ \quad \, \, \, \, -bc_{{1}}c_{{2}}c_{{4}}+dc_{{1}}c_{{3}}c_{{4}}+kmc_{{1}}c_{{3}}+c _{{1}}c_{{2}}c_{{3}}c_{{4}} , \end{array}$

$\begin{array}{l} t_{1} = (c_{{3}}c_{{4}} -bc-bd-bc_{{2}}-bc_{{4}}+cc_{{3}}+cc_{{4}}+dc_{{3}}+dc_{{4}}+2 km+c_{{3}}c_{{2}}+c_{{2}}c_{{4}} ) a+ ( bc_{{1}}+bc_{{4}}-c_{{1}}c_{{3}} \\ \quad \, \, \, \, -c_{{1}}c_{{4}}-c_{{3}}c_{{4}} ) d +b ( km+c_{{1}}c_{{2}}+c_{{1}}c_{{4}}+c_{{2}}c_{{4}}) - ( km+c_{{1}}c_{{2}}+c_{{1}}c_{{4}}+c_{{2}}c_{{4}}) c_{{3} }-kmc_{{1}}-c_{{1}}c_{{2}}c_{{4}} , \end{array}$

$\begin{array}{l} t_{2} = ab-ac-ad-ac_{{2}}-ac_{{3}}-ac_{{4}}-bd-bc_{{1}}-bc_{{2}}-bc_{{4}}+dc_{ {1}}+dc_{{3}}+dc_{{4}}+km+c_{{1}}c_{{2}}+c_{{1}}c_{{3}} \\ \quad \, \, \, \, +c_{{1}}c_{{4}} +c_{{2}}c_{{3}}+c_{{2}}c_{{4}}+c_{{3}}c_{{4}} , \end{array}$

$t_{3} = a+b-d-c_{1}-c_{2}-c_{3}-c_{4} .$

According to Routh-Hurwitz criterion ^[23], the real parts of eigenvalues are negative if and only if

$\begin{equation} t_{3}(t_{1}t_{2}-t_{3}t_{0})-t_{1}^2 > 0, \, t_{3}t_{2}-t_{1} > 0, \, t_{3} > 0, \, t_{0} > 0. \end{equation}$

(4.2)

Case 1. For the parameters $a = 15, \, b = 43, \, c = 1, \, d = 16, \, e = 5, \, m = 5, \, k = 2,$ we assume $c_{j} = -20\, (j = 1, 2, 3, 4)$ , system (4.1) has only one equilibrium point $(0, 0, 0, 0)$ , and the corresponding eigenvalues are

$\lambda_{1} = -63, \, \lambda_{2} = -4.468, \, \lambda_{3} = -35.756, \, \lambda_{4} = -18.774,$

the equilibrium point is asymptotically stable.

Case 2. For the parameters $b = 43, \, a = 15, \, c = 1, \, e = 5, \, k = 5, \, d = 16, \, m = 0.5,$ we assume $c_{j} = -20$ $(j = 1, 2, 3, 4)$ , system (4.1) also has only one equilibrium point $(0, 0, 0, 0)$ , and the corresponding eigenvalues are

$\lambda_{1} = -63, \, \lambda_{2} = -3.747, \, \lambda_{3} = -19.703, \, \lambda_{4} = -35.549,$

the zero equilibrium point is asymptotically stable.

For the two cases, the behaviors of the state $(u, v, w, p)$ of hyperchaotic system (1.2) and controlled hyperchaotic system (4.1) with time are shown in Figures 12 and 13, respectively. These indicate that the controlled hyperchaotic system (4.1) is asymptotically stable at zero-equilibrium point by selecting appropriate feedback coefficients.

Figure 12. Time series of (Ⅰ) system (1.2), (Ⅱ) system (4.1) (Case 1).

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Figure 13. Time series of (Ⅰ) system (1.2), (Ⅱ) system (4.1) (Case 2).

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5. Conclusions

In this paper, we present a new 4D hyperchaotic system by introducing a linear controller to the second equation of the 3D chaotic Qi system. The new system always has two unstable nonzero equilibrium points which are symmetric about $w$ axis. The complex dynamical behaviors, including dissipativity and invariance, equilibria and their stability, quasi-periodicity, chaos and hyperchaos of new 4D system (1.2) are investigated and analyzed. Furthermore, the existence of Hopf bifurcation, the stability and expression of the Hopf bifurcation at zero-equilibrium point are studied using the normal form theory and symbolic computations. Based on Section 3, it can also be seen that the parameters $b$ and $e$ are unrelated to the critical value $k$ of Hopf bifurcation of the new 4D system. In order to analyze and verify the complex phenomena of the system, some numerical simulations are carried out including Lyapunov exponents, bifurcation and Poincaré maps, etc. The results show that the new 4D hyperchaotic system can exhibit complex dynamical behaviors, such as quasi-periodic, chaotic and hyperchaotic. Furthermore, the resistors, capacitors, operational amplifiers and analog multipliers can be used to design the hyperchaotic electronic circuit, to verify the existence of the hyperchaotic attractor of the system from another perspective, embodying the application of the hyperchaotic circuit system. We will investigate the ultimate bound sets of the hyperchaotic system, a new method for hyperchaos control and hyperchaotic electronic circuit in a future study.

Acknowledgments

Project supported by the Doctoral Scientific Research Foundation of Hanshan Normal University (No. QD202130).

Conflict of interest

The authors declare that they have no conflict of interest.

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© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)

通讯作者: 陈斌, bchen63@163.com

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沈阳化工大学材料科学与工程学院沈阳 110142

Applied Computing and Intelligence

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Abstract

1. Introduction

2. Dynamical analysis

2.1. Dissipativity and invariance

2.2. Equilibria

2.3. Lyapunov exponents and Poincaré maps

3. Hopf bifurcation

4. Hyperchaos control

5. Conclusions

Acknowledgments

Conflict of interest

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Applied Computing and Intelligence

Finnish perspective on using synthetic health data to protect privacy: the PRIVASA project

Related Papers:

Abstract

1. Introduction

2. Dynamical analysis

2.1. Dissipativity and invariance

2.2. Equilibria

2.3. Lyapunov exponents and Poincaré maps

3. Hopf bifurcation

4. Hyperchaos control

5. Conclusions

Acknowledgments

Conflict of interest

References

This article has been cited by:

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通讯作者: 陈斌, bchen63@163.com

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