Preventing aircraft from wildlife strikes using trajectory planning based on the enhanced artificial potential field approach

Wenchao Cai; Yadong Zhou; Wenchao Cai; Yadong Zhou

doi:10.3934/mina.2024010

Metascience in Aerospace

2024, Volume 1, Issue 2: 219-245. doi: 10.3934/mina.2024010

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Research article

Preventing aircraft from wildlife strikes using trajectory planning based on the enhanced artificial potential field approach

Wenchao Cai ,
Yadong Zhou ^,

College of Civil Aviation, Nanjing University of Aeronautics and Astronautics, Nanjing, 211100, China

Received: 20 November 2023 Revised: 02 May 2024 Accepted: 27 May 2024 Published: 31 May 2024

Wildlife strikes refer to collisions between animals and aircraft during flight or taxiing. While such collisions can occur at any phase of a flight, the majority occur during takeoff and landing, particularly at lower altitudes. Given that most reported collisions involve birds, our focus was primarily on bird strikes, in line with statistical data. In the aviation industry, aircraft safety takes precedence, and attention must also be paid to optimizing route distances to minimize operational costs, posing a multi-objective optimization challenge. However, wildlife strikes can occur suddenly, even when aircraft strictly adhere to their trajectories. The aircraft may then need to deviate from their planned paths to avoid these collisions, necessitating the adoption of alternative routes. In this article, we proposed a method that combines artificial potential energy (APE) and morphological smoothing to not only reduce the risk of collisions but also maintain the aircraft's trajectory as closely as possible. The concept of APE was applied to flight trajectory planning (TP), where the aircraft's surroundings were conceptualized as an abstract artificial gravitational field. This field exerts a "gravitational force" towards the destination, while bird obstacles exert a "repulsive force" on the aircraft. Through simulation studies, our proposed method helps smooth the trajectory and enhance its security.

Keywords:

bird strike,
path planning,
artificial potential field,
morphological open and closed smoothing principles

Citation: Wenchao Cai, Yadong Zhou. Preventing aircraft from wildlife strikes using trajectory planning based on the enhanced artificial potential field approach[J]. Metascience in Aerospace, 2024, 1(2): 219-245. doi: 10.3934/mina.2024010

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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.

References

[1]	Serious accident database. Database of fatalities and destroyed aircraft due to bird and other wildlife strikes, 1912 to present. Available from: https://avisure.com/serious-accident-database/.
[2]	Skybrary. Bird Strike (2019) Available from: https://www.skybrary.aero/articles/bird-strike.
[3]	Dolbeer RA (2013) The history of wildlife strikes and management at airports. In USDA National Wildlife Research Center-Staff Publications; Johns Hopkins University Press: Baltimore.
[4]	Cleary EC, Dolbeer RA (2005) Wildlife hazard management at airports: a manual for airport personnel. In USDA National Wildlife Research Center-Staff Publications; Johns Hopkins University Press: Baltimore, 133.
[5]	Simplifying. What Happened To The Airbus A320 That Landed On The Hudson? 2022. Available from: https://simpleflying.com/miracle-on-the-hudson-aicraft-fate/.
[6]	News WA, Sita Air Dornier 228 Crashes in Nepal, 19 killed, 2012. Available from: https://worldairlinenews.com/2012/09/29/sita-air-dornier-228-crashes-in-nepal-19-killed/.
[7]	Herald, A. Accident: Ural A321 at Moscow on Aug 15th, 2019, Bird Strike into Both Engines Forces Landing in Cornfield, 2019. Available from: http://avherald.com/h?article=4cb94927&opt=0.
[8]	Allan JR (2000) The costs of bird strikes and bird strike prevention. In Human conflicts with wildlife: economic considerations; Johns Hopkins University Press: Baltimore, 18.
[9]	Dolbeer RA, Begier MJ, Miller PR, et al. (2022) Wildlife Strikes to Civil Aircraft in the United States, 1990–2022. Tech. rep. United States. Department of Transportation. Federal Aviation Administration.
[10]	UK Civil Aviation Authority. CAA Paper 2006/05: The Completeness and Accuracy of Birdstrike Reporting in the UK; CAA Report; UK Civil Aviation Authority: London, UK, 2006.
[11]	Dolbeer RA. Trends in Reporting of Wildlife Strikes with Civil Aircraft and in Identification of Species Struck Under a Primarily Voluntary Reporting System, 1990–2013; Special Report Submitted to the Federal Aviation Administration; DigitalCommons@University of Nebraska–Lincoln: Lincoln, NE, USA, 2015. Available from: https://digitalcommons.unl.edu/cgi/viewcontent.cgi?article=1190&context=zoonoticspu.
[12]	Dolbeer RA, Weller JR, Anderson AL, et al. Wildlife Strikes to Civil Aircraft in the United States 1990–2015; Federal Aviation Administration National Wildlife Strike Database, Serial Report Number 22; Federal Aviation Administration, U.S. Department of Agriculture: Washington, DC, USA, 2016.
[13]	Pitlik TJ, Washburn BE (2012) Using bird strike information to direct effective management actions within airport environments. In Proceedings of the 25th Vertebrate Pest Conference, Monterey, CA, USA, 5–8.
[14]	Systems RR, Three Reasons Why Bird Strikes on Aircraft Are on the Rise, 2021. Available from: https://www.robinradar.com/press/blog/3-reasons-why-bird-strikes-on-aircraft-are-on-the-rise.
[15]	Sabziyan Varnousfaderani E, Shihab SAM (2023) Bird Movement Prediction Using Long Short-Term Memory Networks to Prevent Bird Strikes with Low Altitude Aircraft. AIAA AVIATION 2023 Forum, San Diego, California, USA, 4531. https://doi.org/10.2514/6.2023-4531
[16]	McKee J, Shaw P, Dekker A, et al. (2016) Approaches to wildlife management in aviation. In Angelici, M.F., Eds; Problematic Wildlife, Springer: Cham, Switzerland, 465–488. https://doi.org/10.1007/978-3-319-22246-2_22
[17]	Dolbeer RA (2011) Increasing trend of damaging bird strikes with aircraft outside the airport boundary: implications for mitigation measures. Hum-Wildl Interact 5: 235–248.
[18]	Sabziyan Varnousfaderani E, Shihab SAM, Dulia EF (2023) Deep-Dispatch: A Deep Reinforcement Learning-Based Vehicle Dispatch Algorithm for Advanced Air Mobility. J Air Transp 2024: 1–22. https://doi.org/10.2514/1.D0416 doi: 10.2514/1.D0416
[19]	Avrenli KA, Dempsey BJ (2014) Statistical analysis of aircraft–bird strikes resulting in engine failure. Transp Res Rec 2449: 14–23.
[20]	Metz IC, Ellerbroek J, Mühlhausen T, et al. (2021) Analysis of Risk-Based Operational Bird Strike Prevention. Aerospace 8: 32. https://doi.org/10.3390/aerospace8020032 doi: 10.3390/aerospace8020032
[21]	Filiz E, Öner G, Ahmet U, et al. (2023) An investigation of bird strike cases in the aviation sector with a novel approach within the context of the principal-agent phenomenon: Bird strikes and insurance in the USA. Heliyon 9. https://doi.org/10.1016/j.heliyon.2023.e18115 doi: 10.1016/j.heliyon.2023.e18115
[22]	Zhou Y, Sun Y, Cai W (2019) Bird-striking damage of rotating laminates using SPH-CDM method. Aerosp Sci Technol 84: 265–272. https://doi.org/10.1016/j.ast.2018.10.009. doi: 10.1016/j.ast.2018.10.009
[23]	Zhou Y, Sun Y, Huang T, et al. (2019) SPH-FEM simulation of impacted composite laminates with different layups. Aerosp Sci Technol 95: 105469. https://doi.org/10.1016/j.ast.2019.105469 doi: 10.1016/j.ast.2019.105469
[24]	Metz IC, Mühlhausen T, Ellerbroek J, et al. (2018) Simulation Model to Calculate Bird-Aircraft Collisions and Near Misses in the Airport Vicinity. Aerospace 5: 112. https://doi.org/10.3390/aerospace5040112 doi: 10.3390/aerospace5040112
[25]	Metz IC (2021) Air Traffic Control Advisory System for the Prevention of Bird Strikes. Level of Thesis, Delft University of Technology, Delft, The Netherlands.
[26]	Metz IC, Ellerbroek J, Mühlhausen T, et al. (2020) The bird strike challenge. Aerospace 7: 26. https://doi.org/10.4233/uuid:013fe685-755f-4a76-8428-53be5c67fa51 doi: 10.4233/uuid:013fe685-755f-4a76-8428-53be5c67fa51
[27]	Nicole LM, Keith AH, Christy AM, et al. (2021) Climate variability has idiosyncratic impacts on North American aerial insectivorous bird population trajectories. Biol Conserv 263: 109329. https://doi.org/10.1016/j.biocon.2021.109329. doi: 10.1016/j.biocon.2021.109329
[28]	Lopez-Lago M, Casado R, Bermudez A, et al. (2017) A predictive model for risk assessment on imminent bird strikes on airport areas. Aerosp Sci Technol 62: 19–30. https://doi.org/10.1016/j.ast.2016.11.020 doi: 10.1016/j.ast.2016.11.020
[29]	ABC'S Bird Library, Bird Calls Blog. Available from: https://abcbirds.org/blog/frequent-colliders/.
[30]	Mathaiyan V, Vĳayanandh R, Jung DW (2021) Determination of Strong Factor in Bird Strike Analysis using Taguchis method for Aircraft Manufacturing guide. J Phys Conf Ser 1733: 012002.
[31]	Shao Q, Zhou Y, Zhu P, et al. (2020) Key factors assessment on bird strike density distribution in airport habitats: Spatial heterogeneity and geographically weighted regression model. Sustainability 12: 7235.
[32]	Smojver I, Ivančević D (2011) Bird strike damage analysis in aircraft structures using Abaqus/Explicit and coupled Eulerian-Lagrangian approach. Compos Sci Technol 71: 489–498.
[33]	Riccio A, Cristiano R, Saputo S (2016) A brief introduction to the bird strike numerical simulation. Am J Eng Applied Sci 9: 946–950.
[34]	Company TDB, Airport Bird Control Drone, 2020. Availabl from: https://www.thedronebird.com/safe-humane-and-effective-bird-control/applications/airports/.
[35]	Bishop J, McKay H, Parrott D, et al. (2003) Review of international research literature regarding the effectiveness of auditory bird scaring techniques and potential alternatives. Food Rural Affairs, London, 1–53.
[36]	Seamans TW, Gosser AL (2016) Bird dispersal techniques. Wildlife Damage Management Technical Series.
[37]	Matyjasiak P (2008) Methods of bird control at airports, In Theoretical and applied aspects of modern ecology. J. Uchmański (ed.), Cardinal Stefan Wyszyński University Press, Warsaw, 171–203.
[38]	Blackwell BF, Bernhardt GE (2004) Efficacy of aircraft landing lights in stimulating avoidance behavior in birds. J Wildlife Manage 68: 725–732. https://doi.org/10.2193/0022-541X doi: 10.2193/0022-541X
[39]	Vas E, Lescroe¨l A, Duriez O, et al. (2015) Approaching birds with drones: first experiments and ethical guidelines. Biol Lett 11: 20140754. http://dx.doi.org/10.1098/rsbl.2014.0754
[40]	Paranjape AA, Chung SJ, Kim K, et al. (2018) Robotic herding of a flock of birds using an unmanned aerial vehicle. IEEE Trans Robot 34: 901–915. https://doi.org/10.1109/TRO.2018.2853610 doi: 10.1109/TRO.2018.2853610
[41]	Van Gasteren H, Krĳgsveld KL, Klauke N, et al. (2019) Agroecology meets aviation safety: Early warning systems in Europe and the Middle East prevent collisions between birds and aircraft. Ecography 42: 899–911. https://doi.org/10.1111/ecog.04125 doi: 10.1111/ecog.04125
[42]	Zhao P, Erzberger H, Liu Y (2021) Multiple-aircraft-conflict resolution under uncertainties. J Guid Control Dyn 44: 2031–2049. https://doi.org/10.2514/1.G005825 doi: 10.2514/1.G005825
[43]	Sislak D, Volf P, Komenda A, et al. (2007) Agent-Based Multi-Layer Collision Avoidance to Unmanned Aerial Vehicles. In 2007 International Conference on Integration of Knowledge Intensive Multi-Agent Systems, Waltham, MA, USA, 365–370.
[44]	Sislak D, Rehak M, Pechoucek M, et al. (2007) Negotiation-Based Approach to Unmanned Aerial Vehicles. In IEEE Workshop on Distributed Intelligent Systems: Collective Intelligence and Its Applications (DIS'06), Prague, Czech Republic, 279–284.
[45]	Fasano G, Accardo D, Moccia A, et al. (2007) Multi-sensor-based fully autonomous non-cooperative collision avoidance system for unmanned air vehicles. J Aerosp Comput Inf Commun, 5: 338–360. https://doi.org/10.2514/1.35145 doi: 10.2514/1.35145
[46]	Panchal IM, Riberios M, Armanni S (2022) Urban air traffic management for collision avoidance with noncooperative airspace users, 33rd Congress of the International Council of the Aeronautical Sciences (ICAS 2022), 6801–6818.
[47]	Liu MH, Yang Y, Yue Q (2016) Auxiliary road planning and design based on city engine and ArcGIS. Stud Surv Mapp Sci 64–67.
[48]	Duan HB, Shao S, Su BW (2010) New ideas for the development of unmanned combat aircraft control technology based on bionic intelligence. Chin Sci Tech Sci 40: 853–860.
[49]	Liang XH, Mu YH, Wu BH (2020) Review of related algorithms of path planning. Value Eng 39: 295–299.
[50]	Dong M, Chen TZ, Yang H (2019) Simulation of unmanned vehicle path planning based on improved RRT algorithm. Comput Simul 36: 96–100.
[51]	Wang JJ, Chen LS, Sheng M (2020) Research on robot path planning based on improved artificial potential field method. Eng Agric Environ Food 58: 66–70.
[52]	Yu ZZ, Yan JH, Zhao J (2011) Path planning of mobile robot based on improved artificial potential field method. J Harbin Inst Technol 43: 50–55.
[53]	Wang GC, Wu GX, Zuo YB (2019) Research on trajectory planning of packaging robot based on improved ant colony algorithm. J Electron Meas Instrum 33: 94–100.
[54]	Liu LF, Yang XF (2019) Efficient path solving based on a hybrid genetic algorithm. Comput Appl Eng 55: 244–249.
[55]	Madridano Á , Al-Kaff A, Martín D, et al. (2021) Trajectory planning for multi-robot systems: Methods and applications, Expert Syst Appl 173: 114660. https://doi.org/10.1016/j.eswa.2021.114660 doi: 10.1016/j.eswa.2021.114660
[56]	Rao J, Xiang C, Xi J, et al. (2023) Path planning for dual UAVs cooperative suspension transport based on artificial potential field-A* algorithm. Knowl-Based Syst 277: 110797. https://doi.org/10.1016/j.knosys.2023.110797 doi: 10.1016/j.knosys.2023.110797
[57]	Liu Y, Chen C, Wang Y, et al. (2024) A fast formation obstacle avoidance algorithm for clustered UAVs based on artificial potential field. Aerosp Sci Technol 147: 108974. https://doi.org/10.1016/j.ast.2024.108974 doi: 10.1016/j.ast.2024.108974
[58]	Liang XX, Liu CY, Song XL, et al. (2018) Research on path planning of mobile robots based on improved artificial potential field method. Comput Simul 35: 291–294.
[59]	Liu F (2008) Robot path planning based on artificial potential field and immune algorithm. Software Guide 7: 51–53.
[60]	Han W, Sun KB (2018) Intelligent omnidirectional vehicle path planning based on fuzzy artificial potential field method. Comput Appl Eng 54: 105–109.
[61]	Wu Z, Dai J, Jiang B, et al. (2023) Robot path planning based on an artificial potential field with deterministic annealing. ISA T 138: 74–87. https://doi.org/10.1016/j.isatra.2023.02.018 doi: 10.1016/j.isatra.2023.02.018
[62]	Mbede J B, Huang X, Wang M, (2000). Fuzzy motion planning among dynamic obstacles using artificial potential fields for robot manipulators. Robot Auton Syst 32: 61–72. https://doi.org/10.1016/S0921-8890(00)00073-7 doi: 10.1016/S0921-8890(00)00073-7
[63]	Ren Y, Zhao H (2020) Improved artificial potential field method for robot obstacle avoidance and path planning. Comput Simul 37: 360–364.
[64]	Li L, Guo Y, Zhang X, et al. (2017) Firefly algorithm combined with artificial potential field method for robot path planning. J Inf Sci Eng 7: 8–10.
[65]	Liang XX, Liu CY (2018) Research on improving the artificial potential field method for mobile robot path planning. Comput Simul 35: 291–294,361.
[66]	Khatib O (1986) Real-time obstacle avoidance for manipulators and mobile robots. Int J Rob Re c5: 90–98.
[67]	Yoshida H, Shinohara S, Nagai M (2008) Lane change steering maneuver using model predictive control theory. Vehicle Syst Dyn 46: 669–681.

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