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Multistability and bifurcations in a 5D segmented disc dynamo with a curve of equilibria
Advances in Difference Equations volumeÂ 2019, ArticleÂ number:Â 345 (2019)
Abstract
Multistability, i.e., coexisting attractors, is one of the most exciting phenomena in dynamical systems. This paper presents a new category of coexisting hidden attractor: fivedimensional (5D) systems with a curve of equilibria. Based on the segmented disc dynamo, a new 5D hyperchaotic system is proposed. The paper studies not only coexisting selfexcited attractors but also coexisting hidden attractors in the new system with four types of equilibria: a curve of equilibria, a line equilibrium, a stable equilibrium, and no equilibria. Furthermore, the paper proves that the degenerate Hopf and pitchfork bifurcations occur in the system. Numerical simulations demonstrate the emergence of the two bifurcations.
1 Introduction
Multistability or coexisting different attractors for a given set of parameters is one of the most exciting phenomena in dynamical systems [1]. Complex dynamical systems, ranging from human brain, climate, ecosystems to financial markets and engineering applications, typically have many coexisting attractors [2, 3]. High dimensional hyperchaotic systems describe natural phenomena more explicitly than low dimensional systems [4]. Nowadays the research of high dimensional multistability systems has captured attention of scientists from around the world.
Recently, it has been shown that multistability is connected with the occurrence of unpredictable attractors which have been called hidden attractors. Hidden attractors may cause disastrous events, such as sudden climate changes, serious diseases, financial crises, etc. [2, 5,6,7]. They are often related to dynamical systems with stable equilibria [8, 9], no equilibria [10, 11], a line equilibrium [12, 13], or a curve of equilibria [14,15,16,17]. AÂ selfexcited attractor has a basin of attraction associated with an unstable equilibrium. The classical attractors of Lorenz, RÃ¶ssler, Chua, Chen, Sprott systems (cases B to S) are those excited from unstable equilibria [16]. This paper presents a new category of hidden attractor: fivedimensional (5D) systems with a curve of equilibria.
For the past few years, many dynamical systems have been reported to study curveshaped equilibria [14,15,16,17]. TableÂ 1 in Ref. [15] classifies the chaotic systems with an infinite number of equilibria. However, it is noted that no 5D hyperchaotic systems with a curve of equilibria are reported in the literature.
Motivated by the above findings, the paper proposes a new 5D hyperchaotic system based on the segmented disc dynamo. We study not only coexisting selfexcited attractors but also coexisting hidden attractors in the new system with four types of equilibria: a curve of equilibria, a line equilibrium, a stable equilibrium, and no equilibria. Further, we study the degenerate Hopf bifurcation and pitchfork bifurcation of the system by bifurcation theory [18, 19]. Numerical investigations are performed to verify the corresponding theoretical results for the two bifurcations.
The paper is organized as follows. SectionÂ 2 introduces a new 5D segmented disc dynamo with a curve of equilibria. SectionÂ 3 investigates different types of coexisting attractors. SectionÂ 4 investigates the degenerate Hopf bifurcation, and Sect.Â 5 analyzes the pitchfork bifurcation. SectionÂ 6 concludes the paper.
2 5D segmented disc dynamo with a curve of equilibria
2.1 Presentation of a 5D segmented disc dynamo with a curve of equilibria
Moffatt proposed the segmented disc dynamo which included the current associated with the radial diffusion of the magnetic field and satisfied the Alfven theorem of flux conservation [20]:
Based on (2.1), we translate z to \(zm\) and introduce new parameters, which result in the following 5D segmented disc dynamo:
where r and m are positive parameters, and the others are real parameters.
The divergence of the system is \(\nabla \cdot V =  r  1  m + {k_{5}}+ {k_{6}}\), and the system is dissipative if \({k_{5}}+{k_{6}} < r + m + 1\). System (2.2) is invariant under the transformation \(( {x,y,z,u,v} ) \to ( {  x,  y, z, u,  v} )\).
Now in order to obtain the hyperchaos with three positive Lyapunov exponents (LEs), we need to exclude some parameter sets that cannot make system (2.2) show bounded chaotic solutions.
Theorem 2.1
If the following conditions are satisfied:
then system (2.2) has no bounded chaotic or hyperchaotic solutions.
Proof
From system (2.2), we can get
By condition (2.3), Eq.Â (2.4) becomes
Hence
where c is an arbitrary constant. For \({k_{4}}  r > 0\), system (2.1) is not chaotic because at least one of \(x(t)\), \(y(t)\), \(z(t)\), \(u(t)\), and \(v(t)\) is not bounded.
FigureÂ 1 shows the regions of various dynamical behaviors in the space of the parameters \((m,r) \in [ {1,10} ] \times [ {50,60} ]\) with the other fixed parameters \((g,{k_{1}},{k_{2}},{k_{3}},{k_{4}},{k_{5}},{k_{6}}) =(0, 0.05, 1, 0.6, 10, 1, 0)\) and the initial condition \((2, 7.9221, 1, 0, 15)\). Chaotic and hyperchaotic regions also include hidden chaos and hyperchaos.
â€ƒâ–¡
2.2 Equilibria and stability
For \({k_{4}} = 1 + m\), \({k_{5}} =  1\), and \({k_{1}}{k_{3}} \ne 0\), system (2.2) has a curve of equilibria
We first discuss the case: \({k_{4}} \ne 1 + m\) and \({k_{5}}=1\). For \({k_{1}}{k_{3}} \ne 0\), there is an equilibrium \({E_{2}} ({0, 0, 0, \frac{\mathrm{{g}}}{{{k_{1}}}}, 0} )\). For \({k_{1}}= 0\) and \(g \ne 0\), there is no equilibria. For \({k_{1}}= g = 0\) or \({k_{3}}=0\), the system always has a line equilibrium. The other case, \({k_{4}}=1 + m\) and \({k_{5}} \ne 1\), can be similarly discussed.
Let
and the following theorem is easily proved.
Theorem 2.2

(1)
Suppose \((m,g,{k_{1}},{k_{2}},{k_{3}},{k_{4}}, {k_{5}},{k_{6}}) \in {S_{1}}\). If \({k_{1}}{k_{3}}>0\), then the equilibrium \({E_{1}}\) of system (2.2) has at least threedimensional stable manifold and onedimensional unstable manifold. Otherwise, \({E_{1}}\) has at least fourdimensional stable manifold.

(2)
Suppose \((m,{k_{1}},{k_{3}},{k_{4}},{k_{5}},{k_{6}}) \in {S_{2}}\). If \({k_{1}}{k_{3}}>0\), then the equilibrium \({E_{2}}\) is unstable and has fourdimensional stable manifold and onedimensional unstable manifold. Otherwise, \({E_{2}}\) is stable and has fivedimensional stable manifold.
3 Multistability
The complex dynamics of a hyperchaotic system are usually produced by the bifurcations at equilibria. However, curveshaped equilibria are nonisolated and nonhyperbolic, so it is difficult to obtain the complex dynamics due to the bifurcation at equilibria. The methods of numerical analysis are vital for these systems with a curve of equilibria [14]. In addition, lots of other complex dynamics of system (2.2) are also discovered by means of the detailed numerical analysis.
3.1 Coexisting chaos and hyperchaos for system (2.2) with a curve of equilibria
When \(( {m,r,g,{k_{1}},{k_{2}},{k_{3}},{k_{4}},{k_{5}},{k_{6}}} ) = ( {99, 9, 5, 10, 10, 10, 100, 1, 0} )\), system (2.2) has a curve of equilibria \((x,x,{x^{2}},\frac{1}{2}({x ^{2}}  1),100x(1  {x^{2}}))\). For the initial condition \((2, 7.9221, 1, 0, 15.8407)\), system (2.2) has the LEs \((0.0095, 0.0017, 0, 10.5037, 22.3829)\), and the Kaplanâ€“Yorke dimension is 3.0011. For the initial condition \((0.7513, 0.2551, 0.5060, 0.6991, 0.8909)\), the LEs are \((0.0038, 0, 0.0151, 10.1246, 17.2288)\), and the Kaplanâ€“Yorke dimension is 2.2517. The hyperchaotic and chaotic attractors are shown in Fig.Â 2. FigureÂ 2(c) shows the LEs spectrum with 200 varied initial conditions, and Fig.Â 2(d) shows the LEs spectrum for \(r \in [7, 50]\) and the initial condition \((2, 7.9221, 1, 0, 15.8407)\).
3.2 Coexisting chaos and hyperchaos for system (2.2) with no equilibria
When \(( {m,r,g,{k_{1}},{k_{2}},{k_{3}},{k_{4}},{k_{5}},{k_{6}}} ) = ( {0.11, 8, 1.2, 0, 10, 0,  100,  1,0} )\), system (2.2) has no equilibria. For the initial condition \((7.4047, 10.8076, 8.9184, 0.4523, 0.0641)\), the system has the LEs \((0.0062, 0, 0.0090, 0.0133, 10.0797)\), and the Kaplanâ€“Yorke dimension is 2.6889. The chaos is shown in Fig.Â 3(a), and Fig.Â 3(b) shows the PoincarÃ© map. For the initial condition \((2, 0, 0.1, 0.1, 0)\), system (2.2) has the LEs \((0.0032, 0.0018, 0, 0.0123, 10.0995)\), and the Kaplanâ€“Yorke dimension is 3.4065. A hyperchaotic attractor is shown in Fig.Â 3(c), and Fig.Â 3(d) shows the PoincarÃ© map. FigureÂ 3(e) shows the LEs spectrum with 200 varied initial conditions, and the corresponding Kaplanâ€“Yorke dimensions are shown in Fig.Â 3(f).
3.3 Coexisting chaotic, quasiperiodic and periodic attractors for system (2.2) with a line equilibrium
When \(( {m,r,g,{k_{1}},{k_{2}},{k_{3}},{k_{4}},{k_{5}},{k_{6}}} ) = ( {1.1, 6.1, 12, 0.1, 0, 0, 100, 1, 0} )\), system (2.2) has a line equilibrium \((0,0,z,120,0)\). For the initial condition \((100, 98, 100, 100, 100)\), the system has the LEs \((0.0058, 0, 0.0364, 2.3601, 6.8103)\). FigureÂ 4(a) shows the chaotic attractor. For the initial condition \((0.0506, 5.9116, 3.2069, 20.3950, 1.1477)\), the system has the LEs \((0, 0, 0.4492, 0.4526, 8.2991)\) and displays quasiperiodicity. For the initial condition \((2, 0.1, 0, 0, 0)\), a periodic attractor is shown in Fig.Â 4(b). FigureÂ 4(c) shows the LEs spectrum with 200 varied initial conditions, and the corresponding Kaplanâ€“Yorke dimensions are shown in Fig.Â 4(d).
3.4 Coexisting chaos and hyperchaos for system (2.2) with a stable equilibrium
When \(( {m,r,g,{k_{1}},{k_{2}},{k_{3}},{k_{4}},{k_{5}},{k_{6}}} ) = (1.3, 1, 0.12, 0.01, 0, 0.001, 1, 0, 0.001)\), system (2.2) has a stable equilibrium \((0,0,0,12,0)\). For the initial condition \((0.0326, 0.5612, 0.8819, 0.6692, 0.1904)\), the LEs are \((0.0031, 0, 0.0172, 0.0143, 3.2647)\). The chaotic attractor is shown in Fig.Â 5(a), and Fig.Â 5(b) shows the PoincarÃ© map. For the initial condition \((0, 0, 0, 0, 0)\), the LEs are \((1.8956, 1.4008, 0, 0.0015, 6.5956)\), and the system displays hyperchaos.
3.5 Coexisting selfexcited attractors
When \(( {m,r,g,{k_{1}},{k_{2}},{k_{3}},{k_{4}},{k_{5}},{k_{6}}} ) = (1.3, 1, 12, 0.01, 0.1, 0.001, 2.3, 1, 0.1)\), system (2.2) has an unstable equilibrium \((0,0,0,1200,0)\). For the initial condition \((0.3394, 64.0648, 91.0435, 75.5546, 10.5920)\), system (2.2) has the LEs \((0.0082, 0.0056, 0, 0.5393, 3.5561)\). AÂ hyperchaotic attractor is obtained (see Fig.Â 6(a)), and the PoincarÃ© map is shown in Fig.Â 6(b). For the initial condition \((0.1576, 0.9706, 0.9572, 0.4854, 0.8003)\), the system has the LEs \((0.0047, 0, 0.1228, 0.5298, 3.5469)\). The chaotic attractor is obtained in Fig.Â 6(c). The PoincarÃ© map is shown in Fig.Â 6(d). When \(r \in [1, 10]\), Fig.Â 7(a) shows the LEs spectrum, and the corresponding bifurcation diagram is shown in Fig.Â 7(b).
4 Degenerate Hopf bifurcation in system (2.2)
We utilize the projection method [18] to calculate the Lyapunov coefficients associated with Hopf bifurcation.
Let
For \(({m,r,g,{k_{1}},{k_{2}},{k_{3}},{k_{4}},{k_{5}}} ) \in S\), system (2.2) has only one equilibrium \({E_{2}} ( {0, 0, 0, \frac{g}{{k_{1}}}, 0} )\). \({E_{2}}\)Â has the eigenvalues \(\lambda ({k_{6}})=\frac{{{k_{6}} \pm \sqrt{4{k_{1}}{k_{3}} + {k _{6}}^{2}} }}{2}\), and the other eigenvalues of \({E_{2}}\) satisfy
According to the Routhâ€“Hurwitz criterion, the real parts of the roots Î» are negative if and only if
When \({k_{6}}=0\) and \(( {m,r,g,{k_{1}},{k_{2}},{k_{3}},{k_{4}}, {k_{5}}} ) \in S\), \({E_{2}}\) has a pair of purely imaginary eigenvalues \(\pm \omega i\), and the other three eigenvalues with negative real part.
The transversality condition
is also satisfied, and a Hopf bifurcation at \({E_{2}}\) occurs. We have the following theorem.
Theorem 4.1
Considering system (2.2), for parameter \(( {m,r,g,{k_{1}},{k_{2}},{k_{3}},{k_{4}},{k_{5}}} ) \in S\) and \({k_{6}}=0\), the first and second Lyapunov coefficients \({l_{1}}={l_{2}}=0\) at \({E_{2}}\), and the third Lyapunov coefficient is given by \({l_{3}} =  \frac{{4g ( { ( {m + 2} ) ( {m + 1} ) + r ( {m  3} )} ){\omega ^{4}} + g ( {4{r^{2}} ( {m + 2} ) + r ( {m + 1} ) ( {{k_{4}}  m + 3} )} ){\omega ^{2}} + g{k_{4}}{r^{2}}}}{ { 4 ( {64{\omega ^{6}} + 16 ( {{{ ( {m + 1} )} ^{2}} + {r^{2}}} ){\omega ^{4}} + 4 ( {{{ ( {m + 1} )} ^{2}}{r^{2}} + 2{k_{4}}r ( {m + 1 + r} )} ){\omega ^{2}} + {{k_{4}}^{2}}{r^{2}}} )}}\).

(1)
If \({l_{3}} > 0\), system (2.2) has a transversal Hopf point of codimension three at \({E_{2}}\) which is unstable.

(2)
If \({l_{3}} < 0\), system (2.2) has a transversal Hopf point of codimension three at \({E_{2}}\) which is stable.
Proof
By the changes
system (2.2) becomes the following system (still denoted by x, y, z, u, v):
and the equilibrium \({E_{2}} ( {0,0,0,\frac{g}{{k_{1}}},0} )\) is moved to \(O ( {0,0,0,0,0} )\).
From (4.1), the transversality condition holds. Now we calculate the Lyapunov coefficients, which show the stability of the equilibrium and the periodic orbit which appears.
According to Ref. [18], for the parameters \(({m,r,g,{k_{1}},{k_{2}},{k_{3}},{k_{4}},{k_{5}}} ) \in S\) and \({k_{6}}=0\), we have
where
Therefore
Since \({l_{1}}={l_{2}}=0\), we continue to calculate \({l_{3}}\). Some vector expressions are too complex, and for the convenience of expression, we write the results after calculation as follows:
where
Hence one has \({l_{3}} =  \frac{{4g ( { ( {m + 2} ) ( {m + 1} ) + r ( {m  3} )} ){\omega ^{4}} + g ( {4{r^{2}} ( {m + 2} ) + r ( {m + 1} ) ( {{k_{4}}  m + 3} )} ){\omega ^{2}} + g{k_{4}}{r^{2}}}}{ { 4 ( {64{\omega ^{6}} + 16 ( {{{ ( {m + 1} )} ^{2}} + {r^{2}}} ){\omega ^{4}} + 4 ( {{{ ( {m + 1} )} ^{2}}{r^{2}} + 2{k_{4}}r ( {m + 1 + r} )} ){\omega ^{2}} + {{k_{4}}^{2}}{r^{2}}} )}}\).â€ƒâ–¡
Numerical simulation
For \(m=r=g=1\), \({k_{2}}={k_{5}}=0\), \({k_{1}}=0.001\), \({k_{3}}=0.001\), \({k_{4}}=5\), and \({k_{6}}=0.001\), we have \({l_{3}} = 0.05\). An unstable limit cycle is obtained with the initial condition \((1, 0, 1, 0, 0)\) (see Fig.Â 8).
5 Pitchfork bifurcation in system (2.2)
We utilize the center manifold theorem and the bifurcation theory [18, 19] to study pitchfork bifurcation of system (2.2).
Let
When \(( {{k_{1}},{k_{2}},{k_{3}},{k_{4}},{k_{5}},{k_{6}}} ) \in S\), system (2.2) has only one equilibrium \({E_{2}} (0, 0, 0, \frac{g}{k_{1}}, 0)\).
By the changes (4.2), system (2.2) becomes the following system (still denoted by x, y, z, u, v):
and the equilibrium \({E_{2}}\) is moved to \(O(0, 0, 0, 0, 0)\).
The Jacobian matrix at O is
and the corresponding characteristic equation is
System (2.2) has a zero eigenvalue \({\lambda _{1}}=0\) and the other four eigenvalues
\(O(0,0,0,0,0)\) is nonhyperbolic, and then we can get the following theorem.
Theorem 5.1
For \(( {{k_{1}},{k_{2}},{k_{3}},{k_{4}}, {k_{5}},{k_{6}}} ) \in S\), system (2.2) undergoes a pitchfork bifurcation at \({E_{2}}(0, 0, 0, \frac{g}{{{k_{1}}}}, 0)\). Furthermore, when \({k_{4}} < 0\), there is only one equilibrium \({E_{2}}\) which is stable near the lefthand side of \({k_{4}} = 0\); when \({k_{4}} > 0\), \({E_{2}}\) becomes unstable and the other two equilibria are stable near the righthand side of \({k_{4}} = 0\).
Proof
The corresponding eigenvectors are
Let
By (5.2), system (5.1) becomes
where
From the center manifold theorem, there exists a center manifold for Eqs.Â (5.3), which can be expressed locally as the following set through the variable \({x_{1}}\) and Îµ:
where Î´ and Î´Ì„ are sufficiently small.
Assume that
Considering \(\dot{\varepsilon }\equiv 0\), the center manifold should satisfy
where
Substituting Eqs.Â (5.4) to (5.5) gives
and we obtain
Applying Eqs.Â (5.6) into \({\dot{x}_{1}} = {g_{1}}\) of (5.3) and reducing the vector field to the center manifold, we can get
where
\(F({x_{1}},\varepsilon )\) satisfies
which indicates that the equilibrium \(( {{x_{1}},\varepsilon } ) = ( {0,0} )\) of Eqs.Â (5.7) undergoes a pitchfork bifurcation at \(\varepsilon = 0\) (\(k_{4} = 0\)). Since \( \frac{{{\partial ^{3}}F}}{{\partial {x_{1}}^{3}}}/\frac{ {{\partial ^{2}}F}}{{\partial {x_{1}}\partial \varepsilon }} > 0\), the bifurcation direction is near the righthand side of \(\varepsilon = 0\) (\(k_{4} = 0\)). So TheoremÂ 5.1 is proved.â€ƒâ–¡
Numerical simulation
For \(r = m = {k_{1}} = {k_{2}} = {k_{3}} = 1\) and \(g = {k_{5}} = {k_{6}} =0\), (5.8) becomes
As shown in Fig.Â 9, system (2.2) undergoes a pitchfork bifurcation, which accords with TheoremÂ 5.1.
6 Conclusions
The 5D segmented disc dynamo is very interesting and novel in that there are coexisting hidden attractors with four types of equilibria: a curve of equilibria, a line equilibrium, a stable equilibrium, and no equilibria. The paper studies not only coexisting selfexcited attractors but also coexisting hidden attractors. Hidden hyperchaos with three positive LEs is also displayed. Besides, by choosing an appropriate bifurcation parameter, the paper proves that the degenerate Hopf bifurcation and pitchfork bifurcation occur in the system. The simulation results demonstrate the correctness of the two bifurcations analysis.
The research on the new system may enrich the hyperchaotic theories and engineering applications. It is also hoped that the work is helpful to identify the geometrical characteristics of lower dimensional chaotic attractors. More studies will be explored to reveal the riddled property of the basin of attraction of the hyperchaotic attractors.
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Acknowledgements
The authors would like to express many thanks to Professor Yuming Chen for his corrections and comments on this manuscript.
Funding
The research is supported by the Open Project of Guangxi Colleges and Universities Key Laboratory of Complex System Optimization and Big Data Processing (No. 2017CSOBDP0303) and the National Natural Science Foundation of China (No.Â 11671149).
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Bao, J., Liu, Y. Multistability and bifurcations in a 5D segmented disc dynamo with a curve of equilibria. Adv Differ Equ 2019, 345 (2019). https://doi.org/10.1186/s1366201922840
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DOI: https://doi.org/10.1186/s1366201922840
Keywords
 5D segmented disc dynamo
 Curve of equilibria
 Multistability
 Degenerate Hopf bifurcation
 Pitchfork bifurcation