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Hyperchaos in 3-D piecewise smooth maps
Chaos, Solitons and Fractals 133 (2020) 109681
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Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos
Hyperchaos in 3-D piecewise smooth maps Mahashweta Patra a,∗, Soumitro Banerjee b a b
Department of Applied Mechanics, IIT Madras, Chennai 600 036, India Department of Physical Sciences, Indian Institute of Science Education and Research Kolkata, Mohanpur Campus-741246, West Bengal, India
a r t i c l e
i n f o
Article history: Received 6 October 2019 Revised 1 January 2020 Accepted 4 February 2020
Keywords: Hyperchaotic attractor Lyapunov exponent Bifurcation diagram Coexisting attractors Three dimensional piecewise smooth maps Stable manifold and unstable manifolds
a b s t r a c t In this paper, we show various ways of the occurrence of a hyperchaotic orbit in 3D piecewise linear normal form maps. We show that hyperchaotic orbit can be born from a periodic orbit or a quasiperiodic orbit in various ways like-(a) a direct transition to a hyperchaotic orbit from a periodic orbit or a from a quasiperiodic orbit through border collision bifurcation; (b) a transition from a periodic orbit to a hyperchaotic orbit via quasiperiodic and chaotic orbit; (c) a transition from a mode-locked periodic orbit to a hyperchaotic orbit via higher dimensional torus. We also show bifurcations where a hyperchaotic orbit bifurcates to a different hyperchaotic orbit or a three-piece hyperchaotic orbit. We further show period increment with the coexistence of hyperchaotic attractors. Moreover, we numerically calculate the existence region of a hyperchaotic orbit in the parameter space region.
1. Introduction Three dimensional systems exhibit some attractors such as hyperchaos and higher dimensional torus that cannot be observed in two dimensional systems. Hyperchaos is defined as a chaotic attractor with more than one positive Lyapunov exponent. The behavior and therefore analysis of hyperchaotic systems is much more complicated than the case of systems with just a single positive Lyapunov exponent. Hyperchaos, with more than one positive Lyapunov exponent, has been studied with increasing interest in recent years, in the fields of lasers [21], Colpitts oscillators [6], nonlinear circuits [5], communication [23], control theory [13] and so on. Matsumoto et al. [15] have observed hyperchaos, for the first time, in a real physical system, a fourth order electrical circuit. A few experimental hyperchaotic behaviors have been found; for example in NMR laser (Stopp et al. [1988]), in a semiconductor system (Stoop et al. [1989]) and in a chemical reaction system (Eiswirth et al. [1992]), in hydrodynamics [4] and semiconductor devices [22], in coupled Chua’s circuit [12]. Hyperchaotic orbit is also found in nonsmooth systems [9], fractional order systems [10], Ribosome Autocatalytic Synthesis [14], chemical reactions [2]. Hyperchaotic orbit is highly dependent on initial conditions. Therefore, generating various hyperchaotic systems have applications in information technology such as communication and encryption. Thus hyperchaotic orbit is more useful for applications in chaos and in secure communications, in fluid mixing [1,16,20]. ∗
Corresponding author. E-mail address: [email protected] (M. Patra).
https://doi.org/10.1016/j.chaos.2020.109681 0960-0779/© 2020 Elsevier Ltd. All rights reserved.
© 2020 Elsevier Ltd. All rights reserved.
Therefore, construction of new hyperchaotic systems and its study are useful to explore the nature of hyperchaos. Nowadays it is already well known that many real physical systems, engineering systems, biological systems can be modelled by piecewise smooth linear normal form maps. 1D and 2D PWS systems are well investigated. Very few researches have been reported on 3D PWS systems [3,17,18]. In this paper, we take the 3D piecewise linear normal form map as a system of investigation and the question we address in this paper is: How does the occurrence of hyperchaotic orbit in a nonsmooth system differ from that in a smooth system? The system description is given in Section 2. Occurrence of hyperchaotic orbit is described in Section 3. The conclusion out of the work is given in Section 5. 2. System description A three-dimensional piecewise smooth map in piecewise linear normal form approximation [7,8,17,18] is given by
Xn+1 = Fμ (Xn ) =
Al Xn + μC,
if xn ≤ 0
(1)
Ar Xn + μC, if xn ≥ 0
where Xn = (xn , yn , zn )T ∈ R3 , C = (1, 0, 0 )T ∈ R3 , Al and Ar are real valued 3 × 3 matrices
Al =
τl
−σl
δl
1 0 0
0 1 0
τr and Ar = −σr δr
1 0 0
0 1 0
The phase space of this map is divided by the borderline xb : x = 0 into two regions L := (x, y, z ) ∈ R3 : x ≤ 0 and R :=
2
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Fig. 1. Orbit at (a) μ = 0.1 and (b) μ = −0.1. (c) bifurcation diagram, (d) Lyapunov exponents diagram. Other parameters are: τl = −0.5, σl = 0.95, δl = 0.2, τr = 0.8, σr = −0.6, δr = −0.83. The two positive Lyapunov exponents (0.0457 and 0.0451) are close to each other so that they appears as a single value.
(x, y, z ) ∈ R3 : x > 0. In each region, the dynamics are governed by a linear map and the equations are continuous across xb . Eq. (1) is the piecewise linear approximation of a general piecewise smooth 3D system evaluated in a close neighborhood of the border. If λ1 , λ2 and λ3 are the eigenvalues of the Jacobian matrix evaluated at a fixed point placed on the left side close to the border, then the parameters of the matrix Al are simply the trace τl = λ1 + λ2 + λ3 , the second trace σl = λ1 λ2 + λ2 λ3 + λ3 λ1 and the determinant δl = λ1 λ2 λ3 . The parameters of the matrix Ar depends, in a similar manner, on the eigenvalues of the Jacobian matrix computed at the fixed point placed on the right side. The fixed points of the system in both sides of the boundary are given by
μ μ(−σl + δl ) μδl , , ) 1 − τl + σl − δl 1 − τl + σl − δl 1 − τl + σl − δl μ μ(−σr + δr ) μδr R∗ = ( , , ) 1 − τr + σr − δr 1 − τr + σr − δr 1 − τr + σr − δr L∗ = (
μ = −1 can be one-to-one mapped onto the case μ = +1 using the symmetry
(x, y, z, τl , τr , σl , σr , δl , δr , μ ) ⇐⇒ (−x, −y, −z, τr , τl , σr , σl , δr , δl , −μ ) Therefore any trajectory must have a twin trajectory.
Fig. 2. The unstable manifold of the fixed point L∗ = (xl , yl , zl ) = (−0.0444, 0.0333, −0.0089 ), eigenvalues are −0.3428 ± 0.9797i, 0.1857.
to zero. It has been shown that this dimension is close to other dimensions such as the box-counting and correlation dimensions for typical strange attractors. In a n-dimensional system the Lyapunov exponents are arranged in order from largest to smallest λ1 ≥ λ2 ≥ ≥ λn . Let j be the index for which j
3. Occurrence of hyperchaotic orbit
λi ≥ 0 and
j+1
i=1
In this section, we show hyperchaotic orbit and various ways of its occurrence. We represent some new bifurcation phenomena associated with hyperchaotic attractor that occur in piecewise smooth systems but does not occur in smooth systems. We also offer an explanation of the observed phenomena. One measure of the dimensionality of a strange attractor is the Lyapunov dimension DLY . J. Kaplan and J. A. Yorke conjectured that the dimension of a strange attractor can be approximated from the spectrum of Lyapunov exponents. Such a dimension has been called the Kaplan-Yorke (or Lyapunov) dimension. It is defined as the number of ordered Lyapunov exponents that sum
λi < 0
i=1
Then the dimension of the attractor is
j
DLY = j +
λ i=1 i |λ j+1 |
Lyapunov dimension is an upper bound for Hausdorff and fractal dimensions. 3.1. Periodic orbit to hyperchaos (through border collision bifurcation) Fig. 1 shows a direct transition from a periodic orbit to hyperchaotic orbit through a border collision bifurcation. As μ is varied
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Fig. 3. Orbit at (a) μ = 0.1 and (b) μ = −0.1. (c) bifurcation diagram, (d) Lyapunov exponents diagram. Other parameters are: τl = −0.5, σl = 0.95, δl = 0.2, τr = 0.8, σr = −0.6, δr = −0.93.
Fig. 4. Orbit at (a) δr = −0.83, (b) δr = −0.93, (c) δr = −0.94, (d) δr = −0.95, and (e) δr = −0.97. Other parameters are: τl = −0.5, σl = 0.95, δl = 0.2, τr = 0.8, σr = −0.6, μ = 0.1.
Fig. 5. (a) Unstable (blue) manifold for period-23 saddle fixed point (red) at δr = −0.94. (b) Shows the magnified view of (a). Black shows the eigenplane (it is a 2D plane, from this particular angle it looks like a 1D line). Other parameters are same as in Fig. 4. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
from a positive to a negative value, a periodic orbit bifurcates to a hyperchaotic orbit. The phase portraits are shown in Fig. 1(a) and 1(b). The corresponding bifurcation diagram is shown in Fig. 1(c). Fig. 1(d) shows the highest two Lyapunov exponents plotted against the bifurcation parameter. It confirms the occurrence of bifurcation from a periodic orbit to a hyperchaotic orbit through
a border collision bifurcation. Two-dimensional unstable manifold for L∗ is shown in Fig. 2. For μ = 0.1, L∗ is virtual and R∗ is admissible whose eigenvalues are −0.8979, 0.8489 ± 0.4514i ≡ 0.9615. Therefore we get a stable periodic orbit whose Lyapunov exponents are (−0.04, −0.04, −0.10 ).
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Fig. 6. (a) Unstable (blue) manifold for period-23 saddle fixed point (red) at δr = −0.942. (b) shows magnified view of (a). Cyan shows the stbale eigenplane. Other parameters are same as in Fig. 4. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 7. Stable (cyan) and unstable (blue) manifold for period-11 saddle fixed point (red) at δr = −0.95. Other parameters are same as in Fig. 4. Period-11 saddle focus are shown in black. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 8. Stable (cyan) and unstable (blue) manifold for period-11 saddle fixed point (red) at δr = −0.97. Other parameters are same as in Fig. 4. period-11 saddle focus are shown in black. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
As the bifurcation parameter approaches a negative value μ = −0.1, R∗ becomes virtual and L∗ becomes admissible with eigenvalues −0.3428 ± 0.9797i (modulus 1.0379), and 0.1857, therefore is a saddle-focus. The Lyapunov exponents of the orbit are (0.04, 0.04, −1.50 ). Since the spectrum of the Lyapunov exponents has two positive terms the system is hyperchaotic. We find the sum of the Lyapunov exponents as a negative number, which shows that the hyperchaotic system is dissipative. According to the KaplanYorke definition, the Lyapunov dimension (DLY ) is 2.06.
At μ = 0.1, L∗ is virtual, R∗ is admissible which is a saddle focus with eigenvalues −0.9276, 0.8638 ± 0.5064i (modulus 1.0013). The Lyapunov exponents of the orbit are (0, −0.05, −0.14 ), therefore the orbit is quasiperiodic. As the bifurcation parameter changes to −0.1, L∗ becomes admissible with eigenvalues 0.1857, −0.3428 ± 0.9797i (modulus 1.0379). The Lyapunov exponents are (0.04, 0.04, −1.50 ) and the Kaplan-Yorke dimension is 2.06.
3.3. Periodic orbit to quasiperiodic orbit to chaotic orbit to hyperchaos 3.2. Quasiperiodic orbit to hyperchaos (through border collision bifurcation) Fig. 3 shows a direct transition from a quasiperiodic orbit to a hyperchaotic orbit as μ varies from a positive value to a negative value. Fig. 3(a) and 3(b) show the phase portrait at μ = −0.1 and μ = 0.1 correspondingly and the bifurcation diagram is shown in Fig. 3(c). Fig. 3(d) shows the highest two Lyapunov exponents with the bifurcation parameter which confirms the transition from a quasiperiodic orbit to a hyperchaotic orbit through a border collision bifurcation.
In this section, we show a transition from a periodic orbit to a hyperchaotic orbit via quasiperiodic orbit and chaotic orbit. Fig. 4 shows the orbit at different values of δ r as it varies from −0.83 to −0.97. At δr = −0.83, L∗ is virtual and R∗ is admissible with eigenvalues −0.8979, 0.8489 ± 0.4514i (modulus 0.9615). As the bifurcation parameter changes to −0.93, R∗ undergoes a Neimark-Sacker bifurcation. Eigenvalue of R∗ becomes −0.9276, 0.8638 ± 0.5064i (modulus 1.0013). The periodic orbit bifurcates to a quasiperiodic orbit (Fig. 4b) whose Lyapunov exponents are (0, −0.05, −0.14 ).
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Fig. 9. (a) Bifurcation diagram, (b) Lyapunov diagram with the bifurcation parameter δ r . Other parameters are same as in Fig. 4.
Fig. 10. Orbit at (a) μ = −0.1, (b) μ = 0.1 and (c) bifurcation diagram, (d) Lyapunov diagram. Parameter settings: τl = −0.7, σl = 0.95, δl = 0.2, τr = 0.76, σr = −0.615, δr = −1.2.
Fig. 11. Orbit at (a) τl = −0.7, (b) τl = −0.9 and (c) bifurcation diagram (d) Lyapunov diagram. Parameter settings: σl = 0.95, δl = 0.2, τr = 0.76, σr = −0.615, δr = −1.2, μ = −0.1.
When the bifurcation parameter changes to −0.94, a saddlenode bifurcation occurs on the closed-loop and a period23 saddle and period-23 stable fixed points appear. The quasiperiodic orbit bifurcates to a mode-locked periodic orbit (Fig. 4c). Eigenvalues of the period-23 stable fixed points are (−0.6387, −0.0404, 0.0899 ) and the period-23 saddle fixed points are (1.5427, 0.0365, −0.1939 ). Fig. 5 (a) shows the invariant closed curve formed by the union of the stable fixed points, saddle fixed points and the unstable connection between the saddle fixed points at δr = −0.94. Fig. 5 (b) shows the magnified view of (a) which shows that there is no homoclinic intersection between the unstable and stable manifolds of the saddle fixed points. As the bifurcation parameter changes to −0.942, a homoclinic intersection between the stable and unstable manifolds occurs which is shown in Fig. 6. Fig. 6 (a) shows the unstable manifold for the period-23 saddle fixed point. (b) shows the magnified view which confirms the homoclinic intersection between stable and unstable manifold.
The homoclinic intersection results in a chaotic orbit as the bifurcation parameter δ r changes to −0.95. The Lyapunov exponents of the orbit are (0.03, −0.10, −0.15 ). There exists a period11 saddle cycle and a period-11 saddle-focus cycle. Eigenvalues of the period-11 saddle fixed points are 1.9912, −0.0905 ± 0.2316i (modulus 0.2452). Eigenvalues of the period-11 sadlle focus are −0.0169, 0.3790 ± 1.1608i (modulus 1.2211). The unstable manifold connecting the saddle fixed points is shown in Fig. 7 (a) which has a homoclinic intersection with the stable eigenplane and is shown in the magnified view in Fig. 7 (b). As the bifurcation parameter reaches −0.97, the unstable manifold is contained entirely within the stable manifold Fig. 8. Fig. 8 (b) is the magnified picture of Fig. 8 (a). Therefore, the orbit’s dimension increases by 1. The chaotic orbit becomes hyperchaotic. At that parameter value, eigenvalues of the period-11 saddle fixed points are 1.9803, −0.0719 ± 0.2632i (modulus 0.2728). Eigenvalues of the period-11 sadlle focus are −0.0186, 0.3541 ± 1.2275i (modulus 1.2776). Lyapunov exponents of the orbit are (0.02, 0.01,
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Fig. 12. Unstable manifold (blue) for period-6 saddle fixed point (red) at (a) τl = −0.7, (b) τl = −0.9. Period-6 saddle focus is shown in black. (b) and (d) are the magnified view of (a) and (c) correspondingly. Other parameters are same as in Fig. 11. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
−0.35 ) and the Kaplan-Yorke dimension is 2.10. Fig. 9 shows the bifurcation diagram and the Lyapunov diagram against the bifurcation parameter δ r which demonstrates the transition of a periodic orbit to a hyperchaotic orbit through a quasiperiodic, mode-locked orbit and chaotic orbit sequence. 3.4. Hyperchaos to hyperchaos In this section, we show that a hyperchaotic orbit persists even after the border collision bifurcation and remains hyperchaotic. Fig. 10 (a) shows a hyperchaotic orbit for μ = −0.1. For this parameter values Lyapunov exponents are (0.08, 0.06, −1.54 ), therefore, the orbit is hyperchaotic. Kaplan-Yorke Lyapunov dimension for this hyperchaotic orbit is 2.09. As μ is increased to a positive number μ = 0.1, the Lyapunov exponents are (0.08, 0.04, −0.42 ), Kaplan-Yorke dimension is 2.29, which confirms the attractor as hyperchaotic. Fig. 10 (c) and (d) show the bifurcation diagram and the highest two Lyapunov exponents plotted against μ illustrating the transition from a hyperchaotic orbit to another hyperchaotic orbit as the bifurcation parameter changes from a negative to a positive value. 3.5. One-piece hyperchaos to a three-piece hyperchaos In this section, we show a transition from a one-piece hyperchaotic orbit to a three-piece hyperchaotic orbit as τ l crosses bifurcation point. Fig. 11 (a) and 11(b) show the orbits at different values of τ l . At τl = −0.7, a hyperchaotic orbit exists with Lyapunov exponents (0.08, 0.06, −1.54 ). The Kaplan-Yorke dimension is 2.09. A bifurcation occurs at τl = −0.8 and a three-piece chaotic attractor comes into existence. At τl = −0.9, Lyapunov exponents of the orbit are (0.15, 0.13, −1.13 ) and the Kaplan-Yorke dimension is 2.25. Fig. 11 (c) and 11(d) show the bifurcation diagram and the Lyapunov exponent diagram (highest two Lyapunov exponents) against the bifurcation parameter. The highest two Lyapunov exponents, in this case, are greater than zero. Therefore, we get hyperchaotic attractor throughout the parameter region. An abrupt change in the highest Lyapunov exponent and the second highest exponent occurs at the bifurcation point and the hyperchaotic orbit bifurcates to a three-piece hyperchaotic orbit. At τl = −0.7, a period-6 saddle fixed point and period-6 saddle focus exist with eigenvalues (2.4047, −0.9298, 0.0062 ) and
Fig. 13. Unstbale manifold (blue) for period-3 saddle fixed point (violet) at τl = −0.9. Other parameters are same as in Fig. 11. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
0.0588, 0.3437 ± 1.1370i (modulus 1.1878). Unstable manifold for the period-6 saddle fixed point is shown in Fig. 12(a). Fig. 12(b) shows the magnified view which shows that the one-piece hyperchaotic orbit lies on the unstable manifold. At τl = −0.9, eigenvalues for the period-6 saddle fixed point and period-6 saddle focus become 2.3068, −1.1284, 0.0053 and 0.0593, 0.4611 ± 1.0889i (modulus 1.1825). Unstable manifold for the period-6 saddle fixed point is shown in Fig. 12(c). Fig. 12(d) is the magnified view of (c) which shows that the three-piece hyperchaotic orbit lies on the unstable manifold. Along with this period-6 fixed point, two period-3 saddle fixed point exist with eigenvalues (−1.8496, 1.2235, 0.0212 ) and (0.0883, −1.8896, −1.7266 ) respectively. The unstable manifold for the period-3 saddle fixed point is shown in Fig. 13. The manifold undergoes multiple folds in three regions on which three-piece hyperchaotic attractor lies.
3.6. Different orbits coexisting with hyperchaotic attractor Coexistence of various attractors in 3D piecewise smooth systems was shown in our earlier papers [18,19]. In this section, we show various orbits coexisting with hyperchaotic attractor.
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Fig. 14. Shows period increment with coexistence of hyperchaotic orbit. (a) period-4 orbit with hyperchaotic orbit at σl = 1.1, (b) period-8 orbit with hyperchaotic orbit at σl = 1.05, (c) quasiperiodic orbit with hyperchaotic orbit at σl = 0.98 and (d) chaotic orbit with hyperchaotic orbit at σl = 0.92. Parameter settings are: τl = −0.72, δl = 0.65, τr = −0.76, σr = 0.615, δr = −1.2, μ = 0.35.
Fig. 15. (a) Shows bifurcation diagram. Lyapunov diagram with different initial conditions. Initial conditions are (b) (0.3, −0.5, −0.75 ) and (c) (−00.3, −0.5, −0.75 ). Parameter settings are same as in Fig. 14.
Fig. 17. (a) Mode-locked periodic orbit at τl = 0.65, (b) shows the two-dimensional unstable manifold (blue), saddle fixed point (green) and stable fixed points (red). Parameter settings: σl = −0.785, δl = 0.745, τr = 0.37, σr = 0.755, δr = −0.725, μ = 0.3. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 16. Shows phase space region showing σ l × τ r plane. Other parameters are same as in Fig. 14.
Fig. 14 shows the coexistence of various orbits with hyperchaotic orbit at different parameter values. At σl = 1.1, period-4 orbit coexists with hyperchaotic orbit. As the bifurcation parameter is changed to 1.05, the period-4 orbit is bifurcated to period-8 orbit and coexists with the hyperchaotic orbit. When the bifurcation parameter σ l is changed to 0.98, period-8 orbit bifurcates to 8-loop torus and coexists with hyperchaotic orbit. As the bifurcation parameter changes to 0.92, torus bifurcates to chaotic orbit and coexists with hyperchaotic orbit. Fig. 15 (a) shows the bifurcation diagram showing period increment with coexistence of hyperchaotic orbit. Fig. 15(b) and Fig. 15(c) show Lyapunov exponents for different initial conditions that confirms the coexistence of various orbits with hyperchaotic orbit. Fig. 16 shows phase space region showing σ l × τ r plane. 3.7. Mode-locked periodic orbit to hyperchaotic orbit via higher dimensional torus In this section, we show the occurrence of hyperchaotic orbit from a mode-locked periodic orbit via higher dimensional torus.
Fig. 18. (a) Higher dimensional torus at τl = 0.66. (b) shows the unstable manifold (blue), saddle fixed point (green) and saddle focus (red). Parameter settings are same as in Fig. 17. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 17(a) shows a mode-locked periodic orbit forming a Möbius strip structure at τl = 0.65. The unstable manifold for the saddle point is shown in Fig. 17(b). The Möbius strip structure is formed by period-12 stable fixed points and period-12 saddle fixed points and the unstable connection between them. Eigenvalues of period12 saddle fixed points are (2.8386, −0.8742, −0.0090 ) and eigenvalues of period-12 stable fixed points are λ1,2 = 0.5464 ± 0.8145i (modulus 0.9808), λ3 = −0.0238.
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Fig. 19. Hyperchaotic orbit at τl = 0.69. Other parameters are same as in Fig. 17.
As the parameter τ l is increased to 0.66 the complex conjugate eigenvalues of the stable fixed points go out of the unit circle through a Neimark-Sacker bifurcation and therefore the stable fixed points become saddle focus. Eigenvalues of period-12 saddle fixed point are (2.8783, −0.9003, −0.0086 ). Eigenvalues of period12 saddle focus are λ1,2 = 0.5429 ± 0.8401i (modulus 1.0 0 03), λ3 = −0.0229. We get invariant closed curves around each saddle-focus point and they lie on a Möbius strip structure. The orbit is shown in Fig. 18(a). The Möbius strip structure is therefore formed by the 12 invariant closed curves, period-12 saddle focus points, period12 saddle points and the two-dimensional unstable manifold between them and is shown in Fig. 18(b). The eigenvalues of the saddle fixed points follow the explanation given by [11]. As we increase the bifurcation parameter the higher dimensional torus bifurcates to a hyperchaotic orbit. At τl = 0.69, the orbit is shown in Fig. 19. The Lyapunov exponents are (0.02, 0.004, −0.34 ), the Kaplan-Yorke dimension is 2.08. Fig. 20 shows the highest two Lyapunov exponents showing the transition from a mode-locked periodic orbit to a hyperchaotic orbit via a higher dimensional orbit. Before the bifurcation point, both the highest Lyapunov exponent and second highest Lyapunov exponent are negative, and we get mode-locked periodic orbit lying on a Möbius strip structure. As the bifurcation parameter reaches the bifurcation point τl = 0.66, both the Lyapunov exponents become zero, we get a higher dimensional torus where invariant closed curves lie on a Möbius strip structure. As the bifurcation parameter crosses the bifurcation point two highest Lyapunov exponents become positive, therefore, the orbit becomes hyperchaotic. Transition from a mode-locked orbit to a hyperchaotic orbit is therefore demonstrated by a qualitative change in Lyapunov exponents. Fig. 20 (b) shows the existence region for hyperchaotic orbit (blue), chaotic orbit (green) and quasiperiodic orbit (black). Red star shows the parameter value for which the orbit in Fig. 18(a)
Fig. 21. Schematic diagram showing the structure of the stable and unstable manifolds and their intersection. Blue shows the 2D unstable manifold and red shows the 1d stable manifold. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
has been drawn. The white space shows the region for the periodic orbit. 4. Condition for homoclinic intersection between 2D unstable manifold and 1D stable manifold In all the examples shown in section-3 we show one common feature: the hyperchaotic orbit lies on a two-dimensional unstable manifold. In a discrete dynamical system, for hyperchaotic orbit, two Lyapunov exponents are greater than zero, the orbit has two diverging directions and one converging direction. Therefore in such cases saddle fixed points must have two eigenvalues whose absolute values are more than 1 and one eigenvalue with value less than 1. Thus we get two-dimensional unstable manifold and a one-dimensional stable manifold. Hyperchaotic orbit is born when a homoclinic intersection occurs between a 1D stable manifold and 2D unstable manifold (see Fig. 21). The unstable set folds at every intersection with z = 0 and every image of a fold is also a fold. Similarly, the stable set folds at every intersection with x = 0 and every pre-image of a fold is also a fold. Due to this phenomenon, in the system under consideration, the manifolds undergo numerous folds. That is why it is impossible to analytically obtain the condition of intersection of these two manifolds.
Fig. 20. (a) Lyapunov exponent diagram with bifurcation parameter τ l . Highest two Lyapunov exponents are shown in red and blue color. (b) Parameter space τ l × τ r . Hyperchaotic orbit (blue), chaotic orbit (green), quasiperiodic orbit (black), higher dimensional torus (red) are shown. Other parameters are same as in Fig. 17. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
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5. Conclusion
References
Three dimensional systems exhibit some phenomena which don’t occur in a two dimensional system such as higher dimensional torus (three frequency torus), hyperchaotic orbit. In this paper, we have shown the occurrence of higher dimensional torus and hyperchaotic orbit in three-dimensional piecewise linear maps. We show that hyperchaotic orbit can be born from a periodic orbit or from a quasiperiodic orbit in various ways- (a) a direct transition from a periodic orbit or a quasiperiodic orbit to a hyperchaotic orbit through border collision bifurcation, (b) a transition from a periodic orbit to a hyperchaotic orbit via quasiperiodic and chaotic orbit, (c) a transition from a mode-locked periodic orbit to hyperchaotic orbit through higher dimensional torus. The transition from a periodic orbit to a hyperchaotic orbit through quasiperiodic and chaotic orbit is possible in smooth systems as well as in nonsmooth systems. We have shown some bifurcations that do not occur in a smooth system. A direct transition from a periodic orbit to a hyperchaotic orbit, or from a quasiperiodic orbit to a hyperchaotic orbit is possible in nonsmooth systems, but it is not possible in smooth dynamical systems. We show bifurcations where a hyperchaotic orbit bifurcates to a different hyperchaotic orbit or a three-piece hyperchaotic orbit. Coexistence of hyperchaotic orbit with various orbits is also shown in this paper. Moreover, we numerically calculate the existence region of a hyperchaotic orbit and higher dimensional torus in the parameter space region.
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Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. CRediT authorship contribution statement Mahashweta Patra: Methodology, Software, Validation, Formal analysis, Investigation, Resources, Data curation, Writing - original draft, Writing - review & editing, Visualization. Soumitro Banerjee: Conceptualization, Methodology, Validation, Formal analysis, Investigation, Resources, Writing - review & editing, Supervision, Project administration, Funding acquisition.
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Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos
Hyperchaos in 3-D piecewise smooth maps Mahashweta Patra a,∗, Soumitro Banerjee b a b
Department of Applied Mechanics, IIT Madras, Chennai 600 036, India Department of Physical Sciences, Indian Institute of Science Education and Research Kolkata, Mohanpur Campus-741246, West Bengal, India
a r t i c l e
i n f o
Article history: Received 6 October 2019 Revised 1 January 2020 Accepted 4 February 2020
Keywords: Hyperchaotic attractor Lyapunov exponent Bifurcation diagram Coexisting attractors Three dimensional piecewise smooth maps Stable manifold and unstable manifolds
a b s t r a c t In this paper, we show various ways of the occurrence of a hyperchaotic orbit in 3D piecewise linear normal form maps. We show that hyperchaotic orbit can be born from a periodic orbit or a quasiperiodic orbit in various ways like-(a) a direct transition to a hyperchaotic orbit from a periodic orbit or a from a quasiperiodic orbit through border collision bifurcation; (b) a transition from a periodic orbit to a hyperchaotic orbit via quasiperiodic and chaotic orbit; (c) a transition from a mode-locked periodic orbit to a hyperchaotic orbit via higher dimensional torus. We also show bifurcations where a hyperchaotic orbit bifurcates to a different hyperchaotic orbit or a three-piece hyperchaotic orbit. We further show period increment with the coexistence of hyperchaotic attractors. Moreover, we numerically calculate the existence region of a hyperchaotic orbit in the parameter space region.
1. Introduction Three dimensional systems exhibit some attractors such as hyperchaos and higher dimensional torus that cannot be observed in two dimensional systems. Hyperchaos is defined as a chaotic attractor with more than one positive Lyapunov exponent. The behavior and therefore analysis of hyperchaotic systems is much more complicated than the case of systems with just a single positive Lyapunov exponent. Hyperchaos, with more than one positive Lyapunov exponent, has been studied with increasing interest in recent years, in the fields of lasers [21], Colpitts oscillators [6], nonlinear circuits [5], communication [23], control theory [13] and so on. Matsumoto et al. [15] have observed hyperchaos, for the first time, in a real physical system, a fourth order electrical circuit. A few experimental hyperchaotic behaviors have been found; for example in NMR laser (Stopp et al. [1988]), in a semiconductor system (Stoop et al. [1989]) and in a chemical reaction system (Eiswirth et al. [1992]), in hydrodynamics [4] and semiconductor devices [22], in coupled Chua’s circuit [12]. Hyperchaotic orbit is also found in nonsmooth systems [9], fractional order systems [10], Ribosome Autocatalytic Synthesis [14], chemical reactions [2]. Hyperchaotic orbit is highly dependent on initial conditions. Therefore, generating various hyperchaotic systems have applications in information technology such as communication and encryption. Thus hyperchaotic orbit is more useful for applications in chaos and in secure communications, in fluid mixing [1,16,20]. ∗
Corresponding author. E-mail address: [email protected] (M. Patra).
https://doi.org/10.1016/j.chaos.2020.109681 0960-0779/© 2020 Elsevier Ltd. All rights reserved.
© 2020 Elsevier Ltd. All rights reserved.
Therefore, construction of new hyperchaotic systems and its study are useful to explore the nature of hyperchaos. Nowadays it is already well known that many real physical systems, engineering systems, biological systems can be modelled by piecewise smooth linear normal form maps. 1D and 2D PWS systems are well investigated. Very few researches have been reported on 3D PWS systems [3,17,18]. In this paper, we take the 3D piecewise linear normal form map as a system of investigation and the question we address in this paper is: How does the occurrence of hyperchaotic orbit in a nonsmooth system differ from that in a smooth system? The system description is given in Section 2. Occurrence of hyperchaotic orbit is described in Section 3. The conclusion out of the work is given in Section 5. 2. System description A three-dimensional piecewise smooth map in piecewise linear normal form approximation [7,8,17,18] is given by
Xn+1 = Fμ (Xn ) =
Al Xn + μC,
if xn ≤ 0
(1)
Ar Xn + μC, if xn ≥ 0
where Xn = (xn , yn , zn )T ∈ R3 , C = (1, 0, 0 )T ∈ R3 , Al and Ar are real valued 3 × 3 matrices
Al =
τl
−σl
δl
1 0 0
0 1 0
τr and Ar = −σr δr
1 0 0
0 1 0
The phase space of this map is divided by the borderline xb : x = 0 into two regions L := (x, y, z ) ∈ R3 : x ≤ 0 and R :=
2
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Fig. 1. Orbit at (a) μ = 0.1 and (b) μ = −0.1. (c) bifurcation diagram, (d) Lyapunov exponents diagram. Other parameters are: τl = −0.5, σl = 0.95, δl = 0.2, τr = 0.8, σr = −0.6, δr = −0.83. The two positive Lyapunov exponents (0.0457 and 0.0451) are close to each other so that they appears as a single value.
(x, y, z ) ∈ R3 : x > 0. In each region, the dynamics are governed by a linear map and the equations are continuous across xb . Eq. (1) is the piecewise linear approximation of a general piecewise smooth 3D system evaluated in a close neighborhood of the border. If λ1 , λ2 and λ3 are the eigenvalues of the Jacobian matrix evaluated at a fixed point placed on the left side close to the border, then the parameters of the matrix Al are simply the trace τl = λ1 + λ2 + λ3 , the second trace σl = λ1 λ2 + λ2 λ3 + λ3 λ1 and the determinant δl = λ1 λ2 λ3 . The parameters of the matrix Ar depends, in a similar manner, on the eigenvalues of the Jacobian matrix computed at the fixed point placed on the right side. The fixed points of the system in both sides of the boundary are given by
μ μ(−σl + δl ) μδl , , ) 1 − τl + σl − δl 1 − τl + σl − δl 1 − τl + σl − δl μ μ(−σr + δr ) μδr R∗ = ( , , ) 1 − τr + σr − δr 1 − τr + σr − δr 1 − τr + σr − δr L∗ = (
μ = −1 can be one-to-one mapped onto the case μ = +1 using the symmetry
(x, y, z, τl , τr , σl , σr , δl , δr , μ ) ⇐⇒ (−x, −y, −z, τr , τl , σr , σl , δr , δl , −μ ) Therefore any trajectory must have a twin trajectory.
Fig. 2. The unstable manifold of the fixed point L∗ = (xl , yl , zl ) = (−0.0444, 0.0333, −0.0089 ), eigenvalues are −0.3428 ± 0.9797i, 0.1857.
to zero. It has been shown that this dimension is close to other dimensions such as the box-counting and correlation dimensions for typical strange attractors. In a n-dimensional system the Lyapunov exponents are arranged in order from largest to smallest λ1 ≥ λ2 ≥ ≥ λn . Let j be the index for which j
3. Occurrence of hyperchaotic orbit
λi ≥ 0 and
j+1
i=1
In this section, we show hyperchaotic orbit and various ways of its occurrence. We represent some new bifurcation phenomena associated with hyperchaotic attractor that occur in piecewise smooth systems but does not occur in smooth systems. We also offer an explanation of the observed phenomena. One measure of the dimensionality of a strange attractor is the Lyapunov dimension DLY . J. Kaplan and J. A. Yorke conjectured that the dimension of a strange attractor can be approximated from the spectrum of Lyapunov exponents. Such a dimension has been called the Kaplan-Yorke (or Lyapunov) dimension. It is defined as the number of ordered Lyapunov exponents that sum
λi < 0
i=1
Then the dimension of the attractor is
j
DLY = j +
λ i=1 i |λ j+1 |
Lyapunov dimension is an upper bound for Hausdorff and fractal dimensions. 3.1. Periodic orbit to hyperchaos (through border collision bifurcation) Fig. 1 shows a direct transition from a periodic orbit to hyperchaotic orbit through a border collision bifurcation. As μ is varied
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Fig. 3. Orbit at (a) μ = 0.1 and (b) μ = −0.1. (c) bifurcation diagram, (d) Lyapunov exponents diagram. Other parameters are: τl = −0.5, σl = 0.95, δl = 0.2, τr = 0.8, σr = −0.6, δr = −0.93.
Fig. 4. Orbit at (a) δr = −0.83, (b) δr = −0.93, (c) δr = −0.94, (d) δr = −0.95, and (e) δr = −0.97. Other parameters are: τl = −0.5, σl = 0.95, δl = 0.2, τr = 0.8, σr = −0.6, μ = 0.1.
Fig. 5. (a) Unstable (blue) manifold for period-23 saddle fixed point (red) at δr = −0.94. (b) Shows the magnified view of (a). Black shows the eigenplane (it is a 2D plane, from this particular angle it looks like a 1D line). Other parameters are same as in Fig. 4. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
from a positive to a negative value, a periodic orbit bifurcates to a hyperchaotic orbit. The phase portraits are shown in Fig. 1(a) and 1(b). The corresponding bifurcation diagram is shown in Fig. 1(c). Fig. 1(d) shows the highest two Lyapunov exponents plotted against the bifurcation parameter. It confirms the occurrence of bifurcation from a periodic orbit to a hyperchaotic orbit through
a border collision bifurcation. Two-dimensional unstable manifold for L∗ is shown in Fig. 2. For μ = 0.1, L∗ is virtual and R∗ is admissible whose eigenvalues are −0.8979, 0.8489 ± 0.4514i ≡ 0.9615. Therefore we get a stable periodic orbit whose Lyapunov exponents are (−0.04, −0.04, −0.10 ).
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Fig. 6. (a) Unstable (blue) manifold for period-23 saddle fixed point (red) at δr = −0.942. (b) shows magnified view of (a). Cyan shows the stbale eigenplane. Other parameters are same as in Fig. 4. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 7. Stable (cyan) and unstable (blue) manifold for period-11 saddle fixed point (red) at δr = −0.95. Other parameters are same as in Fig. 4. Period-11 saddle focus are shown in black. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 8. Stable (cyan) and unstable (blue) manifold for period-11 saddle fixed point (red) at δr = −0.97. Other parameters are same as in Fig. 4. period-11 saddle focus are shown in black. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
As the bifurcation parameter approaches a negative value μ = −0.1, R∗ becomes virtual and L∗ becomes admissible with eigenvalues −0.3428 ± 0.9797i (modulus 1.0379), and 0.1857, therefore is a saddle-focus. The Lyapunov exponents of the orbit are (0.04, 0.04, −1.50 ). Since the spectrum of the Lyapunov exponents has two positive terms the system is hyperchaotic. We find the sum of the Lyapunov exponents as a negative number, which shows that the hyperchaotic system is dissipative. According to the KaplanYorke definition, the Lyapunov dimension (DLY ) is 2.06.
At μ = 0.1, L∗ is virtual, R∗ is admissible which is a saddle focus with eigenvalues −0.9276, 0.8638 ± 0.5064i (modulus 1.0013). The Lyapunov exponents of the orbit are (0, −0.05, −0.14 ), therefore the orbit is quasiperiodic. As the bifurcation parameter changes to −0.1, L∗ becomes admissible with eigenvalues 0.1857, −0.3428 ± 0.9797i (modulus 1.0379). The Lyapunov exponents are (0.04, 0.04, −1.50 ) and the Kaplan-Yorke dimension is 2.06.
3.3. Periodic orbit to quasiperiodic orbit to chaotic orbit to hyperchaos 3.2. Quasiperiodic orbit to hyperchaos (through border collision bifurcation) Fig. 3 shows a direct transition from a quasiperiodic orbit to a hyperchaotic orbit as μ varies from a positive value to a negative value. Fig. 3(a) and 3(b) show the phase portrait at μ = −0.1 and μ = 0.1 correspondingly and the bifurcation diagram is shown in Fig. 3(c). Fig. 3(d) shows the highest two Lyapunov exponents with the bifurcation parameter which confirms the transition from a quasiperiodic orbit to a hyperchaotic orbit through a border collision bifurcation.
In this section, we show a transition from a periodic orbit to a hyperchaotic orbit via quasiperiodic orbit and chaotic orbit. Fig. 4 shows the orbit at different values of δ r as it varies from −0.83 to −0.97. At δr = −0.83, L∗ is virtual and R∗ is admissible with eigenvalues −0.8979, 0.8489 ± 0.4514i (modulus 0.9615). As the bifurcation parameter changes to −0.93, R∗ undergoes a Neimark-Sacker bifurcation. Eigenvalue of R∗ becomes −0.9276, 0.8638 ± 0.5064i (modulus 1.0013). The periodic orbit bifurcates to a quasiperiodic orbit (Fig. 4b) whose Lyapunov exponents are (0, −0.05, −0.14 ).
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Fig. 9. (a) Bifurcation diagram, (b) Lyapunov diagram with the bifurcation parameter δ r . Other parameters are same as in Fig. 4.
Fig. 10. Orbit at (a) μ = −0.1, (b) μ = 0.1 and (c) bifurcation diagram, (d) Lyapunov diagram. Parameter settings: τl = −0.7, σl = 0.95, δl = 0.2, τr = 0.76, σr = −0.615, δr = −1.2.
Fig. 11. Orbit at (a) τl = −0.7, (b) τl = −0.9 and (c) bifurcation diagram (d) Lyapunov diagram. Parameter settings: σl = 0.95, δl = 0.2, τr = 0.76, σr = −0.615, δr = −1.2, μ = −0.1.
When the bifurcation parameter changes to −0.94, a saddlenode bifurcation occurs on the closed-loop and a period23 saddle and period-23 stable fixed points appear. The quasiperiodic orbit bifurcates to a mode-locked periodic orbit (Fig. 4c). Eigenvalues of the period-23 stable fixed points are (−0.6387, −0.0404, 0.0899 ) and the period-23 saddle fixed points are (1.5427, 0.0365, −0.1939 ). Fig. 5 (a) shows the invariant closed curve formed by the union of the stable fixed points, saddle fixed points and the unstable connection between the saddle fixed points at δr = −0.94. Fig. 5 (b) shows the magnified view of (a) which shows that there is no homoclinic intersection between the unstable and stable manifolds of the saddle fixed points. As the bifurcation parameter changes to −0.942, a homoclinic intersection between the stable and unstable manifolds occurs which is shown in Fig. 6. Fig. 6 (a) shows the unstable manifold for the period-23 saddle fixed point. (b) shows the magnified view which confirms the homoclinic intersection between stable and unstable manifold.
The homoclinic intersection results in a chaotic orbit as the bifurcation parameter δ r changes to −0.95. The Lyapunov exponents of the orbit are (0.03, −0.10, −0.15 ). There exists a period11 saddle cycle and a period-11 saddle-focus cycle. Eigenvalues of the period-11 saddle fixed points are 1.9912, −0.0905 ± 0.2316i (modulus 0.2452). Eigenvalues of the period-11 sadlle focus are −0.0169, 0.3790 ± 1.1608i (modulus 1.2211). The unstable manifold connecting the saddle fixed points is shown in Fig. 7 (a) which has a homoclinic intersection with the stable eigenplane and is shown in the magnified view in Fig. 7 (b). As the bifurcation parameter reaches −0.97, the unstable manifold is contained entirely within the stable manifold Fig. 8. Fig. 8 (b) is the magnified picture of Fig. 8 (a). Therefore, the orbit’s dimension increases by 1. The chaotic orbit becomes hyperchaotic. At that parameter value, eigenvalues of the period-11 saddle fixed points are 1.9803, −0.0719 ± 0.2632i (modulus 0.2728). Eigenvalues of the period-11 sadlle focus are −0.0186, 0.3541 ± 1.2275i (modulus 1.2776). Lyapunov exponents of the orbit are (0.02, 0.01,
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Fig. 12. Unstable manifold (blue) for period-6 saddle fixed point (red) at (a) τl = −0.7, (b) τl = −0.9. Period-6 saddle focus is shown in black. (b) and (d) are the magnified view of (a) and (c) correspondingly. Other parameters are same as in Fig. 11. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
−0.35 ) and the Kaplan-Yorke dimension is 2.10. Fig. 9 shows the bifurcation diagram and the Lyapunov diagram against the bifurcation parameter δ r which demonstrates the transition of a periodic orbit to a hyperchaotic orbit through a quasiperiodic, mode-locked orbit and chaotic orbit sequence. 3.4. Hyperchaos to hyperchaos In this section, we show that a hyperchaotic orbit persists even after the border collision bifurcation and remains hyperchaotic. Fig. 10 (a) shows a hyperchaotic orbit for μ = −0.1. For this parameter values Lyapunov exponents are (0.08, 0.06, −1.54 ), therefore, the orbit is hyperchaotic. Kaplan-Yorke Lyapunov dimension for this hyperchaotic orbit is 2.09. As μ is increased to a positive number μ = 0.1, the Lyapunov exponents are (0.08, 0.04, −0.42 ), Kaplan-Yorke dimension is 2.29, which confirms the attractor as hyperchaotic. Fig. 10 (c) and (d) show the bifurcation diagram and the highest two Lyapunov exponents plotted against μ illustrating the transition from a hyperchaotic orbit to another hyperchaotic orbit as the bifurcation parameter changes from a negative to a positive value. 3.5. One-piece hyperchaos to a three-piece hyperchaos In this section, we show a transition from a one-piece hyperchaotic orbit to a three-piece hyperchaotic orbit as τ l crosses bifurcation point. Fig. 11 (a) and 11(b) show the orbits at different values of τ l . At τl = −0.7, a hyperchaotic orbit exists with Lyapunov exponents (0.08, 0.06, −1.54 ). The Kaplan-Yorke dimension is 2.09. A bifurcation occurs at τl = −0.8 and a three-piece chaotic attractor comes into existence. At τl = −0.9, Lyapunov exponents of the orbit are (0.15, 0.13, −1.13 ) and the Kaplan-Yorke dimension is 2.25. Fig. 11 (c) and 11(d) show the bifurcation diagram and the Lyapunov exponent diagram (highest two Lyapunov exponents) against the bifurcation parameter. The highest two Lyapunov exponents, in this case, are greater than zero. Therefore, we get hyperchaotic attractor throughout the parameter region. An abrupt change in the highest Lyapunov exponent and the second highest exponent occurs at the bifurcation point and the hyperchaotic orbit bifurcates to a three-piece hyperchaotic orbit. At τl = −0.7, a period-6 saddle fixed point and period-6 saddle focus exist with eigenvalues (2.4047, −0.9298, 0.0062 ) and
Fig. 13. Unstbale manifold (blue) for period-3 saddle fixed point (violet) at τl = −0.9. Other parameters are same as in Fig. 11. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
0.0588, 0.3437 ± 1.1370i (modulus 1.1878). Unstable manifold for the period-6 saddle fixed point is shown in Fig. 12(a). Fig. 12(b) shows the magnified view which shows that the one-piece hyperchaotic orbit lies on the unstable manifold. At τl = −0.9, eigenvalues for the period-6 saddle fixed point and period-6 saddle focus become 2.3068, −1.1284, 0.0053 and 0.0593, 0.4611 ± 1.0889i (modulus 1.1825). Unstable manifold for the period-6 saddle fixed point is shown in Fig. 12(c). Fig. 12(d) is the magnified view of (c) which shows that the three-piece hyperchaotic orbit lies on the unstable manifold. Along with this period-6 fixed point, two period-3 saddle fixed point exist with eigenvalues (−1.8496, 1.2235, 0.0212 ) and (0.0883, −1.8896, −1.7266 ) respectively. The unstable manifold for the period-3 saddle fixed point is shown in Fig. 13. The manifold undergoes multiple folds in three regions on which three-piece hyperchaotic attractor lies.
3.6. Different orbits coexisting with hyperchaotic attractor Coexistence of various attractors in 3D piecewise smooth systems was shown in our earlier papers [18,19]. In this section, we show various orbits coexisting with hyperchaotic attractor.
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Fig. 14. Shows period increment with coexistence of hyperchaotic orbit. (a) period-4 orbit with hyperchaotic orbit at σl = 1.1, (b) period-8 orbit with hyperchaotic orbit at σl = 1.05, (c) quasiperiodic orbit with hyperchaotic orbit at σl = 0.98 and (d) chaotic orbit with hyperchaotic orbit at σl = 0.92. Parameter settings are: τl = −0.72, δl = 0.65, τr = −0.76, σr = 0.615, δr = −1.2, μ = 0.35.
Fig. 15. (a) Shows bifurcation diagram. Lyapunov diagram with different initial conditions. Initial conditions are (b) (0.3, −0.5, −0.75 ) and (c) (−00.3, −0.5, −0.75 ). Parameter settings are same as in Fig. 14.
Fig. 17. (a) Mode-locked periodic orbit at τl = 0.65, (b) shows the two-dimensional unstable manifold (blue), saddle fixed point (green) and stable fixed points (red). Parameter settings: σl = −0.785, δl = 0.745, τr = 0.37, σr = 0.755, δr = −0.725, μ = 0.3. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 16. Shows phase space region showing σ l × τ r plane. Other parameters are same as in Fig. 14.
Fig. 14 shows the coexistence of various orbits with hyperchaotic orbit at different parameter values. At σl = 1.1, period-4 orbit coexists with hyperchaotic orbit. As the bifurcation parameter is changed to 1.05, the period-4 orbit is bifurcated to period-8 orbit and coexists with the hyperchaotic orbit. When the bifurcation parameter σ l is changed to 0.98, period-8 orbit bifurcates to 8-loop torus and coexists with hyperchaotic orbit. As the bifurcation parameter changes to 0.92, torus bifurcates to chaotic orbit and coexists with hyperchaotic orbit. Fig. 15 (a) shows the bifurcation diagram showing period increment with coexistence of hyperchaotic orbit. Fig. 15(b) and Fig. 15(c) show Lyapunov exponents for different initial conditions that confirms the coexistence of various orbits with hyperchaotic orbit. Fig. 16 shows phase space region showing σ l × τ r plane. 3.7. Mode-locked periodic orbit to hyperchaotic orbit via higher dimensional torus In this section, we show the occurrence of hyperchaotic orbit from a mode-locked periodic orbit via higher dimensional torus.
Fig. 18. (a) Higher dimensional torus at τl = 0.66. (b) shows the unstable manifold (blue), saddle fixed point (green) and saddle focus (red). Parameter settings are same as in Fig. 17. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 17(a) shows a mode-locked periodic orbit forming a Möbius strip structure at τl = 0.65. The unstable manifold for the saddle point is shown in Fig. 17(b). The Möbius strip structure is formed by period-12 stable fixed points and period-12 saddle fixed points and the unstable connection between them. Eigenvalues of period12 saddle fixed points are (2.8386, −0.8742, −0.0090 ) and eigenvalues of period-12 stable fixed points are λ1,2 = 0.5464 ± 0.8145i (modulus 0.9808), λ3 = −0.0238.
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Fig. 19. Hyperchaotic orbit at τl = 0.69. Other parameters are same as in Fig. 17.
As the parameter τ l is increased to 0.66 the complex conjugate eigenvalues of the stable fixed points go out of the unit circle through a Neimark-Sacker bifurcation and therefore the stable fixed points become saddle focus. Eigenvalues of period-12 saddle fixed point are (2.8783, −0.9003, −0.0086 ). Eigenvalues of period12 saddle focus are λ1,2 = 0.5429 ± 0.8401i (modulus 1.0 0 03), λ3 = −0.0229. We get invariant closed curves around each saddle-focus point and they lie on a Möbius strip structure. The orbit is shown in Fig. 18(a). The Möbius strip structure is therefore formed by the 12 invariant closed curves, period-12 saddle focus points, period12 saddle points and the two-dimensional unstable manifold between them and is shown in Fig. 18(b). The eigenvalues of the saddle fixed points follow the explanation given by [11]. As we increase the bifurcation parameter the higher dimensional torus bifurcates to a hyperchaotic orbit. At τl = 0.69, the orbit is shown in Fig. 19. The Lyapunov exponents are (0.02, 0.004, −0.34 ), the Kaplan-Yorke dimension is 2.08. Fig. 20 shows the highest two Lyapunov exponents showing the transition from a mode-locked periodic orbit to a hyperchaotic orbit via a higher dimensional orbit. Before the bifurcation point, both the highest Lyapunov exponent and second highest Lyapunov exponent are negative, and we get mode-locked periodic orbit lying on a Möbius strip structure. As the bifurcation parameter reaches the bifurcation point τl = 0.66, both the Lyapunov exponents become zero, we get a higher dimensional torus where invariant closed curves lie on a Möbius strip structure. As the bifurcation parameter crosses the bifurcation point two highest Lyapunov exponents become positive, therefore, the orbit becomes hyperchaotic. Transition from a mode-locked orbit to a hyperchaotic orbit is therefore demonstrated by a qualitative change in Lyapunov exponents. Fig. 20 (b) shows the existence region for hyperchaotic orbit (blue), chaotic orbit (green) and quasiperiodic orbit (black). Red star shows the parameter value for which the orbit in Fig. 18(a)
Fig. 21. Schematic diagram showing the structure of the stable and unstable manifolds and their intersection. Blue shows the 2D unstable manifold and red shows the 1d stable manifold. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
has been drawn. The white space shows the region for the periodic orbit. 4. Condition for homoclinic intersection between 2D unstable manifold and 1D stable manifold In all the examples shown in section-3 we show one common feature: the hyperchaotic orbit lies on a two-dimensional unstable manifold. In a discrete dynamical system, for hyperchaotic orbit, two Lyapunov exponents are greater than zero, the orbit has two diverging directions and one converging direction. Therefore in such cases saddle fixed points must have two eigenvalues whose absolute values are more than 1 and one eigenvalue with value less than 1. Thus we get two-dimensional unstable manifold and a one-dimensional stable manifold. Hyperchaotic orbit is born when a homoclinic intersection occurs between a 1D stable manifold and 2D unstable manifold (see Fig. 21). The unstable set folds at every intersection with z = 0 and every image of a fold is also a fold. Similarly, the stable set folds at every intersection with x = 0 and every pre-image of a fold is also a fold. Due to this phenomenon, in the system under consideration, the manifolds undergo numerous folds. That is why it is impossible to analytically obtain the condition of intersection of these two manifolds.
Fig. 20. (a) Lyapunov exponent diagram with bifurcation parameter τ l . Highest two Lyapunov exponents are shown in red and blue color. (b) Parameter space τ l × τ r . Hyperchaotic orbit (blue), chaotic orbit (green), quasiperiodic orbit (black), higher dimensional torus (red) are shown. Other parameters are same as in Fig. 17. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
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5. Conclusion
References
Three dimensional systems exhibit some phenomena which don’t occur in a two dimensional system such as higher dimensional torus (three frequency torus), hyperchaotic orbit. In this paper, we have shown the occurrence of higher dimensional torus and hyperchaotic orbit in three-dimensional piecewise linear maps. We show that hyperchaotic orbit can be born from a periodic orbit or from a quasiperiodic orbit in various ways- (a) a direct transition from a periodic orbit or a quasiperiodic orbit to a hyperchaotic orbit through border collision bifurcation, (b) a transition from a periodic orbit to a hyperchaotic orbit via quasiperiodic and chaotic orbit, (c) a transition from a mode-locked periodic orbit to hyperchaotic orbit through higher dimensional torus. The transition from a periodic orbit to a hyperchaotic orbit through quasiperiodic and chaotic orbit is possible in smooth systems as well as in nonsmooth systems. We have shown some bifurcations that do not occur in a smooth system. A direct transition from a periodic orbit to a hyperchaotic orbit, or from a quasiperiodic orbit to a hyperchaotic orbit is possible in nonsmooth systems, but it is not possible in smooth dynamical systems. We show bifurcations where a hyperchaotic orbit bifurcates to a different hyperchaotic orbit or a three-piece hyperchaotic orbit. Coexistence of hyperchaotic orbit with various orbits is also shown in this paper. Moreover, we numerically calculate the existence region of a hyperchaotic orbit and higher dimensional torus in the parameter space region.
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Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. CRediT authorship contribution statement Mahashweta Patra: Methodology, Software, Validation, Formal analysis, Investigation, Resources, Data curation, Writing - original draft, Writing - review & editing, Visualization. Soumitro Banerjee: Conceptualization, Methodology, Validation, Formal analysis, Investigation, Resources, Writing - review & editing, Supervision, Project administration, Funding acquisition.