- Email: [email protected]
Elementary chaotic snap flows
Chaos, Solitons & Fractals 44 (2011) 995–1003
Contents lists available at SciVerse ScienceDirect
Chaos, Solitons & Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos
Elementary chaotic snap flows Buncha Munmuangsaen, Banlue Srisuchinwong ⇑ Sirindhorn International Institute of Technology (SIIT), Thammasat University, 131 M5, Tivanont Road, Bangkadi, Muang, Pathum-Thani 12000, Thailand
a r t i c l e
i n f o
Article history: Received 15 October 2010 Accepted 22 August 2011 Available online 25 September 2011
a b s t r a c t ...:
...
_ xÞ have been referred Hyperjerk systems with 4th-order derivative of the form x ¼ f ðx; € x; x; to as snap systems. Five new elementary chaotic snap flows and a generalization of an existing flow are presented through an extensive numerical search. Four of these flows demonstrate elegant simplicity of a single control parameter based on a single nonlinearity of a quadratic, a piecewise-linear or an exponential type. Two others demonstrate elegant simplicity of all unity-in-magnitude parameters based on either a single cubic nonlinearity or three cubic nonlinearities. The chaotic snap flow with a single cubic nonlinearity requires only two terms and can be transformed to its equivalent dynamical form of only five terms which have a single nonlinearity. An advantage is that such a chaotic flow offers only five terms even though the (four) dimension is high. Three of the chaotic snap flows are characterized as conservative systems whilst three others are dissipative systems. Basic dynamical properties are described. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction Chaotic flows can be characterized as either dissipative or conservative [1]. A dissipative chaotic flow ~ F 1 has a property that (bounded) trajectories are attracted to a region of space by a strange attractor and its divergence is negative, i.e. a volume (V) occupied r ~ F 1 ¼ V 1 dV dt by a set of initial conditions collapses in state space. By contrast, a conservative (or equivalently Hamiltonian [2]) chaotic flow ~ F 2does not have phase space attractors and is zero, i.e. a volume (V) its divergence r ~ F 2 ¼ V 1 dV dt in state space occupied by a set of initial conditions remains the same. Early examples [3] of both dissipative and conservative chaotic flows were identified, through a computer search, to be algebraically simpler than the well-known Lorenz [4] and Rössler systems [5]. In response to this early work, Gottlieb [6] suggested a search for chaotic systems of the ... _ xÞ where j is a jerk function and x_ dx form x ¼ jð€ x; x; . The dt term ‘jerk’ comes from the fact that, successive time ⇑ Corresponding author. Tel.: +66 2 501 3505 till 20x1806; fax: +66 2 501 3505 till 20x1801. E-mail address: [email protected] (B. Srisuchinwong). 0960-0779/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2011.08.008
derivatives of displacement are velocity, acceleration, and jerk [7]. Sprott [8] took up Gottlieb’s challenge and discovered the simplest dissipative chaotic flow using a single quadratic nonlinearity in a jerk model of the form ...
x ¼ a€x þ x_ 2 x:
ð1Þ
Chaos is found as ‘a’ is equal to or slightly greater than 2.017. Another simple piecewise exponential nonlinear jerk model [9] in a dissipative system has been of the form ...
_ b x; x ¼ a€x þ jxj
ð2Þ
where ‘a’ and ‘b’ are bifurcation parameters which cannot be unity. In addition, the simplest conservative jerk system with chaotic solutions [10] has been appeared as ...
x ¼ 8€x þ jxj 1:
ð3Þ
On the other hand, hyperjerk systems [11] involve time derivatives of a jerk function. Such derivatives are generally higher than the 3rd-order. In particular, hyperjerk sys...: ... _ xÞ tems with 4th-order derivative of the form x ¼ f ðx; € x; x; have been referred to as snap systems [10]. Early examples of Chaotic Hyperjerks have been demonstrated via an extensive numerical search [11,12]. Two cases of chaotic snap systems have been reported as the algebraically
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simplest Chaotic Hyperjerks. The first case has five terms of the form ...:
...
x ¼ x A€x x_ þ x2 1
ð4Þ
an advantage is that such a chaotic snap flow also offers five (perhaps minimum) terms even though the (four) dimension is higher. Three of all examples are conservative systems and three others are dissipative systems.
ð5Þ
2. A generalization of an existing conservative system based on a single quadratic nonlinearity
and the second case has four terms of the form ...:
...
x ¼ x A€x x_ 2 x:
Both cases have been characterized as the dissipative systems. Very recently, simpler chaotic snap systems of both dissipative and conservative flows have been summarized [10] including a chaotic snap hyperjerk of a conservative flow with three terms and a single quadratic nonlinearity of the form ...:
x ¼ 8€x þ x_ 2 x:
ð6Þ
In this paper, five new examples of elementary chaotic snap flows and a generalization of an existing chaotic snap flow are presented through an extensive numerical search using an equation of the form ...:
...
_ €xÞ; x þa0 x þa1 €x þ a2 x_ þ a3 x ¼ g ðx; x;
ð7Þ
_ € where gðx; x; xÞ is the nonlinearity and coefficients a0, . . . , a3 are control parameters. Chaotic solutions for each of a _ € specified nonlinear function gðx; x; xÞ employ systematic procedures of a numerical search developed in [1– 3,10,12]. In such procedures, the control parameters and initial conditions are scanned to find for a positive Lyapunov exponent, which is a signature of chaos. Although several chaotic snap flows have been found, only the systems with elegant simplicity are selected and reported here. In cases of a candidate for four-dimensional dissipative chaotic snap flows, the Lyapunov exponents are calculated with 107 iterations (to ensure that chaos is neither numerical artifacts nor chaotic transients) using the method described by Wolf et al. [13] with a fixed step size Dt = 0.01. In cases of a candidate for four-dimensional conservative chaotic snap flows, the largest Lyapunov exponent k1 is calculated using the method described in [10] with a 4th-order adaptive-step Runge–Kutta integrator [14], which avoids numerical trajectory divergences observed by the fixed-step integrator in association with occasional large excursions from the chaotic!attractors [15]. In conservative flows (the divergence r F 2 ¼ V 1 dV dt is zero), Lyapunov exponents occur symmetrically in equal pairs with opposite signs and there must be two zero exponents (k2 = k3 = 0) (see Section 1.10 of [10]). Therefore the remaining exponent k4 is equal to the negative value of the largest Lyapunov exponent, i.e. k4 = k1. Four of these chaotic snap flows demonstrate elegant simplicity of a single control parameter based on a single nonlinearity of a quadratic, a piecewise-linear or an exponential type. Two other chaotic snap flows demonstrate elegant simplicity of all unity-in-magnitude parameters based on either a single cubic nonlinearity or three cubic nonlinearities. The chaotic snap flow with a single cubic nonlinearity requires only two terms and can be transformed to its equivalent dynamical form of only five terms which have a single nonlinearity. In comparison with an existing three (lower) dimensional chaotic flow where minimum terms have been identified to be five terms [8],
Through an extensive search, an elementary chaotic snap flow of a conservative system based on a single quadratic nonlinearity has been found with elegant simplicity of a single control parameter ‘A’ as follows: ...:
x ¼ A€x x_ 2 x; _2
ð8Þ
where x is the single quadratic nonlinearity. Although Eq. (8) nearly resembles Eq. (6) except that ‘A’ is specifically equal to 8 with x_ 2 as a quadratic nonlinearity, this section further examines in greater detail for new cases where ‘A’ is generally a single control parameter of various values, with x_ 2 as a single quadratic nonlinearity. Fig. 1 shows the largest Lyapunov exponent k1 [2,10] of Eq. (8) versus the control parameter A using the initial con... _ € ditions ðx; x; x; xÞ ¼ ð1:2; 1:2; 0:1; 0:5Þ. Chaos will be assumed to exist if k1 exceeds 0.001. The value k1 will be set to zero if chaos does not exist or when the trajectory of the solution escapes to a very large value (or infinity). Such an escape will be detected if the calculated value of ... _ þ j€ jxj þ jxj xj þ j x j exceeds 1 106. At A = 4.3765, the value of k1 = 0.03574 was found to be a maximum. As a result, the Lyapunov exponent spectrum of Eq. (8) was found to be (k1, k2, k3, k4) = (0.03574, 0, 0, 0.03574). For the selected initial conditions (1.2, 1.2, 0.1, 0.5), chaos does not exist after A > 4.51 as evidenced in Fig. 1. At A = 8, k1 = 0.0005 (<0.001) and therefore chaos does not exist. Nonetheless, if the set of the selected initial condition is (1.94, 1.43, 0.98, 1.48), then k1 = 0.0055 (>0.001) and therefore chaos does exist at A = 8 as reported in [10]. Equivalently, Eq. (8) can be rewritten as a dynamical system in variables x, y, z, and w of the form
Fig. 1. The largest Lyapunov exponent of Eq. (8) versus A.
B. Munmuangsaen, B. Srisuchinwong / Chaos, Solitons & Fractals 44 (2011) 995–1003
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Fig. 2. A phase-space trajectory for A = 4.3765 of Eq. (8).
!
x_ ¼ y y_ ¼ z z_ ¼ w
ð9Þ
_ ¼ Az y2 x: w Eq. (9) requires only six terms with elegant simplicity of a single quadratic nonlinearity and a single control parameter ‘A’. Eq. (9) has a single equilibrium point at the origin. For the positive sign of the quadratic nonlinearity (+y2), for example, the corresponding Jacobian matrix is
2
0 6 0 6 J¼6 4 0
1 0
0 1
0
0
3 0 07 7 7: 15
ð10Þ
1 2y A 0 At the equilibrium point and for A = 4.3765, the Jacobian matrix (10) has two complex conjugate pairs of eigenvalues at ±2.0334i and ±0.4918i. It has been known [1] that the total sum of all Lyapunov exponents represents an average rate of expansion (along the trajectory) of a small volume (V) in state space occupied by a set of initial conditions and can directly be calculated either from the trace (sum of diagonal elements) of the Jacobian matrix J, or from the divergence of the flow
_ T . For bounded trajectories, the total sum canF ¼ ½x_ y_ z_ w not be positive [1]. If the total sum is negative, the system is dissipative. If the total sum is zero, there is no contraction (no dissipation) and the system is conservative. Existing ‘Chaotic Hyperjerk’ and ‘Hyperchaotic Hyperjerk’ of 4th-order in dissipative flows have the Lyapunov exponent spectrum of the form (+, 0, , ) and (+, +, 0, ), respectively [11]. As mentioned earlier, the Lyapunov exponent spectrum of Eq. (8) is (k1, k2, k3, k4) = (0.03574, 0, 0, 0.03574) and is of the form (+, 0, 0, ) where k1 and k4 occur in an equal pair with opposite sign, i.e. k1 = k4. The total sum of all Lyapunov exponents of Eq. (8) apparently equals zero. As a result, the divergence of the flow is equally zero as follows: !
rF ¼
_ 1 dV @ x_ @ y_ @ z_ @ w ¼ þ þ þ V dt @x @y @z @w
¼ k1 þ k2 þ k3 þ k4 ¼ htraceðJÞi ¼ 0:
ð11Þ
As the trace(J) of Eq. (10) is independent of ‘A’, it follows that Eq. (8) is a conservative system with a single control parameter ‘A’ of various values for a single quadratic nonlinearity x_ 2 , and is a ‘‘Chaotic Hyperjerk’’ system, or equivalently a chaotic snap flow. As an example, the phase space trajectory for A = 4.3765 is shown in Fig. 2. A numerical simulation
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reveals that Eq. (8) also exhibits time-reversible Hamiltonian chaos [2]. 3. A single piecewise-linear nonlinearity 3.1. A conservative system It is natural to wonder whether the quadratic nonlinearity x_ 2 of Eq. (8) can be weakened and can be replaced with an elementary piecewise-linear nonlinearity. Note that a term p can be considered as a piecewise-linear approximation to p2. It has been proven that such a replacement does not enable chaos in the three-dimensional jerk model of Eq. (1) [16]. In this section, another elementary chaotic snap flow of a conservative system is presented with elegant simplicity of a single piecewise-linear nonlinearity and a single control parameter ‘A’ as follows: ...:
_ x: x ¼ A€x jxj
ð12Þ
Chaos was indeed found. Fig. 3 shows the largest Lyapunov exponent (k1) of Eq. (12) versus the parameter A from 2 to ... _ € 5 using the initial conditions ðx; x; x; xÞ ¼ ð0:4; 0:3; 0:6; 0:2Þ. The calculation was in the same manner as described in Section 2. At A = 2.525, the value k1 = 0.05537 was found
Fig. 3. The largest Lyapunov exponent of Eq. (12) versus A.
to be a maximum. As a result, the Lyapunov exponent spectrum was found to be (k1, k2, k3, k4) = (0.05537, 0, 0, 0.05537). Equivalently, Eq. (12) can be rewritten as a dynamical system in variables x, y, z, and w in a similar manner to Eq. (9) except that ±y2 in Eq. (9) is replaced by ±jyj. For the positive sign of the elementary piecewise-linear
Fig. 4. A phase space trajectory for A = 2.525 of Eq. (12).
B. Munmuangsaen, B. Srisuchinwong / Chaos, Solitons & Fractals 44 (2011) 995–1003
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nonlinearity, +jyj, for example, the corresponding Jacobian matrix is
2
0
1
0
0
3
6 0 0 1 07 6 7 J¼6 7; 4 0 0 0 15 1 sgnðyÞ A 0
ð13Þ
where the signum function sgn(y) is defined as:
8 > < 1 for y < 0 sgnðyÞ ¼ 0 for y ¼ 0 > : 1 for y > 0
ð14Þ
Eq. (12) has a single equilibrium point at the origin. At the equilibrium point and for A = 2.525, the Jacobian matrix (13) has two complex conjugate pairs of eigenvalues at ±1.4259i and ±0.7013i. In a similar manner to Eq. (11), the total sum of all Lyapunov exponents of Eq. (12) equals zero and the trace(J) of Eq. (13) is zero and independent of ‘A’. It follows that Eq. (12) is a chaotic snap flow of a conservative system with a single control parameter ‘A’ of various values for a _ As an example, a single piecewise-linear nonlinearity jxj. phase space trajectory for A = 2.525 is shown in Fig. 4. A numerical simulation reveals that Eq. (12) also exhibits time-reversible Hamiltonian chaos [2].
Fig. 5. (a) The largest Lyapunov exponent of Eq. (15) versus A. (b) A bifurcation diagram of Eq. (15) shows a period-doubling route to chaos as A is gradually decreased from 2 to 1.878.
3.2. A dissipative system An elementary chaotic snap flow of a dissipative system is presented with elegant simplicity of a single piecewise-linear nonlinearity and a single control parameter ‘A’ as follows: ...:
...
x ¼ x A€x x_ ðjxj 1Þ:
Fig. 6. A chaotic attractor of Eq. (15).
ð15Þ
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Fig. 7. A phase space trajectory of Eq. (16).
The four-dimensional snap model of Eq. (15) is an extension of an existing three-dimensional jerk model with a piecewise-linear nonlinearity described in [17]. Such a nonlinearity can be implemented electronically with a full-wave rectifier with two diodes and an inverting unity gain amplifier and also with a single diode [18]. Fig. 5(a) illustrates the largest Lyapunov exponent of Eq. (15) versus the control parameter A which is gradually decreased from 2 to 1.6 using the initial conditions ... x ¼ x_ ¼ € x ¼ x ¼ 0: A bifurcation diagram of Eq. (15) for xmax is depicted in Fig. 5(b) as a function of the control parameter A from 2 to 1.6. A period-doubling route to chaos is evident as A is gradually decreased from 2 to 1.878. For positive and negative signs of jxj, the bifurcation and Lyapunov exponents of this system are the same. The only different is that both attractors are 180-degree rotated images to each other. Comparison between Fig. 5(a) and (b) verifies that chaos does exist. At A = 1.686, chaos is maximized with the Lyapunov exponent spectrum (k1, k2, k3, k4) = (0.11107, 0, 0.43926, 0.67184) and the Kaplan–Yorke dimension [19] (or equivalently Lyapunov dimension) Dky = 2.25293. The total sum of the Lyapunov exponent is equivalent to the average rate of fractional volume expansion along the trajectory, and can be calculated in the same manner to that of the conservative systems described by Eq. (11) [1,2]. Since
the total sum of the Lyapunov exponent spectrum or trace(J) of Eq. (15) is a negative value, i.e. trace(J) = 1, Eq. (15) is dissipative with constant state-space contraction of 1. The fluctuations in the largest Lyapunov exponent of this system lie in a small range of 104 for 106 iterations. Such a slight variation in the Lyapunov spectrum, as iteration increases, helps to ensure that the Lyapunov exponent has converged (see Section 4 of [20] for more information). Eq. (15) has two equilibrium points at (±1, 0, 0, 0). At the first equilibrium point (1, 0, 0, 0), the Jacobian matrix has the eigenvalues at 0.6255 ± 0.7802i and 0.1255 ± 0.9921i. At the second equilibrium point (1, 0, 0, 0), the Jacobian matrix has the eigenvalues at 0.1942 ± 1.3861i, 1.0830 and 0.4713. Fig. 6 shows a chaotic attractor of Eq. (15) which resembles the Rössler attractor [5] in a sense that it has a single foldedband structure. 4. Cubic nonlinearity 4.1. A conservative system An elementary chaotic snap flow of a conservative system is presented with elegant simplicity of only two terms and a single cubic nonlinearity of the form ...:
x ¼ €x3 x:
ð16Þ
B. Munmuangsaen, B. Srisuchinwong / Chaos, Solitons & Fractals 44 (2011) 995–1003
1001
Fig. 8. Poincaré sections of Eq. (18).
In addition, all coefficients of Eq. (16) are unity in magnitude without a need for additional parameters on the right hand ... side. By using the initial conditions x ¼ x_ ¼ € x ¼ x ¼ 0:1, the Lyapunov exponent spectrum (k1, k2, k3, k4) = (0.07681, 0, 0, 0.07681) was calculated using the method described in Section 1. The trajectory of Eq. (16) is shown in Fig. 7. Equivalently, Eq. (16) can be rewritten as a dynamical system in variables x, y, z, and w of the form
x_ ¼ y; y_ ¼ z; z_ ¼ w;
ð17Þ
_ ¼ x z3 : w Eq. (17) requires only five terms which have a single nonlinearity. In comparison with an existing three (lower) dimensional chaotic flow where minimum terms have been reported to be five terms [8], an advantage is that such a chaotic snap flow also offers five (perhaps minimum) terms even though the four dimension is higher. Eq. (17) has a single equilibrium point at the origin. At the equilibrium point, pffiffiffi pffiffiffi the Jacobian matrix has eigenvalues at ð1= 2Þ ð1= 2Þi. In a similar manner to Eq. (11), the total sum of all Lyapunov exponents of Eq. (16) is zero and the trace of the Jacobian matrix of Eq. (17) equals zeros. It follows that Eq. (16) is
Fig. 9. The largest Lyapunov exponent of Eq. (19) versus A.
conservative. A numerical simulation reveals that Eq. (16) also exhibits time-reversible Hamiltonian chaos [2]. 4.2. A dissipative system A chaotic snap flow of a dissipative system based on three cubic nonlinearities has been found where all coefficients are unity in magnitude as follows:
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B. Munmuangsaen, B. Srisuchinwong / Chaos, Solitons & Fractals 44 (2011) 995–1003
Fig. 10. Poincaré sections of Eq. (19). ...:
...
€3
_3
3
x ¼ x €x x x x :
ð18Þ ...
_ € By using the initial conditions ðx; x; x; xÞ ¼ ð8:5; 2:3; 5:6; 0:1Þ, the Lyapunov exponent spectrum is (k1, k2, k3, k4) = (0.05351, 0, 0.20296, 0.85054), resulting in the Kaplan–Yorke dimension (or Lyapunov dimension) Dky = 2.26364. Since the total sum of the Lyapunov exponent spectrum and the trace(J) of Eq. (18) equal a negative value, i.e. trace(J) = 1, Eq. (18) is dissipative with constant state-space contraction of 1. Eq. (18) has a single equilibrium point at the origin. At the equilibrium point, the Jacobian matrix has eigenvalues at 0, 0, 0.5 ± 0.866i. Fig. 8 displays the Poincaré sections of Eq. (18) with fractal structures resulting from the fractal dimension Dky = 2.26364. 5. A single exponential nonlinearity for a dissipative system It is also interesting to identify an elementary chaotic snap flow with elegant simplicity of a single exponential nonlinearity and a single control parameter. One example is ...:
...
x ¼ x Ax_ x expð€xÞ:
ð19Þ
Fig. 9 shows the largest Lyapunov exponent versus the con... trol parameter A using initial conditions x ¼ x_ ¼ € x ¼ x ¼ 0. It
is apparent that chaos is maximized at A 6. The Lyapunov exponent spectrum at A = 6 is (k1, k2, k3, k4) = (0.16788, 0, 0.15074, 1.01714) resulting in a Kaplan–Yorke dimension (or Lyapunov dimension) Dky = 3.01685. Eq. (19) has a single equilibrium point at (1, 0, 0, 0). The eigenvalues at the equilibrium point for A = 6 are 1.9408, 0.5558 ± 1.6453i and 0.1708. Since the total sum of the Lyapunov exponent spectrum and the trace(J) of Eq. (19) equal a negative value, i.e. trace(J) = 1, Eq. (19) is dissipative with constant state-space contraction of 1. The Poincaré sections in several planes for A = 6 are shown in Fig. 10. No conservative cases have been found for such an exponential nonlinearity. 6. Conclusions Elementary chaotic snap flows have been presented through an extensive numerical search. They consist of five new examples and a generalization of an existing flow. Four of them have demonstrated elegant simplicity of a single control parameter based on a single nonlinearity of a quadratic, a piecewise-linear or an exponential type. Two others have demonstrated elegant simplicity of all unity-in-magnitude parameters based on either a single cubic nonlinearity or three cubic nonlinearities. The algebraically simplest example found in this study has been
B. Munmuangsaen, B. Srisuchinwong / Chaos, Solitons & Fractals 44 (2011) 995–1003
the chaotic snap flow with a single cubic nonlinearity and all unity-in-magnitude parameters. Although the 4th-order is higher, the equation requires only two terms in jerk representation, and can be transformed to only five terms in its equivalent dynamical form. Such five terms offer algebraically minimum terms comparable to those of existing lower-order chaotic flows. The dissipative chaotic snap flow with a piecewise-linear nonlinearity suggests a possible implementation of electronic circuits using a full-wave rectifier with diodes and an inverting unity-gain amplifier. Basic dynamical properties have been demonstrated in three conservative and three dissipative chaotic flows. Acknowledgements The authors are grateful to Prof. J.C. Sprott for his useful comments. The first author appreciates CIC-Common Fund of SIIT for partially financial support. This work was supported by Telecommunications Research and Industrial Development Institute (TRIDI), NBTC, Thailand, and the National Research University Project of Thailand, Office of Higher Education Commission. References [1] Sprott JC. Some simple chaotic jerk functions. Am J Phys 1997;65:537–43. [2] Sprott JC. Chaos and time-series analysis. Oxford: Oxford University Press; 2003. [3] Sprott JC. Some simple chaotic flows. Phys Rev E 1994;50:R647–50.
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[4] Lorenz EN. Deterministic nonperiodic flow. J Atmos Sci 1963;20:130–41. [5] Rössler OE. An equation for continuous chaos. Phys Lett A 1976;57:397–8. [6] Gottlieb WHP. What is the simplest Jerk function that gives chaos? Am J Phys 1996;64:525. [7] Schot SH. Jerk: the time rate of change of acceleration. Am J Phys 1978;65:1090–4. [8] Sprott JC. Simplest dissipative chaotic flow. Phys Lett A 1997;228:271–4. [9] Sun KH, Sprott JC. A simple Jerk system with piecewise exponential nonlinearity. Int J Nonlinear Sci Numer Simul 2009;10:1443–50. [10] Sprott JC. Elegant chaos: algebraically simple chaotic flows. Singapore: World Scientific; 2010. [11] Chlouverakis KE, Sprott JC. Chaotic hyperjerk systems. Chaos, Solitons Fract 2006;28:739–46. [12] Munmuangsaen B, Srisuchinwong B, Sprott JC. Generalization of the simplest autonomous chaotic system. Phys Lett A 2011;375: 1445–50. [13] Wolf A, Swift JB, Swinney HL, Vastano JA. Determining Lyapunov exponents from a time series. Physica D 1985;16:285–317. [14] Press WH, Teukolsky SA, Vetterling WP, Flannery B. Numerical recipes in C: the art of scientific computing. 2nd ed. Cambridge: Cambridge University Press; 1992. [15] Marshall D, Sprott JC. Simple driven chaotic oscillators with complex variables. Chaos 2009;19. pp. 013124-1–7. [16] Sprott JC, Linz SJ. Algebraically simple chaotic flows? Int J Chaos Theory Appl 2000;5:3–22. [17] Linz SJ, Sprott JC. Elementary chaotic flow. Phys Lett A 1999;259: 240–5. [18] Sprott JC. Simple chaotic system and circuits. Am J Phys 2000;68:758–63. [19] Kaplan J, Yorke J. Chaotic behavior of multidimensional difference equations. In: Peitgen H-O, Walther H-O, editors. Functional differential equations and approximations of fixed points. Lecture notes in mathematics, vol. 730. Berlin: Springer; 1979. p. 228–37. [20] Albers DJ, Sprott JC. Structural stability and hyperbolicity violation in high-dimensional dynamical systems. Nonlinearity 2006;19: 1801–47.
Contents lists available at SciVerse ScienceDirect
Chaos, Solitons & Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos
Elementary chaotic snap flows Buncha Munmuangsaen, Banlue Srisuchinwong ⇑ Sirindhorn International Institute of Technology (SIIT), Thammasat University, 131 M5, Tivanont Road, Bangkadi, Muang, Pathum-Thani 12000, Thailand
a r t i c l e
i n f o
Article history: Received 15 October 2010 Accepted 22 August 2011 Available online 25 September 2011
a b s t r a c t ...:
...
_ xÞ have been referred Hyperjerk systems with 4th-order derivative of the form x ¼ f ðx; € x; x; to as snap systems. Five new elementary chaotic snap flows and a generalization of an existing flow are presented through an extensive numerical search. Four of these flows demonstrate elegant simplicity of a single control parameter based on a single nonlinearity of a quadratic, a piecewise-linear or an exponential type. Two others demonstrate elegant simplicity of all unity-in-magnitude parameters based on either a single cubic nonlinearity or three cubic nonlinearities. The chaotic snap flow with a single cubic nonlinearity requires only two terms and can be transformed to its equivalent dynamical form of only five terms which have a single nonlinearity. An advantage is that such a chaotic flow offers only five terms even though the (four) dimension is high. Three of the chaotic snap flows are characterized as conservative systems whilst three others are dissipative systems. Basic dynamical properties are described. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction Chaotic flows can be characterized as either dissipative or conservative [1]. A dissipative chaotic flow ~ F 1 has a property that (bounded) trajectories are attracted to a region of space by a strange attractor and its divergence is negative, i.e. a volume (V) occupied r ~ F 1 ¼ V 1 dV dt by a set of initial conditions collapses in state space. By contrast, a conservative (or equivalently Hamiltonian [2]) chaotic flow ~ F 2does not have phase space attractors and is zero, i.e. a volume (V) its divergence r ~ F 2 ¼ V 1 dV dt in state space occupied by a set of initial conditions remains the same. Early examples [3] of both dissipative and conservative chaotic flows were identified, through a computer search, to be algebraically simpler than the well-known Lorenz [4] and Rössler systems [5]. In response to this early work, Gottlieb [6] suggested a search for chaotic systems of the ... _ xÞ where j is a jerk function and x_ dx form x ¼ jð€ x; x; . The dt term ‘jerk’ comes from the fact that, successive time ⇑ Corresponding author. Tel.: +66 2 501 3505 till 20x1806; fax: +66 2 501 3505 till 20x1801. E-mail address: [email protected] (B. Srisuchinwong). 0960-0779/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2011.08.008
derivatives of displacement are velocity, acceleration, and jerk [7]. Sprott [8] took up Gottlieb’s challenge and discovered the simplest dissipative chaotic flow using a single quadratic nonlinearity in a jerk model of the form ...
x ¼ a€x þ x_ 2 x:
ð1Þ
Chaos is found as ‘a’ is equal to or slightly greater than 2.017. Another simple piecewise exponential nonlinear jerk model [9] in a dissipative system has been of the form ...
_ b x; x ¼ a€x þ jxj
ð2Þ
where ‘a’ and ‘b’ are bifurcation parameters which cannot be unity. In addition, the simplest conservative jerk system with chaotic solutions [10] has been appeared as ...
x ¼ 8€x þ jxj 1:
ð3Þ
On the other hand, hyperjerk systems [11] involve time derivatives of a jerk function. Such derivatives are generally higher than the 3rd-order. In particular, hyperjerk sys...: ... _ xÞ tems with 4th-order derivative of the form x ¼ f ðx; € x; x; have been referred to as snap systems [10]. Early examples of Chaotic Hyperjerks have been demonstrated via an extensive numerical search [11,12]. Two cases of chaotic snap systems have been reported as the algebraically
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B. Munmuangsaen, B. Srisuchinwong / Chaos, Solitons & Fractals 44 (2011) 995–1003
simplest Chaotic Hyperjerks. The first case has five terms of the form ...:
...
x ¼ x A€x x_ þ x2 1
ð4Þ
an advantage is that such a chaotic snap flow also offers five (perhaps minimum) terms even though the (four) dimension is higher. Three of all examples are conservative systems and three others are dissipative systems.
ð5Þ
2. A generalization of an existing conservative system based on a single quadratic nonlinearity
and the second case has four terms of the form ...:
...
x ¼ x A€x x_ 2 x:
Both cases have been characterized as the dissipative systems. Very recently, simpler chaotic snap systems of both dissipative and conservative flows have been summarized [10] including a chaotic snap hyperjerk of a conservative flow with three terms and a single quadratic nonlinearity of the form ...:
x ¼ 8€x þ x_ 2 x:
ð6Þ
In this paper, five new examples of elementary chaotic snap flows and a generalization of an existing chaotic snap flow are presented through an extensive numerical search using an equation of the form ...:
...
_ €xÞ; x þa0 x þa1 €x þ a2 x_ þ a3 x ¼ g ðx; x;
ð7Þ
_ € where gðx; x; xÞ is the nonlinearity and coefficients a0, . . . , a3 are control parameters. Chaotic solutions for each of a _ € specified nonlinear function gðx; x; xÞ employ systematic procedures of a numerical search developed in [1– 3,10,12]. In such procedures, the control parameters and initial conditions are scanned to find for a positive Lyapunov exponent, which is a signature of chaos. Although several chaotic snap flows have been found, only the systems with elegant simplicity are selected and reported here. In cases of a candidate for four-dimensional dissipative chaotic snap flows, the Lyapunov exponents are calculated with 107 iterations (to ensure that chaos is neither numerical artifacts nor chaotic transients) using the method described by Wolf et al. [13] with a fixed step size Dt = 0.01. In cases of a candidate for four-dimensional conservative chaotic snap flows, the largest Lyapunov exponent k1 is calculated using the method described in [10] with a 4th-order adaptive-step Runge–Kutta integrator [14], which avoids numerical trajectory divergences observed by the fixed-step integrator in association with occasional large excursions from the chaotic!attractors [15]. In conservative flows (the divergence r F 2 ¼ V 1 dV dt is zero), Lyapunov exponents occur symmetrically in equal pairs with opposite signs and there must be two zero exponents (k2 = k3 = 0) (see Section 1.10 of [10]). Therefore the remaining exponent k4 is equal to the negative value of the largest Lyapunov exponent, i.e. k4 = k1. Four of these chaotic snap flows demonstrate elegant simplicity of a single control parameter based on a single nonlinearity of a quadratic, a piecewise-linear or an exponential type. Two other chaotic snap flows demonstrate elegant simplicity of all unity-in-magnitude parameters based on either a single cubic nonlinearity or three cubic nonlinearities. The chaotic snap flow with a single cubic nonlinearity requires only two terms and can be transformed to its equivalent dynamical form of only five terms which have a single nonlinearity. In comparison with an existing three (lower) dimensional chaotic flow where minimum terms have been identified to be five terms [8],
Through an extensive search, an elementary chaotic snap flow of a conservative system based on a single quadratic nonlinearity has been found with elegant simplicity of a single control parameter ‘A’ as follows: ...:
x ¼ A€x x_ 2 x; _2
ð8Þ
where x is the single quadratic nonlinearity. Although Eq. (8) nearly resembles Eq. (6) except that ‘A’ is specifically equal to 8 with x_ 2 as a quadratic nonlinearity, this section further examines in greater detail for new cases where ‘A’ is generally a single control parameter of various values, with x_ 2 as a single quadratic nonlinearity. Fig. 1 shows the largest Lyapunov exponent k1 [2,10] of Eq. (8) versus the control parameter A using the initial con... _ € ditions ðx; x; x; xÞ ¼ ð1:2; 1:2; 0:1; 0:5Þ. Chaos will be assumed to exist if k1 exceeds 0.001. The value k1 will be set to zero if chaos does not exist or when the trajectory of the solution escapes to a very large value (or infinity). Such an escape will be detected if the calculated value of ... _ þ j€ jxj þ jxj xj þ j x j exceeds 1 106. At A = 4.3765, the value of k1 = 0.03574 was found to be a maximum. As a result, the Lyapunov exponent spectrum of Eq. (8) was found to be (k1, k2, k3, k4) = (0.03574, 0, 0, 0.03574). For the selected initial conditions (1.2, 1.2, 0.1, 0.5), chaos does not exist after A > 4.51 as evidenced in Fig. 1. At A = 8, k1 = 0.0005 (<0.001) and therefore chaos does not exist. Nonetheless, if the set of the selected initial condition is (1.94, 1.43, 0.98, 1.48), then k1 = 0.0055 (>0.001) and therefore chaos does exist at A = 8 as reported in [10]. Equivalently, Eq. (8) can be rewritten as a dynamical system in variables x, y, z, and w of the form
Fig. 1. The largest Lyapunov exponent of Eq. (8) versus A.
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Fig. 2. A phase-space trajectory for A = 4.3765 of Eq. (8).
!
x_ ¼ y y_ ¼ z z_ ¼ w
ð9Þ
_ ¼ Az y2 x: w Eq. (9) requires only six terms with elegant simplicity of a single quadratic nonlinearity and a single control parameter ‘A’. Eq. (9) has a single equilibrium point at the origin. For the positive sign of the quadratic nonlinearity (+y2), for example, the corresponding Jacobian matrix is
2
0 6 0 6 J¼6 4 0
1 0
0 1
0
0
3 0 07 7 7: 15
ð10Þ
1 2y A 0 At the equilibrium point and for A = 4.3765, the Jacobian matrix (10) has two complex conjugate pairs of eigenvalues at ±2.0334i and ±0.4918i. It has been known [1] that the total sum of all Lyapunov exponents represents an average rate of expansion (along the trajectory) of a small volume (V) in state space occupied by a set of initial conditions and can directly be calculated either from the trace (sum of diagonal elements) of the Jacobian matrix J, or from the divergence of the flow
_ T . For bounded trajectories, the total sum canF ¼ ½x_ y_ z_ w not be positive [1]. If the total sum is negative, the system is dissipative. If the total sum is zero, there is no contraction (no dissipation) and the system is conservative. Existing ‘Chaotic Hyperjerk’ and ‘Hyperchaotic Hyperjerk’ of 4th-order in dissipative flows have the Lyapunov exponent spectrum of the form (+, 0, , ) and (+, +, 0, ), respectively [11]. As mentioned earlier, the Lyapunov exponent spectrum of Eq. (8) is (k1, k2, k3, k4) = (0.03574, 0, 0, 0.03574) and is of the form (+, 0, 0, ) where k1 and k4 occur in an equal pair with opposite sign, i.e. k1 = k4. The total sum of all Lyapunov exponents of Eq. (8) apparently equals zero. As a result, the divergence of the flow is equally zero as follows: !
rF ¼
_ 1 dV @ x_ @ y_ @ z_ @ w ¼ þ þ þ V dt @x @y @z @w
¼ k1 þ k2 þ k3 þ k4 ¼ htraceðJÞi ¼ 0:
ð11Þ
As the trace(J) of Eq. (10) is independent of ‘A’, it follows that Eq. (8) is a conservative system with a single control parameter ‘A’ of various values for a single quadratic nonlinearity x_ 2 , and is a ‘‘Chaotic Hyperjerk’’ system, or equivalently a chaotic snap flow. As an example, the phase space trajectory for A = 4.3765 is shown in Fig. 2. A numerical simulation
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reveals that Eq. (8) also exhibits time-reversible Hamiltonian chaos [2]. 3. A single piecewise-linear nonlinearity 3.1. A conservative system It is natural to wonder whether the quadratic nonlinearity x_ 2 of Eq. (8) can be weakened and can be replaced with an elementary piecewise-linear nonlinearity. Note that a term p can be considered as a piecewise-linear approximation to p2. It has been proven that such a replacement does not enable chaos in the three-dimensional jerk model of Eq. (1) [16]. In this section, another elementary chaotic snap flow of a conservative system is presented with elegant simplicity of a single piecewise-linear nonlinearity and a single control parameter ‘A’ as follows: ...:
_ x: x ¼ A€x jxj
ð12Þ
Chaos was indeed found. Fig. 3 shows the largest Lyapunov exponent (k1) of Eq. (12) versus the parameter A from 2 to ... _ € 5 using the initial conditions ðx; x; x; xÞ ¼ ð0:4; 0:3; 0:6; 0:2Þ. The calculation was in the same manner as described in Section 2. At A = 2.525, the value k1 = 0.05537 was found
Fig. 3. The largest Lyapunov exponent of Eq. (12) versus A.
to be a maximum. As a result, the Lyapunov exponent spectrum was found to be (k1, k2, k3, k4) = (0.05537, 0, 0, 0.05537). Equivalently, Eq. (12) can be rewritten as a dynamical system in variables x, y, z, and w in a similar manner to Eq. (9) except that ±y2 in Eq. (9) is replaced by ±jyj. For the positive sign of the elementary piecewise-linear
Fig. 4. A phase space trajectory for A = 2.525 of Eq. (12).
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nonlinearity, +jyj, for example, the corresponding Jacobian matrix is
2
0
1
0
0
3
6 0 0 1 07 6 7 J¼6 7; 4 0 0 0 15 1 sgnðyÞ A 0
ð13Þ
where the signum function sgn(y) is defined as:
8 > < 1 for y < 0 sgnðyÞ ¼ 0 for y ¼ 0 > : 1 for y > 0
ð14Þ
Eq. (12) has a single equilibrium point at the origin. At the equilibrium point and for A = 2.525, the Jacobian matrix (13) has two complex conjugate pairs of eigenvalues at ±1.4259i and ±0.7013i. In a similar manner to Eq. (11), the total sum of all Lyapunov exponents of Eq. (12) equals zero and the trace(J) of Eq. (13) is zero and independent of ‘A’. It follows that Eq. (12) is a chaotic snap flow of a conservative system with a single control parameter ‘A’ of various values for a _ As an example, a single piecewise-linear nonlinearity jxj. phase space trajectory for A = 2.525 is shown in Fig. 4. A numerical simulation reveals that Eq. (12) also exhibits time-reversible Hamiltonian chaos [2].
Fig. 5. (a) The largest Lyapunov exponent of Eq. (15) versus A. (b) A bifurcation diagram of Eq. (15) shows a period-doubling route to chaos as A is gradually decreased from 2 to 1.878.
3.2. A dissipative system An elementary chaotic snap flow of a dissipative system is presented with elegant simplicity of a single piecewise-linear nonlinearity and a single control parameter ‘A’ as follows: ...:
...
x ¼ x A€x x_ ðjxj 1Þ:
Fig. 6. A chaotic attractor of Eq. (15).
ð15Þ
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Fig. 7. A phase space trajectory of Eq. (16).
The four-dimensional snap model of Eq. (15) is an extension of an existing three-dimensional jerk model with a piecewise-linear nonlinearity described in [17]. Such a nonlinearity can be implemented electronically with a full-wave rectifier with two diodes and an inverting unity gain amplifier and also with a single diode [18]. Fig. 5(a) illustrates the largest Lyapunov exponent of Eq. (15) versus the control parameter A which is gradually decreased from 2 to 1.6 using the initial conditions ... x ¼ x_ ¼ € x ¼ x ¼ 0: A bifurcation diagram of Eq. (15) for xmax is depicted in Fig. 5(b) as a function of the control parameter A from 2 to 1.6. A period-doubling route to chaos is evident as A is gradually decreased from 2 to 1.878. For positive and negative signs of jxj, the bifurcation and Lyapunov exponents of this system are the same. The only different is that both attractors are 180-degree rotated images to each other. Comparison between Fig. 5(a) and (b) verifies that chaos does exist. At A = 1.686, chaos is maximized with the Lyapunov exponent spectrum (k1, k2, k3, k4) = (0.11107, 0, 0.43926, 0.67184) and the Kaplan–Yorke dimension [19] (or equivalently Lyapunov dimension) Dky = 2.25293. The total sum of the Lyapunov exponent is equivalent to the average rate of fractional volume expansion along the trajectory, and can be calculated in the same manner to that of the conservative systems described by Eq. (11) [1,2]. Since
the total sum of the Lyapunov exponent spectrum or trace(J) of Eq. (15) is a negative value, i.e. trace(J) = 1, Eq. (15) is dissipative with constant state-space contraction of 1. The fluctuations in the largest Lyapunov exponent of this system lie in a small range of 104 for 106 iterations. Such a slight variation in the Lyapunov spectrum, as iteration increases, helps to ensure that the Lyapunov exponent has converged (see Section 4 of [20] for more information). Eq. (15) has two equilibrium points at (±1, 0, 0, 0). At the first equilibrium point (1, 0, 0, 0), the Jacobian matrix has the eigenvalues at 0.6255 ± 0.7802i and 0.1255 ± 0.9921i. At the second equilibrium point (1, 0, 0, 0), the Jacobian matrix has the eigenvalues at 0.1942 ± 1.3861i, 1.0830 and 0.4713. Fig. 6 shows a chaotic attractor of Eq. (15) which resembles the Rössler attractor [5] in a sense that it has a single foldedband structure. 4. Cubic nonlinearity 4.1. A conservative system An elementary chaotic snap flow of a conservative system is presented with elegant simplicity of only two terms and a single cubic nonlinearity of the form ...:
x ¼ €x3 x:
ð16Þ
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Fig. 8. Poincaré sections of Eq. (18).
In addition, all coefficients of Eq. (16) are unity in magnitude without a need for additional parameters on the right hand ... side. By using the initial conditions x ¼ x_ ¼ € x ¼ x ¼ 0:1, the Lyapunov exponent spectrum (k1, k2, k3, k4) = (0.07681, 0, 0, 0.07681) was calculated using the method described in Section 1. The trajectory of Eq. (16) is shown in Fig. 7. Equivalently, Eq. (16) can be rewritten as a dynamical system in variables x, y, z, and w of the form
x_ ¼ y; y_ ¼ z; z_ ¼ w;
ð17Þ
_ ¼ x z3 : w Eq. (17) requires only five terms which have a single nonlinearity. In comparison with an existing three (lower) dimensional chaotic flow where minimum terms have been reported to be five terms [8], an advantage is that such a chaotic snap flow also offers five (perhaps minimum) terms even though the four dimension is higher. Eq. (17) has a single equilibrium point at the origin. At the equilibrium point, pffiffiffi pffiffiffi the Jacobian matrix has eigenvalues at ð1= 2Þ ð1= 2Þi. In a similar manner to Eq. (11), the total sum of all Lyapunov exponents of Eq. (16) is zero and the trace of the Jacobian matrix of Eq. (17) equals zeros. It follows that Eq. (16) is
Fig. 9. The largest Lyapunov exponent of Eq. (19) versus A.
conservative. A numerical simulation reveals that Eq. (16) also exhibits time-reversible Hamiltonian chaos [2]. 4.2. A dissipative system A chaotic snap flow of a dissipative system based on three cubic nonlinearities has been found where all coefficients are unity in magnitude as follows:
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Fig. 10. Poincaré sections of Eq. (19). ...:
...
€3
_3
3
x ¼ x €x x x x :
ð18Þ ...
_ € By using the initial conditions ðx; x; x; xÞ ¼ ð8:5; 2:3; 5:6; 0:1Þ, the Lyapunov exponent spectrum is (k1, k2, k3, k4) = (0.05351, 0, 0.20296, 0.85054), resulting in the Kaplan–Yorke dimension (or Lyapunov dimension) Dky = 2.26364. Since the total sum of the Lyapunov exponent spectrum and the trace(J) of Eq. (18) equal a negative value, i.e. trace(J) = 1, Eq. (18) is dissipative with constant state-space contraction of 1. Eq. (18) has a single equilibrium point at the origin. At the equilibrium point, the Jacobian matrix has eigenvalues at 0, 0, 0.5 ± 0.866i. Fig. 8 displays the Poincaré sections of Eq. (18) with fractal structures resulting from the fractal dimension Dky = 2.26364. 5. A single exponential nonlinearity for a dissipative system It is also interesting to identify an elementary chaotic snap flow with elegant simplicity of a single exponential nonlinearity and a single control parameter. One example is ...:
...
x ¼ x Ax_ x expð€xÞ:
ð19Þ
Fig. 9 shows the largest Lyapunov exponent versus the con... trol parameter A using initial conditions x ¼ x_ ¼ € x ¼ x ¼ 0. It
is apparent that chaos is maximized at A 6. The Lyapunov exponent spectrum at A = 6 is (k1, k2, k3, k4) = (0.16788, 0, 0.15074, 1.01714) resulting in a Kaplan–Yorke dimension (or Lyapunov dimension) Dky = 3.01685. Eq. (19) has a single equilibrium point at (1, 0, 0, 0). The eigenvalues at the equilibrium point for A = 6 are 1.9408, 0.5558 ± 1.6453i and 0.1708. Since the total sum of the Lyapunov exponent spectrum and the trace(J) of Eq. (19) equal a negative value, i.e. trace(J) = 1, Eq. (19) is dissipative with constant state-space contraction of 1. The Poincaré sections in several planes for A = 6 are shown in Fig. 10. No conservative cases have been found for such an exponential nonlinearity. 6. Conclusions Elementary chaotic snap flows have been presented through an extensive numerical search. They consist of five new examples and a generalization of an existing flow. Four of them have demonstrated elegant simplicity of a single control parameter based on a single nonlinearity of a quadratic, a piecewise-linear or an exponential type. Two others have demonstrated elegant simplicity of all unity-in-magnitude parameters based on either a single cubic nonlinearity or three cubic nonlinearities. The algebraically simplest example found in this study has been
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the chaotic snap flow with a single cubic nonlinearity and all unity-in-magnitude parameters. Although the 4th-order is higher, the equation requires only two terms in jerk representation, and can be transformed to only five terms in its equivalent dynamical form. Such five terms offer algebraically minimum terms comparable to those of existing lower-order chaotic flows. The dissipative chaotic snap flow with a piecewise-linear nonlinearity suggests a possible implementation of electronic circuits using a full-wave rectifier with diodes and an inverting unity-gain amplifier. Basic dynamical properties have been demonstrated in three conservative and three dissipative chaotic flows. Acknowledgements The authors are grateful to Prof. J.C. Sprott for his useful comments. The first author appreciates CIC-Common Fund of SIIT for partially financial support. This work was supported by Telecommunications Research and Industrial Development Institute (TRIDI), NBTC, Thailand, and the National Research University Project of Thailand, Office of Higher Education Commission. References [1] Sprott JC. Some simple chaotic jerk functions. Am J Phys 1997;65:537–43. [2] Sprott JC. Chaos and time-series analysis. Oxford: Oxford University Press; 2003. [3] Sprott JC. Some simple chaotic flows. Phys Rev E 1994;50:R647–50.
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