Hyperchaos
| Christophe Letellier and Otto E. Rossler (2007), Scholarpedia, 2(8):1936. | doi:10.4249/scholarpedia.1936 | revision #91369 [link to/cite this article] |
A hyperchaotic attractor is typically defined as chaotic behavior with at least two positive Lyapunov exponents. Combined with one null exponent along the flow and one negative exponent to ensure the boundness of the solution, the minimal dimension for a (continuous) hyperchaotic system is 4.
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The first hyperchaotic system
The first 4D flow was proposed in 1979 by Rössler. It reads
\[ \dot{x} = -y -z \] \[ \dot{y} = x+ay+w \] \[ \dot{z} = b+xz \] \[ \dot{w} = -cz+dw \] It has hyperchaotic behavior (Fig. 1) as a solution with the parameter values a=0.25, b=3, c=0.5 and d=0.05. Initial conditions can be \( \left( x_0 =-10;y_0=-6;z_0=0;w_0=10.0 \right) \ .\) The corresponding four Lyapunov exponents are \( \lambda_1=0.112 \ ,\) \( \lambda_2=0.019 \ ,\) \( \lambda_3=0 \) and \( \lambda_4=-25.188 \ .\)
The hyperchaotic character of this behaviour is not so obvious from this plane projection which looks a bit like a "noisy" chaotic attractor. When a first-return map to the Poincaré section \[ P = \left\{ (x_n,y_n,z_n) \in R^3 ~|~ \dot{y}_n = 0, \ddot{y}_n < 0 \right\} \] is computed (Fig. 2), its structure is obviously two-dimensional. Such a feature results from the conjugation of two expanding folding-overs.
Hyperchaotic maps
The folded-towel map (Fig. 3)
\[ x_{n+1} = a x_n (1-x_n) -0.05 (y_n +0.35) (1-2z_n) \] \[ y_{n+1} = 0.1 \left( \left( y_n +0.35 \right)\left( 1+2z_n\right) -1 \right) \left( 1 -1.9 x_n \right) \] \[ z_{n+1} = 3.78 z_n (1-z_n) + b y_n \] was indicated by Rössler in 1979 as an invertible map with two positive and one negative Lyapunov exponents\[ \lambda_1=0.430 \ ,\] \( \lambda_2=0.377 \) and \( \lambda_3=-3.299 \ .\)
In 1985, the generalized Hénon map (Fig. 4)
\[ x_{n+1} = a -y_n^2 -b z_n \] \[ y_{n+1} = x_n \] \[ z_{n+1} = y_n \] was introduced by Baier and Klein. In this case, a hyperchaotic behavior is obtained with a=3.4 and b=0.1. The corresponding Lyapunov exponents are \( \lambda_1=0.276 \ ,\) \( \lambda_2=0.257 \) and \( \lambda_3=-4.040 \ .\)
A 9D model for a chaos-hyperchaos transition
A 9D model for a Rayleigh-Bénard convection in a square cell was proposed by Reiterer in 1998. It reads
\[ \dot{C}_1 = -\sigma b_1C_1 - C_2C_4 + b_4C_4{}^2 + b_3C_3C_5-\sigma b_2C_7 \] \[ \dot{C}_2 = -\sigma C_2 + C_1C_4 - C_2C_5 + C_4C_5-\sigma C_9/2 \] \[ \dot{C}_3 = -\sigma b_1C_3 + C_2C_4 - b_4C_2{}^2 - b_3C_1C_5+\sigma b_2C_8 \] \[ \dot{C}_4 = -\sigma C_4 - C_2C_3 - C_2C_5 + C_4C_5+\sigma C_9/2 \] \[ \dot{C}_5 = -\sigma b_5C_5 + C_2{}^2/2 - C_4{}^2/2 \] \[ \dot{C}_6 = -b_6C_6 + C_2C_9 - C_4C_9 \] \[ \dot{C}_7 = -b_1C_7 - RC_1 + 2C_5C_8 - C_4C_9 \] \[ \dot{C}_8 = -b_1C_8 + RC_3 - 2C_5C_7 + C_2C_9 \] \[ \dot{C}_9 = -C_9 - RC_2 + RC_4 - 2C_2C_6 + 2C_4C_6 + C_4C_7 - C_2C_8 \]
where the constant parameters \(b_i\) are a measure for the geometry of the square cell, defined by
\[ b_1 := 4\frac{1+a^2}{1+2a^2} ~~ b_2 := \frac{1+2a^2}{2\left(1+a^2\right)} ~~ b_3 := 2\frac{1-a^2}{1+a^2} \] \[ b_4 := \frac{a^2}{1+a^2} ~~ b_5 := \frac{8a^2}{1+2a^2} ~~ b_6 := \frac{4}{1+2a^2} \] In previous studies, this system was investigated with a=0.25 for which the route to chaos is a period-doubling cascade. The transition from chaos to hyperchaos (Fig. 5) is observed around R=43.3 where a second Lyapunov exponent becomes positive (see the evolution of the Lyapunov spectrum shown in Fig. 6). This "continuous" route from chaos to hyperchaos does not correspond to the route proposed by Kapitaniak et al for driven systems. Such an almost smooth transition from chaos to hyperchaos is still not completely understood.
Experimental hyperchaotic behaviors
Very few experimental hyperchaotic behaviors have been identified so far. This possibly results from the side-effect that the measurement function may affect the observability of the underlying dynamics (see Letellier & Aguirre, 2002). Indeed, since (continuous) hyperchaotic systems are necessarily four-dimensional, the effect of the measurement function \(h: R^m \mapsto R\) becomes more critical than for three-dimensional (continuous chaotic) systems for its possibly introducing spurious foldings since the projection of an object in \(R^m\) unto \(R\) will most likely result in false neighbors forming that it is not always possible to remove by increasing the embedding dimension. This is a consequence of the observability problem of a dynamics through the measurement of a single (scalar) time series.
In particular, when the largest Lyapunov exponent \(\lambda_1\) is sufficiently larger than the second positive Lyapunov exponent \(\lambda_2\ ,\) there are two different time scales in the hyperchaotic dynamics. As a consequence, it becomes quite difficult to reconstruct an adequate - sufficiently unfolded - phase portrait, even when the phase space dimension is increased as suggested by Takens' theorem. Many time series must be measured to reconstruct in a non ambiguous way the whole phase portrait. Moreover, due to the large value of \(\lambda_1\ ,\) measurements sufficiently far in the past are irrelevant to the present. As a result, the strength of distortion observed in the reconstructed state portrait diverges at a rate that depends exponentially on the dimension m (see Casdagli, 1991).
Thus, only few experimental hyperchaotic behaviors have been identified. The occurrence of hyperchaotic behavior has been found in an electronic circuit (Matsumoto et al, 1986), NMR laser (Stoop et al, 1988), in a semi-conductor system (Stoop et al, 1989) and in a chemical reaction system (Eiswirth et al, 1992). In the 9D model, observability is significantly reduced in the hyperchaotic regime when the first positive Lyapunov exponent is an order of magnitude larger than the second positive Lyapunov exponent.
Reminiscences about the discovery of hyperchaos
"Hyperchaos is a step towards the X-attractor. The X-attractor is the radically new class of attractors that arises with the fourth dimension (just as chaos arises in the third, oscillation in the second, and steady state in the first). It is still elusive. Hyperchaos (Paul Rapp suggested in 1977 that this name was superior to "superchaos", as I first tended to call it) arises if a 2D cross section through a 3D flow is re-injected (via the fourth dimension) in an expanding manner, expanding in two directions rather than a single one. In other words, it is a straightforward generalization of the reinjection which yields chaos in three dimensions (reinjection, after expansion, back towards the middle of a spiral). Now, it is reinjection towards the tip of an expanding screw. The expanded and folded-over "walking-stick" map in a cross section is hence replaced by a "folded-towel map" (the product of 2 folding-overs). And the former Cantor sets become Julia sets, if you like scary terms. The first example was a terribly complicated, singular-perturbation generalization of two stacked van der Pol-Liénard oscillators, presented in 1978 in Madison. Then, a stepwise simplification on a fast analogue computer (with several 3D projections displayed simultaneously) eventually led to an "irregularly breathing version" of the 3D spiral chaos, again with but a single nonlinearity. The switching variable hereby no longer sounded merely hoarse (or screeching) as in chaos, but rather resembled the noise made by rain drops on a car roof: that is the sound of hyperchaos. But could one trust the analogue machine ? The "translation" into a digital version, with exactly known parameter values, took months of intense work punctured by desperation, Edison-type. It would have been impossible without the first desk-top computer (HP 9845 B) which was programmable in Basic and had the unique feature of, on the touch of a button, printing out the whole momentary screen content (including the plots, equations, axes, initial and final conditions), on a thermo-printer. Art Winfree had recommended its use (Mitch Feigenbaum and Rob Shaw also later quoted this machine as their main co-author).
Some years later (1995), Gerold Baier and Sven Sahle found the n-dimensional generalization of the equation, still with maximal hyperchaos (n-2 positive exponents) and still with one quadratic term only. Think in the limit of a linear continuum (first-order wave equation) coupled to a single nonlinear variable, producing the highest possible nonlinear complexity ! But : How about the X-attractor ? Is there a new hierarchy starting with the fourth dimension, just like chaos with hyperchaos etc. following started with 3 and oscillation with the quasi-periodicity hierarchy started with 2 ?
The Kaplan-Yorke phenomenon does give rise to a fat, superfat, etc. hierarchy, starting with the fourth dimension, as Jack Hudson and I found in 1991. These attractors are nowhere differentiable on a Cantor set ("singular-continuous nowhere differentiability") but they are still close to chaos (adding Weierstrass-type non differentiability). To find out if this is all there is to hope for, a big-science project involving toddlers was proposed: give them play-dough and document their activities on video; for the fourth flow dimension can still be completely covered in this fashion, focusing on repeatable operations in 3D (3D diffeomorphisms). And the creative power of toddlers exceeds any other in the universe (by attributing benevolence out of nothing, they create the "person attractor", as Detlev Linke calls it."
Acknowledgements
We wish to thank Jean-Marc Malasoma for his computations of the Lyapunov exponents.
References
G. Baier & M. Klein, Maximum hyperchaos in generalized Hénon map, Physics Letters A, 151, 281-284, 1990.
G. Baier & S. Sahle, Design of hyperchaotic flows, Physical Review E, 51, R2712-R2714, 1995.
M. Casdagli, S. Eubank, J. D. Farmer & J. Gibson, State space reconstruction with noise, Physica D, 51, 52-98, 1991.
M. Eiswirth, Th.-M. Kruel, G. Ertl & F. W. Schneider, Hyperchaos in a chemical reaction, Chemical Physics Letters, 193 (4), 305, 1992.
T. Kapitaniak, K. E. Thylwe, I. Cohen & J. Wjewoda, Chaos-hyperchaos transition, Chaos, Solitons & Fractals, 5 (10), 2003-2011, 1995.
C. Letellier & L. A. Aguirre, Investigating nonlinear dynamics from time series: the influence of symmetries and the choice of observables, Chaos, 12, 549-558, 2002.
T. Matsumoto, L. O. Chua & K. Kobayashi, Hyperchaos: laboratory experiment and numerical confirmation, 'IEEE Transactions on Circuits and Systems, CAS-33 (11), 1143-1147, 1986.
P. Reiterer, C. Lainscsek, F. Schürrer, C. Letellier & J. Maquet, A nine-dimensional Lorenz system to study high-dimensional chaos, Journal of Physics A, 31, 7121-7139, 1998.
O. E. Rössler, An equation for hyperchaos, Physics Letters A, 71, 155-157, 1979.
O. E. Rössler & J. L. Hudson, A superfat attractor with a Singular-Continuous 2-D Weierstrass function in a cross-section, Zeitschrift fur Naturforschung A, 48, 673-678, 1993.
K. Stefanski, Modelling chaos and hyperchaos with 3D maps, Chaos, Solitons & Fractals, 9 (1-2), 83-93, 1998.
R. Stoop & P. F. Meier, Evaluation of Lyapunov exponents and scaling functions from time series, Journal of the Optical Society of America B, 5 (5), 1037, 1988.
R. Stoop, J. Peinke, J. Parisi, B. Röhricht & R. P.Hübener, A p-Ge semiconductor experiment showing chaos and hyperchaos, Physica D, 35, 425-435, 1989.
Internal references
- John W. Milnor (2006) Attractor. Scholarpedia, 1(11):1815.
- Olaf Sporns (2007) Complexity. Scholarpedia, 2(10):1623.
- Yuri A. Kuznetsov (2007) Conjugate maps. Scholarpedia, 2(12):5420.
- James Meiss (2007) Dynamical systems. Scholarpedia, 2(2):1629.
- Jeff Moehlis, Kresimir Josic, Eric T. Shea-Brown (2006) Periodic orbit. Scholarpedia, 1(7):1358.
