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Attractor of Smale-Williams type in autonomous distributed system

We consider an autonomous system of partial differential equations for one-dimensional distributed medium with periodic boundary conditions. Dynamics in time consists of alternating birth and death of patterns with spatial phases transformed from one stage of activity to another by the doubly expanding circle map. So, attractor in the Poincar\'e section is uniformly hyperbolic, a kind of Smale-Williams solenoid. Finite-dimensional models are derived as ordinary differential equations for amplitudes of spatial Fourier modes (the 5D and 7D models). Correspondence of the reduced models to the original system is demonstrated numerically. Computational verification of the hyperbolicity criterion is performed for the reduced models: the distribution of angles of intersection for stable and unstable manifolds on the attractor is separated from zero, i.e. the touches are excluded. The considered example gives a partial justification to the old hopes that chaotic behavior of autonomous distributed systems may be associated with uniformly hyperbolic attractors.


Introduction
Uniformly hyperbolic chaotic attractors such as Smale-Williams solenoid or Plykin attractor were introduced in mathematical theory of dynamical systems several decades ago [1][2][3][4][5]. After many steps, asymptotically, the solenoid arises having an infinite number of turns and Cantor-like transversal structure. Once it was believed that such attractors may describe chaos and turbulence in many cases, but later it turned out that chaotic dynamics commonly occurring in applications do not fit the class of uniformly hyperbolic attractors. Physically implementable systems with hyperbolic chaos were discovered (or, rather constructed) only very recently [6][7][8][9][10]. Their principle of operation was based on chaotic nature of maps for the angular variables characterizing phases of oscillations in time, at successive stages of activity of oscillatory elements constituting the system. An alternative general approach to elaboration of systems with hyperbolic chaos, appropriate for spatially extended systems, was advanced in Ref. [11]: instead of phases of oscillations in time, it was suggested to deal with spatial phases of patterns generated in a distributed medium.
In computations, this principle was demonstrated and illustrated with a model based on a modified Swift-Hohenberg equation. Due to externally forced periodic modulation of a parameter controlling the characteristic spatial scale, the model system generates long-wave and short-wave Turing patterns alternately, and operates in such way that the spatial phases at successive stages of the dynamical evolution are governed by an expanding circle map. One more example with analogous dynamics of the spatial phases relates to alternating parametric excitation of long-scale and short-scale standing-wave patterns due to pump modulation in a medium described by a wave equation with nonlinear dissipation [12,13]. Such examples justify, at least particularly, the old expectations for applicability of the hyperbolic theory to chaotic dynamics in spatially extended systems. Some dissatisfaction is the fact that these examples relate to non-autonomous systems, with time-dependent coefficients in the partial differential equations. It would be interesting to discover autonomous extended systems manifesting the hyperbolic chaos. Elaboration of such an example is a goal of the present article; we will present an autonomous set of nonlinear partially differential equations manifesting attractor of Smale-Williams type in the Poincaré map. At the moment we do not pretend to relate this construction to a concrete physical system, but we suppose that it may be implementable, in particular, on a base of electronic circuits (say, in a kind of nonlinear transmission line).

The main model and results of computer simulation
Following [11], let us start with the one-dimensional Swift-Hohenberg equation A trivial solution of this equation 0 ≡ u becomes unstable at 0 μ > . In linear approximation we evaluate the increment of the perturbation growth as It is maximal at that corresponds roughly to the wave number of the Turing pattern, which grows up in the medium from arbitrarily small random initial perturbations. Due to the cubic nonlinear term in (1) the pattern, which is nearly periodic in space, saturates at some finite level of magnitude.
To modify the model, we add one more variable v depending on the spatial coordinate x and time t, and turn to the following set of equations . , 3 cos ε μ ) 1 ( For our purposes it is appropriate to postulate circular geometry of the medium: where L is the system length. The first relation in (2)  implies periodic spatially inhomogeneous coupling between two components involved in the dynamics.
In accordance with the periodic boundary conditions, the functions u and v describing the pattern are spatially periodic and may be represented by Fourier series expansions.
Qualitatively, the functioning of the systems may be explained as follows. Suppose at some moment the multiplier in the first equation (2) is positive; hence, the variable u grows, giving rise to a spatial pattern with appropriate wave-number about k 0 =1. The most significant is the contribution of the first spatial harmonic component, which is characterized naturally by some phase φ: . After a while, due to the growth of the pattern magnitude, the coefficient ( ) in the second equation becomes positive (except narrow spatial neighborhoods of nodes of the pattern), and then the amplitude of the variable v starts to grow. In the function v the second spatial harmonic component is relevant, and at the initial stage of this instability it accepts the doubled spatial phase 2φ as it is stimulated by the quadratic nonlinear term u 2 in the second equation (2). So, we have ) 2 2 cos( μ v u − + once more becomes positive, and the variable u starts to grow again. The excitation of the first subsystem on this stage is stimulated by the term composed as a product of the second harmonic component of v and the function x 3 cos ε accounting the spatial inhomogeneity. It provides transfer of the doubled spatial phase back to the pattern of u (with the opposite sign), as follows from the relation . Then, the process repeats itself over and over again. So, the spatial phases on successive stages of the pattern formation evolve according to the expanding circle map const 2 Numerical simulation of the dynamics was performed at , the explicit-implicit difference scheme on a grid with temporal and spatial steps, respectively, . One can observe relatively long-time stages of growth of the patterns alternating with their fast decay almost to zero (Fig. 2). The mean time period between the successive stages of excitation in this regime according to the computations is 37 . 50 = τ .   For the sustained chaotic regime we computed Lyapunov exponents for the Poincaré map.
It was done using the Benettin algorithm adapted for the distributed system [14][15][16]9], with . The fractional part of the dimension is rather small because of strong transversal compression of the phase volume nearby the attractor.

The 5D model
Having in mind that the functioning of the system according to the above qualitative explanation is based on interaction of spatial Fourier harmonics, one can try to reduce the dynamics to finitedimensional ordinary differential equations for amplitudes of the most significant modes of the components u and v.
The simplest set of shortened equations, which correctly reflects the qualitative character of the dynamics, may be derived by substitution of the following ansatz:   Figure 5 illustrates some results of numerical solution of (5) by the Runge-Kutta fourth order method. Here the time dependences are plotted for amplitudes and arguments of the complex variables 1 U and 2 V as well as that for the real variable w. One can see that the plots look similar to those corresponding to the original model governed by the partial differential equations. Jumps on the plots for the phases correspond to short time intervals of transfer of the excitation between the spatial modes as explained.

The 7D model
One can increase the number of accounting spatial Fourier components to obtain more accurate finite-dimensional models of larger phase space dimension. The system (5) where 4 V is the complex amplitude of the fourth spatial Fourier component of the variable v(x, t). Then, from the manipulations similar to those in derivation of the model (5), we obtain the set of ordinary differential equations (We do not account additional harmonics in the Fourier expansion for the function u(x, t) as they appear to have small amplitudes and do not influence notably the accuracy of the description.) which is six-dimensional for this model. Figure 8a shows the diagram for the phases of 1 U at successive crossing the surface of the Poincaré section. It corresponds to the expanding circle map, like in the main model of Section 2 and the low-dimensional model of Section 3. Observe that the undulating in the form of the curve has decreased essentially in comparison with the five-dimensional model, and the form corresponds much better to the original distributed system (cf. Fig. 4a). Figure 8b shows attractor in the Poincaré cross-section in projection onto a plane of real and imaginary parts of the variable 1 U .

Hyperbolicity test for the finite-dimensional models
For attractors of the 5D and 7D models discussed in two previous sections, we have conducted a numerical test for hyperbolicity following the approach suggested in [17,18] and used in application to hyperbolic attractors in [6][7][8][9].
The method is based on estimate of distribution of angles between stable and unstable manifolds on an attractor. The stable and unstable manifolds for the hyperbolic attractor can meet only with nonzero angle; the touches must be excluded.  [19], where this modification of the method was suggested. If zero values of the angle do not occur, i.e. the statistical distribution of the angles is separated from zero, one concludes that the dynamics is hyperbolic. If the statistics show nonvanishing probability for zero angles, it implies non-hyperbolic behavior because of presence of the homoclinic tangencies of the stable and unstable manifolds.
In our 5D and 7D models we generate, first, a sufficiently long representative orbit on the attractor from the numerical solution of Eqs. (5)

Conclusion
In this article, we have discussed an example of autonomous distributed system with ring geometry (periodic boundary condition), which implements chaotic dynamics corresponding to uniformly hyperbolic attractor, a kind of Smale-Williams solenoid in the Poincaré map. The dynamics consists of sequential birth and death of the spatial patterns; the Smale-Williams attractor occurs due to the fact that the spatial phases of these patterns on each next stage of activity are transformed according to the doubly expanding circle map. Also we derived and studied numerically the truncated models represented by a five-dimensional and a seven-dimensional set of ordinary differential equations. Their dynamics are found to correspond qualitatively to the original distributed system. For these models, the hyperbolicity of the attractor is confirmed by a computer-based test, which indicates the lack of touches for stable and unstable manifolds of the orbits on the attractor.
Thus, we get the first example of an autonomous distributed system with a uniformly hyperbolic attractor. As believed, it revives the old hope that such attractors may be relevant to some cases of complex dynamics of spatially extended systems (like the hydrodynamic turbulence). We can assume that the considered system may be implemented in electronics based on a kind of nonlinear transmission line. Attractiveness of systems with uniformly hyperbolic attractors in a frame of possible practical application of chaos is determined by their structural stability, or robustness: the generated chaos is insensitive to variations of parameters, imperfection of fabrication, technical fluctuations in the system, etc.