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Robust chaos in autonomous time-delay system

We consider an autonomous system constructed as modification of the logistic differential equation with delay that generates successive trains of oscillations with phases evolving according to chaotic maps. The system contains two feedback loops characterized by two generally distinct retarding time parameters. In the case of their equality, chaotic dynamics is associated with the Smale-Williams attractor that corresponds to the double-expanding circle map for the phases of the carrier of the oscillatory trains. Alternatively, at appropriately chosen two different delays attractor is close to torus with Anosov dynamics on it as the phases are governed by the Fibonacci map. In both cases the attractors manifest robustness (absence of regularity windows under variation of parameters) and presumably relate to the class of structurally stable hyperbolic attractors.


Introduction
The concept of uniformly hyperbolic chaotic dynamics was advanced in mathematical theory of dynamical systems half a century ago [1][2][3][4][5][6]. It deals with invariant sets in state space of systems composed exclusively of orbits of saddle type with such a restriction that their stable and unstable manifolds do not touch each other (intersect transversally). The hyperbolic invariant sets are structurally stable. It implies their robustness in respect to (at least small) variation of functions and parameters in the dynamical equations. Henceforth, the hyperbolic dynamics may be regarded as preferable for any possible practical application of chaos and as deserving prior research in a frame of physical and technical disciplines [7,8].
One important class of the uniform hyperbolicity is Anosov dynamics in systems where the hyperbolic invariant set occupies the whole accessible phase space; usually it is considered in the context of the phase volume preserving maps or flows (conservative dynamics). Alternatively, in the context of dissipative systems, the uniformly hyperbolic attractors were introduced; their mathematical examples are Smale -Williams solenoid, Plykin type attractors, DA-attractor of 2 Smale. 1 Although originally they relate to artificial discrete-time systems, analogous attractors can occur as well in Poincaré maps associated with continuous time systems governed by differential equations. In the last case it is used to speak about suspension of these attractors provided by the appropriate flows.
Till a short time ago, realistic examples of uniformly hyperbolic dynamics which could relate to real-world systems were poorly presented and discussed in the literature. One example is a hinge mechanism with Anosov dynamics described and studied in [9]. An artificial example of suspension of Plykin-type attractor was constructed in PhD thesis of Hunt [10], but it is surely too complicated to allow a real physical implementation. Possible occurrences of suspension of Plykin-type attractors in neuron model [11] and in modified Lorenz model [12] were discussed, but no convincing data on a level of concrete equations and numerical simulation were provided.
Recently a number of implementable systems with uniformly hyperbolic attractors were advanced and studied due to efforts of Saratov group of nonlinear dynamics [13][14][15][16][17][18][19][20]. It is essential that these examples are constructed on a base of the physical rather than the mathematical toolbox exploiting such entities as oscillators, nonlinear elements, interactions, self-oscillatory and parametric excitation etc.
One particular productive method for design of systems with uniformly hyperbolic attractors is based on using time-delay nonlinear feedback loops [21][22][23][24]. Most of the examples suggested were non-autonomous systems functioning in presence of the external periodic driving [21][22][23]. An exception is an autonomous system with attractor of Smale-Williams type considered in Ref. [24]. The present article is inspired by that work. We construct and study numerically an autonomous time-delay system manifesting different types of uniformly hyperbolic attractors depending on two parameters of time delay. 1 The abbreviation DA stands for "derived from Anosov".

Basic equations
Let us start with logistic delay equation offered in due time in population biology [25]: (1) Here the population is characterized by the positive variable r evolving in time and normalized in such way that the saturation occurs at r =1. A positive parameter μ is the birth rate, τ 1 is the delay time characterizing lag of the effect of saturation.
According to Ref. [25], under condition μ π < τ 2 / Solutions of the equations (2) instead of the solitary pulses manifest successive oscillatory trains for x and y while the envelope evolves exactly according to (1) (Fig.1b).
Next, let us add terms proportional to and with small coefficient ε in the first and the second where we always suppose that the delays satisfy the inequality . Now, in the case of generation of pulses with low enough level of minimal amplitude between 1 2 τ ≥ τ them, just these additional terms will initiate formation of each next pulse of oscillations. Due to this, the phase of the oscillations for the new-born pulse will be determined by the phases of the previous pulses via some mapping as discussed in detail in the following sections. In presence of this initiation of the oscillations the characteristic repetition period of the pulses is reduced comparing to the original system (2).

Attractor of Smale -Williams type
Let us start with the simplest particular case setting both delay time parameters equal: To clarify the principle of functioning of the system suppose that during some current stage of activity the variables behave as ) cos( 0 ϕ + ω t x and ; then the additional terms are expressed as These relations determine actually the stimulating signal for oscillations on the immediate next stage of activity. The phase shift of this signal is transferred to the oscillations arising on this stage. Hence, the phase undergoes the doubling transformation const 2 It is the expanding circle map, or the Bernoulli map, which is chaotic and is characterized by the Lyapunov exponent In accordance with the above qualitative considerations, the process looks like a sequence of trains of oscillations. Note that the average period of their appearance is less than that in models (1), (2), and in the regime under discussion it is T≈10. 6 Although represented by nearly periodic alternation of excitation and suppression of the oscillations, the process is actually chaotic (in contrast to the regular one observed in Fig.1b). Chaos reveals itself in variations of phases on successive stages of activity. More accurate analysis shows that these phases obey approximatelt the Bernoulli map.
, and the second on the plane of variables (x, y).
For computation of Lyapunov exponents we use the Benettin algorithm [26,27] adapted for the time-delayed systems [28,29]. It is based on simultaneous numerical solutions of the equations (4) and the variation equations Formally, there are an infinite number of Lyapunov exponents in the time-

Attractor associated with with Anosov-type dynamics on torus
Now return to the model with two different time delays t 1 and t 2 : (10) and select these parameters to get such situation that the delayed signals stimulating excitation of a new n+1-th pulse of oscillations arrive being emitted from the previous two pulses n (delay t 1 ) and n-1 (delay t 2 ).
Suppose that at the previous two activity stages the phases were φ n and φ n-1 , . Then, the additional terms in the right-hand parts are expressed as ) sin( ) As follows, on the next, (n+1)-th stage of activity the phase will be determined (up to a constant) by the Fibonacci map ) 2 (mod const Of what kind attractor is in this case? For the infinite-dimensional Poincaré map of our time-delay system attractor is an object close geometrically to a twodimensional torus, and the discrete-time dynamics of the angular variables on this torus obey the map (12).
The process looks like a sequence of trains of oscillations for which the phases of the high-frequency carrier vary in a random-like way from one to the next stage of activity. The average period of repetition of the pulses evaluated numerically is roughly Tº10.85.    One more way to illustrate the Anosov dynamics for the phases is similar to the famous picture of Arnold's cat [30,15,16]. For each pair of phases n ϕ and on the n-th and n-1 stages of activity, we determine either the point with such coordinates fit or not in the picture of the cat face drawn on the plane. If yes, then the point is marked on the graph, and on the following two plots the corresponding points are depicted, respectively after 3 and 6 iteration steps. If the initial point does not fall within the specified cat face area, the dots are not marked, and we carry on with further iterations. Figure 8 The perturbation vectors are represented in computations as finite arrays of and  Viewing the graphics in Fig. 9 one can distinguish two somewhat different situations. Namely, in the left parts of the diagrams the first negative exponent is nearly equal to the positive one in absolute value, and the next negative exponent is lesser. As we go to the right some crossover occurs, and the two negative exponents exchange their places. After that, obviously, approach of trajectories to the attractor in time is slower than the convergence and the divergence of orbits on the attractor itself. (Apparently, it implies different nature of the attractors in these two situations, but we leave the comparative analysis of them outside the frame of the present article.) We outline again the smooth character of dependences of the Lyapunov exponents on parameters that confirms robust nature of chaos. It may be thought that the attractors we deal with in this section are uniformly hyperbolic, close to the so-called "non-strange chaotic attractors" briefly mentioned and discussed in the literature [31][32][33].

Conclusion
In this paper, we introduce an autonomous system built on the basis of the logistic differential equation with delay. As shown, the dynamics of the phases of generated successive oscillatory trains corresponds to expanding circle map or Anosov map on a torus depending on selection of two time retarding parameters.
Numerical simulations show robust chaos generation in these cases, and we conjecture the uniformly hyperbolic nature on the attractors in the infinitedimensional state space of the time-delay system.
The suggested system and analogous constructions may be realized on a basis of electron devices as generators of robust chaos; physically they may be simpler in implementation comparing, for example, with systems constructed on the basis of coupled oscillators [13][14][15][16].
Due to the insensitivity of chaos to variations in parameters, the chaos generators of this kind are of interest from a practical point of view, for example in