Simple examples of strange nonchaotic attractors

In modern nonlinear dynamics, systems with continuous and discrete time are studied on equal rights. The first are those defined by differential equations, and the second by iterative maps. In a continuous-time system appearance of SNA becomes possible in a case of quasiperiodic time dependence of coefficients in the differential equation, which demand presence at least two incommensurate frequencies. In iterative maps there is an internal own 'rhythm', the steps of discrete time, so, it is sufficient to have one frequency of driving incommensurate with the frequency of steps. The simplest class of systems manifesting SNA is represented by mappings with one coordinate variable and one phase. The external driving is characterized by one parameter of frequency, which is often taken to be equal to the 'golden mean', the number (5^1/2-1)/2.

Remark. This selection may be explained not only by simplification of the theoretical analysis, as usually mentioned, but also by the fact that the subtle structures examined in experiments and numerics are distinguishable better that at other frequency ratios.

Pitchfork bifurcation model

Let us take a map x_n+1=lth x_n, which demonstrates a pitchfork bifurcation: at l<1 a single attractor is a fixed point at origin; at l>1 it becomes unstable, but instead two symmetric stable points appear x=±x₀(l). Following Grebogi et al., let us modify the map introducing a quasiperiodic modulation of the coefficient:

where q is a phase variable. Parameter w=(5^1/2-1)/2 corresponds to a frequency of the external driving, which is of multiplicative nature.

It is easy to estimate Lyapunov exponent for the solution x=0. In linear approximation,

and

Due to irrationality of w, the phase values q_n at N® ¥ are uniformly and dense distributed over the unit interval. Therefore, we change the sum to the integral, and for the Lyapunov exponent get

At l<1 we have L<0, i.e. the solution is stable, and the line x=0 on the plane (q,x) is an attractor. For l>1 it is unstable, has a positive Lyapunov exponent, and cannot be an attractor.

What can one say about an attractor at l>1? First, it must contain points with |x|>0. Second, due to the relation |th x|<1, the attractor is placed entirely in a band |x|<2l. Third, the axis x=0 contains an infinite set of points of the attractor. Indeed, at q=1/4 cosine vanishes, therefore, for all q_n=1/4+nw, where n=1, 2,...,¥ we have x_n=0. This subset of the attractor is infinite and dense, so, the attractor obviously is a nontrivial geometric object, the 'strange set'. In figure, portraits of SNA in the model under consideration are shown on the phase plane (q, x) and on the iteration diagram in coordinates (x_n,x_n+1).

Lyapunov exponent of this attractor is negative. Indeed, for the function f(x)=th x the relation is valid: |f'(x)|£|f(x)/x|, and the equality is reached only at x=0. Hence, for the Lyapunov exponent we have

Lyapunov exponent of this attractor is negative. Indeed, for the function f(x)=th x the relation is valid: |f'(x)|£|f(x)/x|, and the equality is riched only at x=0. Hence, for the Lyapunov exponent we have

Driven quadratic map

One of popular and rich models in nonlinear dynamics is a quadratic map x_n+1=l-x_n², manifesting a transition to chaos via period doubling bifurcation cascade under increase of the control parameter l. Introducing modulation of the parameter with an irrational frequency w and amplitude e, we get the model

which can demonstrate SNA. In the following figure, we show a chart of regimes in the parameter plane (l, e), and on periphery examples of iteration diagrams for attractors at several characteristic points. Yellow, orange and light blue areas are those of torus of type Т1, Т2, Т4, which are depicted by a respective number of closed curves in iteration diagrams. Green designates SNA, and dark blue - chaos.

Model with tangent bifurcation

Let us consider a map x_n+1=f(x_n)+b, where a function f is selected in such way that a tangent bifurcation occurs: in a course of increase of b two fixed points, one stable and another unstable, get closer, collide and disappear, leaving a narrow 'channel', through which an orbit travels for a long time. If the dynamics after the passage is of such kind that a return to the input of the channel is ensured, a type of dynamical behavior known as intermittency (Pomeau and Manneville, 1980) takes place. To perform a local analysis of the bifurcation, we may exclude the reinjection mechanism and consider function f(x)=x/(1–x). Assuming that the parameter b oscillates quasiperiodically with amplitude e, we come to the model

It is appropriate for a study of the transition, which is an analog of the tangent bifurcation al gives birth to SNA, but for description of the SNA itself it is needed in a modification introducing the reinjection. It is interesting that a variable change x_n=1–Y_n/Y_n–1 reduces the equation to a linear difference equation of Harper

known in solid-state physics in the context of analysis of localization and delocalization of quantum states in one-dimensional model of quasiperiodic medium.

Circle map with quasiperiodic driving

The circle map is defined as x_n+1=x_n+r+(K/2p)sin2px_n, where the variable x_n is defined up to an integer part ('modulo 1'). It may be interpreted as an equation for self-oscillator under periodic pulses: x_n is a phase for the oscillations just before the n-th kick, K characterizes intensity of the kicks, and r – detuning of the frequency of pulses and of self-oscillations in the device. With a term corresponding to an additional external driving with incommensurate frequency we get a model

Without the additional term, in the parameter plane (r,K) one can see the Arnold tongues, the regions of periodic dynamics, or 'mode locking' (gray), quasiperiodicity between them (white), and chaos (blue), see the left diagram.

The bottom part of the diagram, where the regular dynamics take place, and the top one, where chaos is possible, are separated by the critical line K=1 (dotted). In presence of the quasiperiodic force, instead of the critical line one observes a critical zone (Ding et al., Phys.Rev. A39, 1989, 2593). Its width grows with the amplitude of the driving (see the right diagram). The Arnold tongues transform into zones of two-frequency quasiperiodic dynamics (one is the frequency of steps of discrete time, and the second - the frequency of driving). Regimes between the tongues correspond now to the three-frequency quasiperiodicity, or three-dimensional tori (a frequency of 'rotation' of the cyclic variable x is the third one). These two types of regular dynamics take place below the critical zone. Above it the chaotic dynamics occur. Finally, in the critical zone itself, in dependence of parameters one can observe either two-frequency quasiperiodicity, or SNA (see an example in the figure below for e=0.6, K=1, r=0.3, w=(5^1/2-1)/2).