Limit cycle attractor

Suppose that an experimental system has a limit-cycle attractor. Given that one of its variables has a time series x(t) = sin r, plot the time-delayed trajectory x(Z) = (x(f),x(f -t- T)) for different values of T. Which value of T would be best if the data were noisy ... 

Quasi-periodicity arises if there are two or more disjoint stable limit cycle attractors in which none of the variables of one of the cycles receive inputs from the variables involved in the other cycles, and the periods of all cycles are noncommensurate. 

Limit cycles in networks of dimension as small as four can be surprisingly long and complicated, even when the focal point coordinates are all 1. For example, the network with logical structure depicted in Fig. 4 has an asymptotically stable hmit cycle in which 174 switches take place before the cycle repeats. Projections of the phase portrait of this limit cycle attractor are shown in Fig. 5. The existence and stability of this limit cycle were confirmed analytically by the methods outlined above. We have found four-dimensional networks with confirmed limit cycles involving as many as 252 switches before repeating. Gedeon [48] has constructed four-dimensional networks with focal point coordinates tuned so that limit cycles of arbitrary length can be obtained. 

The phase space representation of trajectories computed numerically, as described above, has been introduced in another chapter of this volume. TTie systems considered there are Hamiltonian systems which arise in chemistry in the context of molecular dynamics problems, for example. The difference between Hamiltonian systems and the dissipative ones we are considering in this chapter is that, in the former, a constant of the motion (namely the energy) characterizes the system. A dissipative system, in contrast, is characterized by processes that dissipate rather than conserve energy, pulling the trajectory in toward an attractor (where in refers to the direction in phase space toward the center of the attractor). We have already seen two examples of attractors, the steady state attractor and the limit cycle attractor. These attractors, as well as the strange attractors that arise in the study of chaotic systems, are most easily defined in the context of the phase space in which they exist. 

In dissipative systems, many trajectories with different choices ot initial conditions will be attracted to the same region in phase space and end up, asymptotically, on the same attractor. The phase space portrait, then, will usually consist only of the asymptotic state, that is, a trajectory that represents the final state for many different initial conditions. This asymptotic trajectory traces over the attractor and reveals its shape. Figure 22(a) shows a steady state attractor, whereas Figure 22(b) shows a limit cycle attractor. 

A Poincare surface of section is defined for dissipative systems in the same way as for Hamiltonian ones but, again, will look somewhat different because the phase space trajeaory consists only of the asymptotic state, a single attractor. To construct the Poincare seaion, the phase space portrait is cut with a surface to create a cross-sectional view of the attractor. Hence, for a simple limit cycle attractor which is a single loop in phase space, the Poincare section consists of a single point. For a more complex attractor, the Poincare section will be more elaborate, as we will see. 

Figure 22 Examples of attractors (a) shows a steady state attractor (here, a node), whereas (b) shows a limit cycle attractor.

Chaotic behavior in nonlinear dissipative systems is characterized by the existence of a new type of attractor, the strange attractor. The name comes from the unusual dimensionality assigned to it. A steady state attractor is a point in phase space, whereas a limit cycle attractor is a closed curve. The steady state attractor, thus, has a dimension of zero in phase space, whereas the limit cycle has a dimension of one. A torus is an example of a two-dimensional attractor because trajectories attracted to it wind around over its two-dimensional surface. A strange attractor is not easily characterized in terms of an integer dimension but is, perhaps surprisingly, best described in terms of a fractional dimension. The strange attractor is, in fart, a fractal object in phase space. The science of fractal objects is, as we will see, intimately connected to that of nonlinear dynamics and chaos. 

Another route to chaos that is important in chemical systems involves a torus attractor which arises via bifurcation from a limit cycle attractor. Again chaos is found to be associated with periodic behavior and to arise from it through a sequence of transformations and associated bifurcations of a periodic state of the system. The specific sequence is different in this case, however, and somewhat more complex. 

By integrating particular examples, it is discovered that the cycles in Eq. (22) can be of two types, unstable cycles, in which the trajectories spiral into an intersection point of the threshold axes, and stable cycles, in which there is a stable limit cycle attractor in concentration space. Unstable oscillations are found for Eq. (22) in structure IV, Fig. 3, and stable limit cycle oscillations are found for Eq. (22) in Fig. 4 (see Section V.3). It is not known whether other types of asymptotic behavior besides extremal steady states and stable and unstable cycles are possible in Eq. (22). 

Fig. 7. (a) The amplitudes and (b) the period T of the oscillations found from integrating Eq. (27), N= 3, for varying values of n. Except for the value n = 4, global limit cycle attractors were found. The arrows on the right-hand side of the diagrams represent the theoretical values for the piecewise linear equation in the limit The arrow on the left-hand ride of (b) is the period predicted by the Hopf bifurcation theorem. 

Figs. 3d and 4, since it represents the cyclic attractor found in the state transition diagram of Eq. (27), iV = 4. For any number of dimensions there will always be a cyclic attractor through 2N volumes corresponding to the cyclic attractor found for Eq. (27). Numerical integration of Eq. (27) for n = 8, N = 5,6, 7 has indicated stable limit cycle attractors for each case where both the period of the oscillation and the amplitude increase as N increases. 

For the trajectories defined in Equation (51) it is possible to prove that there is a unique limit cycle attractor. If p, q are two points on the boundary between regions I and II, and the distance between the two points d[p, ] = /, it is possible to show that the distance between the images of these points on the boundary between regions II and III is smaller than 4 II 20 + 5). If f ip) is the map that takes point p on the border between regions I and II back to this border after one complete circuit through the four regions, we compute... 

Fig. 9. Phase portrait illustrating the destruction of the limit cycle attractor by the inhomogeneous fluctuations under the conditions of Figure 7 and 8.

The effects of internal noise on a limit-cycle attractor in a well-stirred system are shown in Figure 4, which compares the period-2 attractors obtained from the automaton simulation for two different system sizes (middle and right panels) with the deterministic period-2 orbit (left panel). For small system sizes, below about Af = (300), the noisy attractor bears little resemblance to the deterministic period-2 orbit (cf. right panel rather it looks like a noisy... 

Attractors can be simple time-independent states (points in F), limit cycles (simple closed loops in F) corresponding to oscillatory variations of tire chemical concentrations with a single amplitude, or chaotic states (complicated trajectories in F) corresponding to aperiodic variations of tire chemical concentrations. To illustrate... 

The simplest possible attraetor is a fixed point, for which all trajectories starting from the appropriate basin-of-attraction eventually converge onto a single point. For linear dissipative dynamical systems, fixed-point attractors are in fact the only possible type of attractor. Non-linear systems, on the other hand, harbor a much richer spectrum of attractor-types. For example, in addition to fixed-points, there may exist periodic attractors such as limit cycles for two-dimensional flows or doubly periodic orbits for three-dimensional flows. There is also an intriguing class of attractors that have a very complicated geometric structure called strange attractors [ruelleSO],... 

In contrast to dissipative dynamical systems, conservative systems preserve phase-space volumes and hence cannot display any attracting regions in phase space there can be no fixed points, no limit cycles and no strange attractors. There can nonetheless be chaotic motion in the sense that points along particular trajectories may show sensitivity to initial conditions. A familiar example of a conservative system from classical mechanics is that of a Hamiltonian system. 

Since the phase space of a dissipative dynamical system contracts with time, we know that, in the long time limit, t oo, the motion will be confined to some fixed attractor, A. Moreover, becaust of the contraction, the dimension, D, of A, must be lower than that of the actual phase space. While D adds little information in the case of a noiichaotic attractor (we know immediately, and trivially, for example, that all fixed-points have D = 0, limit cycles have D = 1, 2-tori have D = 2, etc.), it is of significant interest for strange attractors, whose dimension is typically non-integer valued. Three of the most common measures of D are the fractal dimension, information dimension and correlation dimension. 

The spatial and temporal dimensions provide a convenient quantitative characterization of the various classes of large time behavior. The homogeneous final states of class cl CA, for example, are characterized by d l = dll = dmeas = dmeas = 0 such states are obviously analogous to limit point attractors in continuous systems. Similarly, the periodic final states of class c2 CA are analogous to limit cycles, although there does not typically exist a unique invariant probability measure on... 

Yet another type of complex oscillatory behavior involves the coexistence of multiple attractors. Hard excitation refers to the coexistence of a stable steady state and a stable limit cycle—a situation that might occur in the case of circadian rhythm suppression discussed in Section VI. Two stable limit... 

Controlled chaos may also factor into the generation of rhythmic behavior in living systems. A recently proposed modeL describes the central circadian oscillator as a chaotic attractor. Limit cycle mechanisms have been previously offered to explain circadian clocks and related phenomena, but they are limited to a single stable periodic behavior. In contrast, a chaotic attractor can generate rich dynamic behavior. Attractive features of such a model include versatility of period selection as well as use of control elements of the type already well known for metabolic circuitry. 

The frequency of modulation il is now the main parameter, and we are able to switch the system of SHG between different dynamics by changing the value of il. To find the regions of where a chaotic motion occurs, we calculate a Lyapunov spectrum versus the knob parameter il. The first Lyapunov exponent A,j from the spectrum is of the greatest importance its sign determines the chaos occurrence. The maximal Lyapunov exponent Xj as a function of is presented for GCL in Fig. 6a and for BCL in Fig. 6b. We see that for some frequencies il the system behaves chaotically (A-i > 0) but orderly Ck < 0) for others. The system in the second case is much more damped than in the first case and consequently much more stable. By way of example, for = 0.9 the system of SHG becomes chaotic as illustrated in Fig. 7a, showing the evolution of second-harmonic and fundamental mode intensities. The phase point of the fundamental mode draws a chaotic attractor as seen in the phase portrait (Fig. 7b). However, the phase point loses its chaotic features and settles into a symmetric limit cycle if we change the frequency to = 1.1 as shown in Fig. 8b, while Fig. 8a shows a seven-period oscillation in intensities. To avoid transient effects, the evolution is plotted for 450 < < 500. 

Stationary-state solutions correspond to conditions for which both numerator and denominator of (3.54) vanish, giving doc/dp = 0/0, and so are singular points in the phase plane. There will be one singular point for each stationary state each of the different local stabilities and characters found in the previous section corresponds to a different type of singularity. In fact the terms node, focus, and saddle point, as well as limit cycle, come from the patterns on the phase plane made by the trajectories as they approach or diverge. Stable stationary states or limit cycles are often refered to as attractors , unstable ones as repellors or sources . The different phase plane patterns are shown in Fig. 3.4. 

Another useful rule which can frequently guide us to situations where oscillatory solutions will be found is the Poincare-Bendixson theorem. This states that if we have a unique stationary state which is unstable, or multiple stationary states all of which are unstable, but we also know that the concentrations etc. cannot run away to infinity or become negative, then there must be some other non-stationary atractor to which the solutions will tend. Basically this theorem says that the concentrations cannot just wander around for an infinite time in the finite region to which they are restricted they must end up somewhere. For two-variable systems, the only other type of attractor is a stable limit cycle. In the present case, therefore, we can say that the system must approach a stable limit cycle and its corresponding stable oscillatory solution for any value of fi for which the stationary state is unstable. 

The limit cycle is an attractor. A slightly different kind occurs in the theory of the laser Consider the electric field in the laser cavity interacting with the atoms, and select a single mode near resonance, having a complex amplitude E. One then derives from a macroscopic description laced with approximations the evolution equation... 

A Jacobi or Gauss-Seidel iteration on (6) will provide us with the coordinates of the steady state, or it will cycle indefinitely, depending on the slope of the functions. On the other hand, determining the trajectory by numerical integration of equation (5) will lead to a stable steady state or to a limit cycle depending on the slope of the functions. There is thus an obvious formal similarity between the two situations. However, the steepness corresponding to the transition from a punctual to a cyclic attractor is much smaller in the first case (in which the cyclic attractor is an iteration artifact) as in the second case (in which the cyclic attractor is close to the real trajectory). 

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