Limit cycles examples

Stability, Bifurcations, Limit Cycles Some aspects of this subject involve the solution of nonlinear equations other aspects involve the integration of ordinaiy differential equations apphcations include chaos and fractals as well as unusual operation of some chemical engineering eqmpment. Ref. 176 gives an excellent introduction to the subject and the details needed to apply the methods. Ref. 66 gives more details of the algorithms. A concise survey with some chemical engineering examples is given in Ref. 91. Bifurcation results are closely connected with stabihty of the steady states, which is essentially a transient phenomenon. 

Chapter 5 provides some examples of purely analyti( al tools useful for describing CA. It discusses methods of inferring cycle-state structure from global eigenvalue spectra, the enumeration of limit cycles, the use of shift transformations, local structure theory, and Lyapunov functions. Some preliminary research on linking CA behavior with the topological characteristics of the underlying lattice is also described. 

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... 

One can easily find differential equations having limit cycles as solutions. For example, a system ... 

In this example the situation is a little more complicated. In fact, here we have three roots the root p0 = 0 and the roots plt 2 = pj2 VOp2/4) - ff- For stability, limit cycles, and one ascertains easily that the smaller of these two roots is unstable and the larger is stable the state of rest is also stable. We have thus the configuration SU8 of the notation used previously. 

Example 14.9 This example cites a real study of a laboratory CSTR that exhibits complex d5mamics and limit cycles in the absence of a feedback controller. We cite the work of Vermeulen, and Fortuin, who studied the acid-catalyzed hydration of 2,3-epoxy-1-propanol to glycerol ... 

Edgar The endoreplication cycles might be a better example of growth-limited cycles. Christian Lehner has done experiments in which cyclin E is overexpressed in endoreduplicating cells. They get stuck in Gl. We have then expressed Myc in these Gl-arrested cells it doesn t make them grow much. They absolutely require endoreplication for growth. 

The experiments and the simulation of CSTR models have revealed a complex dynamic behavior that can be predicted by the classical Andronov-Poincare-Hopf theory, including limit cycles, multiple limit cycles, quasi-periodic oscillations, transitions to chaotic dynamic and chaotic behavior. Examples of self-oscillation for reacting systems can be found in [4], [17], [18], [22], [23], [29], [30], [32], [33], [36]. The paper of Mankin and Hudson [17] where a CSTR with a simple reaction A B takes place, shows that it is possible to drive the reactor to chaos by perturbing the cooling temperature. In the paper by Perez, Font and Montava [22], it has been shown that a CSTR can be driven to chaos by perturbing the coolant flow rate. It has been also deduced, by means of numerical simulation, that periodic, quasi-periodic and chaotic behaviors can appear. 

Figure 1. In most examples of biological rhythms, sustained oscillations correspond to the evolution toward a hmit cycle. The limit cycle shown here was obtained in a model for circadian oscillations of the PER protein and per mRNA in Drosophila [107].

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. 

Within the region of order (k < 0) we see intricate symmetric and non-symmetric limit cycles in phase diagrams. For example, for 7j =4.1 we see in... 

Fig. 16a symmetric limit cycles for the second-harmonic mode (GCL) and in Fig. 16b, an nonsymmetric phase portrait example for 7) = 0.5 for BCL. In both cases the phase point settles down into a closed-loop trajectory, although not earlier than about x > 200. An intricate limit cycle is usually related to multiperiod oscillations. For example, the cycle in Fig. 16a corresponds to five-period oscillations of the fundamental and SHG modes intensity, and the phase portrait in Fig. 16b resembles the four-period oscillations (see Fig. 17). Generally, for 7) > 0.5, we observe many different multiperiod (even 12-period) oscillations in intensity and a rich variety of phase portraits.

The single Kerr anharmonic oscillator has one more interesting feature. It is obvious that for Cj = 0 and y- = 0, the Kerr oscillator becomes a simple linear oscillator that in the case of a resonance 00, = (Do manifests a primitive instability in the phase space the phase point draws an expanding spiral. On adding the Kerr nonlinearity, the linear unstable system becomes highly chaotic. For example, putting A t = 200, (D (Dq 1, i = 0.1 and yj = 0, the spectrum of Lyapunov exponents for the first oscillator is 0.20,0, —0.20 1. However, the system does not remain chaotic if we add a small damping. For example, if yj = 0.05, then the spectrum of Lyapunov exponents has the form 0.00, 0.03, 0.12 1, which indicates a limit cycle. 

The reduction of chaos for 9 = 1.45 is presented in the intensity portraits of Fig. 39. However, as is seen in Fig. 38a, there is a small region (1.68 < 9 < 1.80) where the system behaves orderly in the classical case but the quantum correction leads to chaos. By way of an example for 9=1.75, the classical system, after quantum correction, loses its orderly features and the limit cycle settles into a chaotic trajectory. Generally, Lyapunov analysis shows that the transition from classical chaos to quantum order is very common. For example, this kind of transition appears for 9 = 3.5 where chaos is reduced to periodic motion on a limit cycle. Therefore a global reduction of chaos can be said to take place in the whole region of the parameter 0 < 9 < 7. 

For the particular example in Fig. 9.10, the Hopf point occurs for /i0 1.105. The two turning points are located at n0 = 0.72 and 0.636. This means that for reactant concentrations in the range 0.72 < fi0 < 1.105, the system has a unique stationary-state profile which is unstable. Under such conditions, the reaction will exhibit time-dependent as well as spatially dependent solutions, i.e. there is a limit cycle. Some representative non-stationary profiles are shown in Fig. 9.12. 

Figure 13.10 shows a representation of the phase plane behaviour appropriate to small-amplitude forcing. There are two basic cycles which make up the full motion first, there is the natural limit cycle, corresponding for example to Fig. 13.9(a) around which the unforced system moves secondly, there is a small cycle, perpendicular to the limit cycle, corresponding to the periodic forcing term. The overall motion, obtained as the small cycle is swept around the large one, gives a torus and the buckled limit cycle oscillations at low rf in Fig. 13.9 draw out a path over the surface of such a torus. 

The oscillations produced by a frequency generator are of this type and have inspired Van der Pol to construct his classic example of a differential equation having a limit cycle. The best known example in chemistry is the Zhabotinskii reaction. In biology many periodic phenomena are known that can presumably be described in this way. ... 

In the case of a chemical clock, the asymptotic (f -A oo) solution depends on time, there are not only singular points but also singular trajectories. An example is the stable limit cycle - Fig. 2.4, i.e., a closed trajectory to which all phase trajectories existing in its vicinity strive. 

Of interest is the study of conditions under which such a limit cycle emerges in a system. The chemical clock serves as an example of the so-called temporary structures study which was stimulated by a fundamental problem of order emerging from chaos. In the last decade it became a central part of a new discipline called synergetics [1, 21, 22],... 

The principal role of diffusion in these processes could be established considering rather simple examples [2]. If the kinetic equations for a well-stirred system are able to reproduce self-oscillations (the limit cycle), the extended system could be presented as a set of non-linear oscillators continuously distributed in space. Diffusion acts to conjunct these local oscillations and under certain conditions it can result in the synchronisation of oscillations. Thus, autowave solutions could be interpreted as a result of a weak coupling (conjunction) of local oscillators when they are not synchronised completely. The stationary spatial distributions in an initially homogeneous systems can also arise due to diffusion, which makes homogeneous solutions unstable. 

When the forcing amplitude is very small and the midpoint of the forcing oscillation scans the autonomous bifurcation diagram, the qualitative response of the forced system for all frequencies can be deduced from the autonomous system characteristics. As the amplitude of the forcing becomes larger, one cannot predict a priori what will occur for a particular system. For this example, the most complicated phenomenon possible is a turning point bifurcation on a branch of periodic solutions where two limit cycles, one stable and one unstable, collide and disappear. This will appear as a pinch on the graph of the map [Fig. 1(d)],... 

This work is centred around the study of the response to periodic forcing of systems that, when autonomous, had a stable limit cycle surrounding an unstable steady state in their phase plane. For the sake of simplicity—and since many of the fundamental phenomena are the same—we studied two-dimensional systems. We chose two examples of isothermal reactor models the first is an autocatalytic homogeneous Brusselator (Glansdorff and Prigog-ine, 1971) ... 

The model that will be used for forced oscillation studies is one which was first proposed by Takoudis et al. (1981) as a simple example of an isothermal surface reaction without coverage dependent parameters in which limit cycles can occur. The bimolecular reaction between species A and B is presumed to occur as a Langmuir-Hinshelwood bimolecular process except that two adjacent vacant sites on the surface are required for the reaction to take place. 

For even larger values of hf, the system eventually reaches a unique fixed steady state that is stationary and involves no limit cycle at all, just as we have seen to be the case for small values of hf in Figure 4.52. For example, for hf = 0.0065, the phase plot starts at sn = 1.27 and S12 = 0.2 in the bottom plot of Figure 4.63 and moves in two spiral loops toward the asymptotic steady state with sn 1.285 and si2 0.17, as depicted in Figure 4.63. 

Fig. 1. Examples of w-limit sets, (a) Rest point (b) limit cycle (c) Lorenz attractor [projection on the (Cj, c3) plane, a = 10, r = 30, b = 8/3].

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