Lyapunov exponents bifurcated

Figure 9. The two largest Lyapunov exponents (a) and the bifurcation diagram (the maxima of yi) (b) versus the modulation parameter Q. Parameters are/o — 1. — y2 — 0.01 and the initial...

Figure 38. The calssical (dashed) and quantum (solid line) maximal Lyapunov exponents (a) and the appropriate bifurcation maps (b,c) versus the modulated parameter Q. The parameters are...

For periodic orbits that undergo a bifurcation, some Lyapunov exponents may vanish so that the orbit becomes of neutral linear stability in the critical directions [32]. In such cases, the dynamics transverse to the periodic orbit... 

Moreover, the Poincare mappings of (3.14) at values of Pc) fixed by the existence of (5, 6) and I4 + L5 = 0 show the presence of classically chaotic motions with a bifurcation at E - 6900 cm-1 (see Fig. 6). At this bifurcation, the periodic orbit (5, 6) becomes unstable because one of its Lyapunov exponents turns positive, as shown in Fig. 7. The periodic orbit (6, 7) destabilizes by a similar scenario around E - 7200 cm-1. These results show that the interaction between the bending modes leads to classically chaotic behaviors that destabilize successively the periodic orbits. For the bulk peri-... 

Differences between the lifetimes obtained from equilibrium point quantization and periodic-orbit quantization appear as the bifurcation is approached. The lifetimes are underestimated by equilibrium point quantization but overestimated by periodic-orbit quantization. The reason for the upward deviation in the case of periodic-orbit quantization is that the Lyapunov exponent vanishes as the bifurcation is approached. The quantum eigenfunctions, however, are not characterized by the local linearized dynamics but extend over larger distances that are of more unstable character. 

Thus, we speculate that, in the processes where qualitatively different flows emerge, the system would experience the situation where normal hyperbolicity breaks down. This speculation is based on the argument that, in order for the flows along the normal directions of NHIMs to bifurcate, one of the Lyapunov exponents of the normal directions must change its sign from plus to minus or from minus to plus. In the middle of these changes, normal hyperbolicity breaks down. 

Sacker bifurcation curve emerging from the Zero-Hopf point, (b) Parameter values, for which an attractor with positive largest Lyapunov exponent exists. 

Fig. 6.17. Behavior of the Lyapunov exponents, (a) Transverse Lyapunov exponents and (b) the largest longitudinal Lyapunov exponent. The parameters are the same as in Fig. 6.16. Bifurcation points A-E are explained in the text, (c) Zoom of the interval...

Fig. 15.9. Dynamics of the foodweb model (15.8) in the phase coherent regime as a function of the control parameter 6. (a) Bifurcation diagram, plotted are the maxima of w, (b) largest Lyapunov exponent A (c) mean frequency cj (solid line). Further indicated is the approximation wo b) = Vb (dotted line).

Computational techniques are centrally important at every stage of investigation of nonlinear dynamical systems. We have reviewed the main theoretical and computational tools used in studying these problems among these are bifurcation and stability analysis, numerical techniques for the solution of ordinary differential equations and partial differential equations, continuation methods, coupled lattice and cellular automata methods for the simulation of spatiotemporal phenomena, geometric representations of phase space attractors, and the numerical analysis of experimental data through the reconstruction of phase portraits, including the calculation of correlation dimensions and Lyapunov exponents from the data. 

Assuming Sxi = (5xo exp[(t - 1)A] for large i, we may compute A via A = limi cc lnX5,=o 4 (1 - 2x ). Figure .12 shows A = A(r). Negative values mean that the iteration approaches a stable fix point or limit cycle. Positive values mean that a small perturbation grows exponentially. We notice that the bifurcations are associated with A = 0. A is called Lyapunov-exponent. 

In this situation, a periodic variation of coolant flow rate into the reactor jacket, depending on the values of the amplitude and frequency, may drive to reactor to chaotic dynamics. With PI control, and taking into account that the reaction is carried out without excess of inert (see [1]), it will be shown that it the existence of a homoclinic Shilnikov orbit is possible. This orbit appears as a result of saturation of the control valve, and is responsible for the chaotic dynamics. The chaotic d3mamics is investigated by means of the eigenvalues of the linearized system, bifurcation diagram, divergence of nearby trajectories, Fourier power spectra, and Lyapunov s exponents. 

For cases having an extra degeneracy (for example an equilibrium state with zero characteristic exponent and zero first Lyapunov value) the boundary of the stability region may lose smoothness at the point There may also exist situations where the boimdary is smooth but bifurcations in different nearby one-parameter families are different (i.e. there does not exist a versal one-parameter family, for example, such as the case of an equilibrium state with a pair of purely imaginary exponents and zero first Lyapunov value). In such cases the procedure is as follows. Consider a surface 971 of a smaller dimension (less than (p — 1)) which passes through the point and is a part of the stability boundary, selected by some additional conditions in the above examples the condition is that the first Lyapunov value be zero. If (fc — 1) additional conditions are imposed, then the surface 971 will be (P fc)-dimensional and it is defined by a system of the form... 

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