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8. Selected nonlinear phenomena

8. Selected nonlinear phenomena. 8.1 Robustness and limit cycles Though sensitive to the initial conditions, typical feature of nonlinearsystems is the existence of stable limit cycles which is often robust against all perturbations.

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8. Selected nonlinear phenomena

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  1. 8. Selected nonlinear phenomena • 8.1 Robustness and limit cycles • Though sensitive to the initial conditions, typical feature of nonlinearsystems is the existence of stable limit cycles which is often robust against all perturbations. • Data of limit cycle type lead to two different difficulties. • If data is restricted by limit cycles, it is not possible to construct unique dynamical nonlinear model ( they fill a very small part of phase space). • Limit cycles may be of different complexity. Noise driven limit cycle may be easily mistaken for chaotic motion. • 8.2 Coexistence of attractors • Repetition of the experiment for nonlinear systems with the same parameters may yield different results. • The region of initial conditions on which the trajectory will settle down is called the basin of attractor.

  2. 8. Selected nonlinear phenomena • Unless reset externally during the observation period, a single time series can represent only one possible attractor. • 8.3 Transients • An attractor in dissipative nonlinear systems has Lebesgue measure zero in phase space. Therefore the probability that an arbitrarily chosen intial conditions already lies on the attractor is zero. • During transient time the attractor has entirely different properties than the attractor itself. • Long transients may be a problem for time series work since dynamics is non stationary. • 8.4 Intermittency • Intermittency means that the signal alternates between periodic and chaotic behavior in irregular fashion. • The alternation between two different dynamical regimes involves new time scales related to the duration of two phases.

  3. 8. Selected nonlinear phenomena • Chaotic bursts due to rare appearances have insufficient information about them. Inside the bursts the dynamics is much faster. The in homogeneity of the series may have additional difficulties with some methods of analysis. • Unbiased dimension estimation becomes difficult. In order to study intermittency we need a record of many switching events. • 8.5 Structural stability • Feature of nonlinear systems is their lack of robustness against slight changes of parameters. • Structural stability means that every tiny perturbation of any of the system parameters can be compensated by smooth transformation of the variables. • The system with perturbed parameters is conjugate of the original one. • Severe problems can arise when we try to setup model equations for structurally unstable systems.

  4. 8. Selected nonlinear phenomena • 8.6 Bifurcations • Bifurcations are the abrupt changes of the attractor geometry or even topology at a critical value of the control parameter. This may happen due to change of stability. • One of the striking features of bifurcations is their universality. • Period – doubling bifurcation, pitch fork bifurcation, Feigenbaum bifurcation, tangent bifurcation and Hopf bifurcation are the various types.

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