By L. Schimansky-geier, B. Fiedler, J. Kurths, E. Scholl

Nonlinear dynamics of advanced techniques is an lively examine box with huge numbers of guides in uncomplicated study, and large functions from varied fields of technological know-how. Nonlinear dynamics as manifested through deterministic and stochastic evolution versions of complicated habit has entered statistical physics, actual chemistry, biophysics, geophysics, astrophysics, theoretical ecology, semiconductor physics and -optics, and so on. This box of analysis has triggered a brand new terminology in technological know-how hooked up with new questions, difficulties, suggestions and techniques. New situations have emerged for spatio-temporal constructions in dynamical platforms faraway from equilibrium. Their research and attainable keep watch over are exciting and hard points of the present study. The duality of primary and utilized examine is a focus of its major attractivity and fascination. easy themes and foundations are consistently associated with concrete and particular examples. types and measurements of advanced nonlinear techniques evoke and impress new basic questions that diversify and increase the mathematical options and instruments. In go back, new mathematical methods to modeling and research amplify the scope and potency of utilized study.

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**Sample text**

The spike rate of (z) increases at first but then the synchronization between the individual elements is lost and the spike rate decreases again. 1 x lop4. 1. [41] to the diameter of the small amplitude oscillations. For further increasing noise the chaotic attractor quickly but continuously increases in size. This phenomenon resembles the well-known Canard explosion. Here though, it is not a single limit cycle whose size increases quickly but the chaotic attractor as a whole. The moment dynamics method allows us to quickly discover the bifurcation diagram.

Lo5 5. lo5 I Fig. 3. The output z ( t ) of a bistable system as described by eq. 025. The reduced two state description is illustrated by the gray line: s ( t ) 0. 50 T . Prager and L . Schimansky-Geier In order to simplify the description of this system one neglects the fast dynamics in the potential wells and considers only the transitions from one well to the other which happen on a much slower time scale. Under the assumption that the potential barrier AU between the two wells is large compared t o the noise strength D and the relaxation in the wells is fast compared to the time scale of the jumps between the wells, the transitions can be considered as a rate process.

Eq. 3 reveals, that the mean time between phase slips is determined by two effects, namely a drift part, caused by different frequencies and a diffusion part which persists even if the mean frequency of the system agrees with the signal frequency. This diffusive part is due to the stochastic nature of the system. e. Au + 0 the mean locking episodes are solely determined by the effective diffusion coefficient Synchronization in periodically driven discrete systems 47 On the other hand if the effective diffusion coefficient vanishes, as is the case for deterministic systems, we have 1 Au' Stochastic m : n synchronization can thus be characterized as an adjustment of frequencies, fl M $ u and a reduction in the diffusivity of the pro.