Nonlinear Dynamics New Directions: Models and Applications by Hernán González-Aguilar, Edgardo Ugalde

By Hernán González-Aguilar, Edgardo Ugalde

This booklet, in addition to its better half quantity, Nonlinear Dynamics New instructions: Theoretical Aspects, covers themes starting from fractal research to very particular functions of the speculation of dynamical structures to biology. This moment quantity includes often new functions of the speculation of dynamical structures to either engineering and biology. the 1st quantity is dedicated to primary facets and encompasses a variety of very important new contributions in addition to a few evaluate articles that emphasize new improvement customers. the themes addressed within the volumes comprise a rigorous therapy of fluctuations in dynamical structures, subject matters in fractal research, stories of the brief dynamics in organic networks, synchronization in lasers, and keep watch over of chaotic structures, between others.

This publication also:

· Develops purposes of nonlinear dynamics on a range of subject matters reminiscent of styles of synchrony in neuronal networks, laser synchronization, regulate of chaotic platforms, and the learn of temporary dynamics in biological

· contains a examine of self-organized regularity in long-range systems

· Explains use of Levenstein's distance for measuring lexical evolution rates

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Then y¯ from (22) can run values from y¯min = λk (Cˆ cos (kψ + θ ) + O( ξ )) till y¯max = maxϕ(s)|s|≤ε1 + O(λk ). However, values of the coordinate y on the strip σk0 satisfy the inequality γ −k (y − − ε1 ) < y < γ −k (y − + ε1 ). Evidently, there are such δ0 > 0 and δ1 > 0 that (i) y¯max > δ0 for all sufficiently large k and (ii) for any ψ, since Cˆ > 0 and ξ is small, there are infinitely many such k that (Cˆ cos (kψ + θ ) + O( ξ )) < −δ1 . Thus, the first return map T1 T k for such values of k transforms the strip σk0 into the horseshoe Tk (σk0 ) such that its top is s posed below σk0 (and even below Wloc ) and the horseshoe intersects σk0 forming (at least) two connected components.

Phys. J. E 3(3), 205–219 (2000) 35. : Neural excitability, spiking, and bursting. Int. J. Bifurc. Chaos 10, 1171–1266 (2000) 36. : Transition between tonic spiking and bursting in a neuron model via the blue-sky catastrophe. Phys. Rev. Lett. 94(4), 048101 (2005) 37. : Mechanism of bistability: Tonic spiking and bursting in a neuron model. Phys. Rev. E 71, 056214 (2005) 38. : Coexistence of tonic firing and bursting in cortical neurons. Phys. Rev. E 74, 031922 (2006) 39. : Cooperative oscillatory behavior of mutually coupled dynamical systems.

It means that the system (0, u − u+ ) = ax + b(y − y − ), 0 = cx has a unique solution. Thus, condition C reads as b1 = 0, c = 0 in case A1 (14) b12 + b22 = 0, c12 + c22 = 0 in case A2. (15) or as Note that Fig. 5b and c correspond to the case A1 with b1 = 0 and c = 0, respectively. 4 On Simple Homoclinic Tangencies in the Sectionally Saddle Case In this section we consider, essentially, the multidimensional sectionally saddle case σ > 1. Concerning a type of the homoclinic tangencies, we assume in this section that they are isolated and one-sided.

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