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...ork is dynamically equivalent to a one-parameter full family of S-unimodal maps on the interval [0; 1], which is well-known to become chaotic through the period-doubling route as the parameter varies =-=[10, 13, 17]-=-. We note that the discrete-time neural networks are similar to cellular automata [11, 26, 42] but with infinite number of possible states. Much simple two-cell-state cellular automata with high w 21 ...

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...ll persist because trajectories cannot cross each other. In [ 12] this and the fact that this surface cannot intersect itself was used to deduce the chaos cannot occur in three-dimensional systems given by Eq. (2.1). Aperiodic dynamics may however be possible in higher dimensions. Now consider the intersection of iterates of a ray with a hypersurface S of dimension (n - 2) that is transverse to the rays in a Poincar6 section, Fig. 3. This projection preserves essential features of the dynamics. For instance, one can assume that S is a unit sphere, to generate a linear automorphism on a sphere [34], i.e. M r (X) = A x / I A x , I, where M r (X) is a reduced map. A ray is represented in this (n - 2)-dimensional surface $ as a point. That is, iterations give a sequence of points on this surface S which may approach the fixed point. The intersection of W S ( F i ) , from Eq. (4.6), with this surface S gives the representation of the stable manifold of a fixed point in S. The representation of the unstable manifold W~ in S, is defined by the intersection of span {Fi W~4(Fi)}, see Fig. 3. T. Mestl et al./Physica D 98 (1996) 33-52 43 To analyze our four-dimensional example, we will take the p...

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...nd 11), and Table 8. The ordinary CGL equation corresponds to the case F = W − (1 + iα)W |W |2,m = 2 and admits only the basic symmetries (5.1). • The nonautonomous dynamical systems in a phase space =-=[8]-=- ∂u1 ∂t − ∂2u1 ∂x2 −A(u1, u2) = h1(t, x), ∂u2 ∂t + α ∂u1 ∂t − ∂2u2 ∂x2 − νu1 = h2(t, x) (12.8) are also equivalent to a system of the type (1.3) at least in the case of constant h1 and h2. The related...

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...8. A typical example is the system generated by standard logistic map. M. Feigenbaum studied the dynamics of the logistic map for λ ∈ (0, 4]. Taking λ = λ∞ ≈ 3.570 the corresponding invariant set A ⊂ =-=[0, 1]-=- has both Hausdorff and box dimensions equal to ≈ 0.538 (Grassberger [3], Grassberger and Procaccia [4]). Here we compute precise values of box dimension of trajectories corresponding to period-doubli...

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...he famous set that bears Mandelbrot’s name. Peitgen et al. [30] undertake a further analysis of fractals. Mathematical analysis of complex iterations may be found in Peitgen and Richter [29], Devaney =-=[3]-=-, and Falconer [4]. While the fractal nature of the Julia sets corresponding to the individual members of the Basic Family follows from the Julia theory on rational iteration functions, that theory do...

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...ystems so far in biological modelings are near-equilibrium limit-cycle oscillations under weak couplings. In these cases, individual limit cycles can be regarded as resulted from the Hopf bifurcation =-=[6, 12]-=- from some equilibria, and the dominant effect of the coupling is to influence the phase of each oscillator around the limit cycle, without effecting its amplitude. So the collective synchronization p...