E. Atlee Jackson, Perspectives of Nonlinear Dynamics, Cambridge University Press, Cambridge, 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....This system can be viewed, as in Fig. 1, where the hysteresis is realized by the fast bistable subsystem in the first equation. The van der Pol system was originally used to model a tunnel diode circuit, while analogous models have been used to study synchronization in biological systems [6], 8] In fact, the same hysteresis feedback paradigm for an oscillatory system has been extensively used in designing a variety of chemical oscillators [3] 9] In this paper, we consider relay relaxation oscillators which consist of a relay hysteresis (cf. 10, p. 262] and a linear system in ....

....Synchrony is a state wherein all oscillators in the network oscillate with the same phase. In phase locking, each oscillator maintains a constant phase difference with respect to the others. This subject has been studied in great detail in many physical, chemical and biological systems (see [6] [8] 27] 36] 37] and the references therein) In the remainder of the paper, we focus on a similar analysis for the relay relaxation oscillator. We treat in detail the simple but representative case where the linear system consists of an integrator (see Fig. 9) We demonstrate entrainment ....

E. A. Jackson, Perspectives of Nonlinear Dynamics. New York: Cambridge Univ. Press, 1991, vol. 1--2.

....Mathematically, this question is addressed by Topological Orbital Equivalence (TOE) which tests whether one dynamical system can be continuously transformed into another one. A formal way of establishing TOE is to find 11 an orientation preserving homeomorphism between two dynamical systems (Jackson, 1989; Arnol d, 1983) The following scaling relation h : h : x B,n = c x B,n x P,n = c x P,n x P,n = c 2 x P,n t n = ct n ( t = ct) c 0 (7) fulfills the requirements of TOE for Equation 3. For any constant, c , the primed variables also fulfill Equation 3, which can be verified ....

Jackson, E. A. (1989). Perspectives of nonlinear dynamics, Vol.1. New York: Cambridge University Press.

....[22] for a review on activity dependent neural network development. 3 Turing patterns In 1952 Alan Turing published a paper [21] showing how patterns might grow from a nearly homogeneous situation and how di usion could drive an instability. In the following, I am borrowing the description from [11]) Turing considered a one dimensional chain of identical cells (k = 1; 2; N) containing various chemicals he called morphogens (form producers) If there are m morphogens, the internal dynamics of each cell is controlled m coupled linear di erential equations, while the coupling between the ....

Jackson, E. A. Perspectives of Nonlinear Dynamics, vol. 2. Cambridge University Press, 1990.

....F i do not depend explicitly on time, the system is said to be autonomous, otherwise it is non autonomous. In a non autonomous system, more than one solution, say y (1) t) y (2) t) can pass through the same point of the phase space (i.e. y (1) t ) y (2) t ) for some t ) (Jackson, 1991). To ensure that only one solution passes through each point, we will consider only autonomous 6 systems y = F (y) y 2 n (6) The general solution y(t; y 0 ) of (6) with y 0 2 n at a xed time t can be viewed as a continuous mapping in the phase space Y t : n n , which ....

Jackson E. A. 1991. Perspective of nonlinear dynamics. Cambridge University Press.

....the schemes in software. Also, while for most common chaotic maps there are numerous exact results guaranteeing aperiodic, chaotic sequences for parameters from a set of nonzero Lebesgue measure, we cannot directly transfer the results to computer approximations. It has been pointed out by Jackson [1991] and Wheeler [1989] that computer implementations of chaotic maps can exhibit surprisingly different behavior, e.g. very short cycles, depending on the particular numerical representation. While it is probably true that the typical behavior of finite approximations of chaotic systems should ....

Jackson, E. A. [1991] Perspectives in Nonlinear Dynamics, Vol. 2 (Cambridge Univ. Press, Cambridge), p. 33.

....jy0 Gammax 0 j 0 log jy n Gamma xn j jy 0 Gamma x 0 j = lim n 1 1 n n X j=1 log fi fi fi fi df(x j ) dx j fi fi fi fi (8) where is the Lyapunov exponent. This quantity is often calculated using log 2 since is then in units of bits of information. For an introduction, see [Jackson 1989]. 8 The Lyapunov exponent is a measure of the local rate of spreading, averaged over the actual trajectory of the system. There are three possibilities for : 0, the system is periodic; 0, marginally stable; and 0, chaotic. For any system, there are as many Lyapunov exponents as ....

Jackson, E. A. 1989, Perspectives of Nonlinear Dynamics, (Cambridge: Cambridge University Press)

.... It is by now well established that low order dynamical systems constitute an alternative to conventional numerical methods as we strive to understand the nonlinear behavior of flow [14] The simplicity of the Lorenz equations and the rich sequence of flow phenomena exhibited by their solution [3, 4, 16] have been the major contributing factors to their widespread use as a model for examining the onset of chaotic motion. Despite the severe level of truncation involved in the formulation of these equations, some of the basic qualitative elements of the onset of thermal convection and the ....

Jackson, E.A., Perspectives of Nonlinear Dynamics, vols. 1 and 2, Cambridge University Press, 1991.

....Pol equation (VDPE) u u (u 2 Gamma 1) u u = 0 (5) arises as a typical model of self sustained oscillations in many physical, chemical and biological applications. Much work has been done to investigate the properties of the harmonically driven VDPE theoretically and experimentally (see Jackson, 1990, for recent review) Very detailed bifurcation analysis of the sinusoidally driven VDPE is given by Mettin et al. 1993) However, much less interest has caught the VDPE with constant bias p u , u u (u 2 Gamma 1) u u p u = 0 (6) Abraham and Simo, 1986) An asymmetric VDPE v v ....

E. A. Jackson. Perspectives of Nonlinear Dynamics, vols. 1, 2. Cambridge Univ. Press, 1990.

....cusp singularity, the vector field above is also and equivalently a versal unfolding of a vector field invariant under rotation of the plane by 2 =q with q = 1. Apart from the original references given above, the system is considered in textbooks by Arnold [1983] Guckenheimer Holmes [1983] Jackson [1989] , Arrowsmith Place [1990b] and Wiggins [1990] amongst other places. The system of Eq. 4) allows the fixed points to move relative to each other and eventually coalesce at a saddle node bifurcation. If we restrict our attention to the region away from the saddle node bifurcations where ....

Jackson, E. A. [1989] Perspectives of Nonlinear Dynamics, vol. 1 (Cambridge University Press).

....Lyapunov dimension. Estimates which give the shape of the attractor are important as they lead to a good upper bound on the dimension of the Lorenz attractor [6] Until now, approximate locations of the Lorenz s attractor in the phase space have been obtained by the method of Lyapunov functions [1, 5, 6, 8, 9]. Very recently, thanks to this method, it has been shown that the global attractor of the Lorenz equations is contained in a volume bounded by a sphere, a cylinder, the volume between two parabolic sheets, an ellipsoide and a cone [6] In this paper, we apply a di#erent method for obtaining ....

....in the zone of the phase space where : z # 2 3b b 2 c2 2# b# 2r# 2# x 2 1 4# x 4 (2 b)xy #y 2 x 2 k 3 (21) for Lorenz s canonical values k 3 = 1131.56) We have also found several semipermeable families of ellipsoids. In fact, we have generalised results given in [6, 5, 9]. Surfaces like S = c 3 r # x 2 y 2 (z c 3 ) 2 = R (22) are semipermeable for the following cases : c 3 r for arbitrary #, b, r and for values of R as in fig. 10. The interest of 3 2 ) c 3 2 b 2 R 4 (b 1) c 3 2 R c 3 2 c 3 2 c ) 2 ( b 2 ( R 1 s b Figure ....

Jackson, E.A., Perspectives of Nonlinear Dynamics (Cambridge University Press, 1990).

....Because all bounded orbits are necessarily periodic on a computer, true chaotic behaviour seems out of reach. 388 Chaitin Chatelin F. Is Finite Precision Arithmetic Useful For Physics In order to examine the basis for this pessimism, we introduce the lattice logistic considered in Jackson (1991, pp. 216 221) The variable x in [0; 1] is discretized by (j) j N 1 ; j = 1; Delta Delta Delta ; N and f(x) is approximated by Phi( 1 N 1 b(N 1)r (1 Gamma )c ; 2) where bac denotes the integer part of a and 2 Phi (1) Delta Delta Delta (N) Psi . The ....

E. A. Jackson, (1991), Perspectives of nonlinear dynamics, Cambridge University Press, Cambridge. Tome 1.

.... of laissez faire is when not only the network is fine grained and uniform, but the initial data are random (at least on a short scale) In this case, the behavior that emerges can only be the macroscopic expression of the microscopic law built into the node, i.e. is an attractor of the dynamics[32]. Though the attractors are in principle completely determined by the microscopic dynamics, their specific form is not easibly deducible from it; the whole point of the computation is to make the attractors manifest (cf. x2.3 and Fig. 11) In terms of applications, emergent computation is relevant ....

Jackson, E. Atlee, Perspectives of nonlinear dynamics, Cambridge Univ. Press, 1991.

....lies in the random selection of one of two possible periods for each relaxation oscillation, so this picture, which shows only part of one cycle, cannot display chaos. This forced van der Pol equation exhibits chaotic behaviour (see, for example, Tomita [1986] Thompson Stewart [1986] or Jackson [1989]) and is also stiff, as can be seen in Fig. 8. The presence of fast and slow time scales in a problem is a characteristic of stiffness. Stiff problems are not mere curiosities, but are common in dynamics and elsewhere [Aiken, 1985] When integrating a stiff problem with a variable step ....

Jackson, E. A. [1989] Perspectives of Nonlinear Dynamics, vol. 1 (Cambridge University Press).

....dt = f i x j x(t) z(t) 15) where z(t) is the perturbation of the path x(t) In this case, we know that jz(t)j A exp( 1 t) or 1 (x) lim t 1 1 t ln jz(t)j: 16) A straightforward numerical implementation of (16) will result in an accurate approximation of 1 . Refer to Jackson[10] for a discussion on how to approximate the remaining i . 2.3 Features of Lyapunov Exponents There are two important features of Lyapunov exponents that limit their usefulness as a practical tool for characterizing chaotic systems. Firstly, Lyapunov exponents are local in nature. Lyapunov ....

....of a system. If an experimenter can only measure the state of a system to a finite degree of accuracy, then topological entropy measures the rate at which the experimentor can gain more information about the system as it evolves over time. 3. 1 The Calculation of Topological Entropy (Jackson[10]) Consider a compact space X and a continuous map f that takes the space into itself, f : X X (26) Denote by A = fa i g a cover of X if a i 2 X 8 i; X ae [ i a i : 27) This cover is a partition if (a i a j = 8 j; k) Denote by N (A) the number of regions in the partition. We now ....

E. Atlee Jackson. Perspectives of nonlinear dynamics, Volume II. Cambridge University Press, Cambridge, Great Britain, 1991.

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E. Atlee Jackson, Perspectives of Nonlinear Dynamics, Cambridge University Press, Cambridge, 1991.

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E. A. Jackson, Perspectives of Nonlinear Dynamics. Cambridge, U.K.: Cambrige Univ. Press, 1989.

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E. Atlee Jackson. Perspectives of Nonlinear Dynamics, volume 2. Cambridge University Press, 1990.

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Jackson, E.A., Perspectives of nonlinear dynamics, 2 vols., Cambridge University Press (1991).

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Jackson, E.A. [1991] Perspectives in Nonlinear Dynamics (Cambridge University Press, New York).

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Jackson, E.A. (1991) Perspectives of Nonlinear Dynamics, Cambridge University Press, Cambridge.

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JACKSON E.A. Perspective of nonlinear dynamics, Cambridge University Press, (1991).

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Jackson, E.A., Perspectives of nonlinear dynamics, 2 vols., Cambridge University Press (1991).

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Jackson, E.A. (1991) Perspectives of Nonlinear Dynamics, Cambridge University Press, Cambridge.

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JACKSON E.A. Perspective of nonlinear dynamics, Cambridge University Press, (1991).

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Jackson, E.A. (1979). Perspectives of Nonlinear Dynamics, Vol. 1. Cambridge University Press.

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