R. Artuso, E. Aurell and P. Cvitanovi'c, Recycling of strange sets: I. Cycle expansions, Nonlinearity 3 (1990) 325-359.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....product (1.2) are purely formal. In general, they may not even exist without regularization. An exception is the nonchaotic 2 disk system, as it has only one periodic orbit, t 0 (k) Therefore, the semiclassical resonance poles are normally computed from ZGV (z=1; k) in the curvature expansion [6, 17, 5] up to a given topological length m. This procedure corresponds to a Taylor expansion of ZGV (z; k) in z around z = 0 up to order z (with z taken to be one at the end) ZGV (z; k) z Gamma z (1.3) 2t p (t p ) j 2 Gamma n p 0 =1 t p 0 p 0 ....

R. Artuso, E. Aurell and P. Cvitanovi'c, Recycling of strange sets: I. Cycle expansions, Nonlinearity 3 (1990) 325-359.

....calculations in sections 7.3 and 7.4 below. It is this geometrical fall o in matrix coecients [3,29] that will allow us to approximate S by a nite matrix below. It is also why zeta function methods work in obtaining growth rates using periodic orbit sums in SFS and similar fast dynamo models [1 5,29]. 6.4 The adjoint operator S Now we have de ned the adjoint space B , we can obtain the adjoint operator S to S, de ned using (6.17) by hS b; ci = hb; Sci = 1 2 i I D b(z) e i 1 2 (z 1) c( 1 2 (z 1) e i 1 2 (1 z) c( 1 2 (1 z) dz; 6.21) so that, changing ....

Artuso, R., Aurell, E. & Cvitanovic, P. 1990a Recycling of strange sets: I. Cycle expansions. Nonlinearity 3, 325-359.

....of the Lyapunov exponents. These methods are Monte Carlo simulation, weak disorder expansion [1] the microcanonical method[2] and the cycle expansion method[3] Cycle expansions have already been very successful in obtaining non perturbative results for for example chaotic dynamical systems[4]. For a recent interesting application, see [5] After discussing the methods, we use the formalism developed on explicit examples, namely the 1 dimensional Ising model and the 2 dimensional Ising model on a strip with quenched disorder. 2 The Lyapunov Exponents Consider the product of n m Theta ....

Roberto Artuso, Erick Aurell and Predrag Cvitanovic, Recycling of strange sets, Nonlinearity, 3:325-386, 1990

....cycle, p] its topological length and z is a bookkeeping variable for keeping track of the topological order #1 . L p is the length of the p th orbit, p its Maslov index (plus the group theoretical weight) and p its stability (the leading eigenvalues of the stability matrix) see refs. [11, 12, 13] for details and definitions. The characteristic determinant and the Gutzwiller Voros zeta function are related as detM(k) s:c: Gamma ZGV (k) 2.3) where the exact meaning of the semi classical limes s.c. still has to be defined (see below) Let Qm (k) denote the m th cumulant of ....

....(2.4) has been found so far. Thus the corresponding semi classical m th order curvature term, c m (k) of ZGV (k) can only be constructed from the semi classical equivalent of the Plemelj Smithies recursion relation [10] which exactly corresponds to the standard curvature expansion of refs. [11, 12, 8]) and is therefore inherently plagued by large cancellations: c m (k) Gamma 1 m m X j=1 c m Gammaj (k) X p;r 0 with [p]r=j [p] t p (k) r 1 Gamma i 1 p j r for m 1 (2.5) c 0 (k) j 1 ; where the periodic orbits result from the following semiclassical limit [10] Tr(A j ....

R. Artuso, E. Aurell and P. Cvitanovi'c, Recycling of strange sets: I. Cycle expansions, Nonlinearity 3 (1990) 325-359.

....p th primitive periodic orbits. Note that the expression (1.1) for the Gutzwiller Voros zeta function is purely formal as it stands. In general, it may not even exist without regularization. Therefore, the semi classical resonance poles are computed from ZGV (z=1; k) in the curvature expansion [16, 15, 17] up to a given topological length m. This procedure corresponds to a Taylor expansion of ZGV (z; k) in z around z = 0 up to order z m (with z taken to be one at the end) ZGV (z; k) z 0 Gamma z X [p] 1 t p 1 Gamma 1 p (1.2) Gamma z 2 2 8 : X [p] 2 2t p 1 Gamma 1 ....

R. Artuso, E. Aurell and P. Cvitanovi'c, Recycling of strange sets: I. Cycle expansions, Nonlinearity 3 (1990) 325-359.

....For the present, we shall concentrate on establishing our method in the concrete case of computing the dimension of E 2 , and closely related sets. The approach we use is motivated by related calculations for quantities arising in the study of the linearized Feigenbaum renormalization operator [A A C], Pol] and the functional determinant of the Laplacian [Pol R] A common feature is an explicit expression as a power series, whose coefficients can readily be computed in terms of closed orbits for an underlying dynamical system and which tend to zero very rapidly, allowing very good ....

R. Artuso, E. Aurell, P. Cvitanovi'c, Recycling of strange sets. II. Applications., Nonlinearity 3 (1990), 361--386.

....hand branch. Summarising, the winding number of arg(O(z) along a finite segment determines how many eigenvalues are located to the right hand side of the segment. Finally, let me mention that from the point of view of its analytical structure expression (17) resembles dynamical zeta functions [16], but no deeper relationship seems to be available at the moment. 6 0.4 0.2 0 0.2 0.4 Re(z) 0 0.1 0.2 0.3 0.4 0.5 0 0.2 0.4 0.6 0.8 1 Im(z) b Fig. 2. Dependence of the real and imaginary part of the rescaled eigenvalue z on the dimensionless driving amplitude fi for ffi = 2=3. ....

R. Artuso, E. Aurell, and P. Cvitanovic, Recycling of strange sets: I. Cycle expansion, Nonlin. 3 325 (1990)

.... dynamics of the map F , and can be a very useful way to classify the system; often the partition can be chosen so that the map between points and sequences is one to one, allowing us to enumerate its periodic points [20] and calculate quantities like entropies, escape rates and Liapunov exponents [1]. As another example, consider a cellular automaton (CA) in one dimension. This is a dynamical system on sequences where each site is updated according to some local rule, as a function of its state and those of its neighbors; for instance, suppose the state at each site is 0 or 1, and F (a) i = ....

R. Artuso, E. Aurell and P. Cvitanovic, \Recycling strange sets." Nonlinearity 3 (1990) 325{.

.... This point of view is argued in [ Hunt and Ott, 1996b ] Hunt and Ott, 1996a ] However, this conclusion does not hold generally and some applications may require long period orbits also [ Zoldi and Greenside, 1997a ] In this case it may be possible to apply the cycles expansion theory [ Arutso et al. 1990a ] Arutso et al. 1990b ] in order to manage all the periodic orbits set. The instability of periodic orbits of chaotic systems make the problem of their numerical search more complicated. They can not be found by a simple stabilisation method used generally to nd stable orbits. Therefore, ....

.... in [ Hunt and Ott, 1996b ] Hunt and Ott, 1996a ] However, this conclusion does not hold generally and some applications may require long period orbits also [ Zoldi and Greenside, 1997a ] In this case it may be possible to apply the cycles expansion theory [ Arutso et al. 1990a ] Arutso et al. 1990b ] in order to manage all the periodic orbits set. The instability of periodic orbits of chaotic systems make the problem of their numerical search more complicated. They can not be found by a simple stabilisation method used generally to nd stable orbits. Therefore, some special algorithm must ....

[Article contains additional citation context not shown here]

R. Arutso, E. Aurell, and P. Cvitanovic. Recycling of strange sets: 2. applications. Nonlinearity, 3:361, 1990.

.... This point of view is argued in [ Hunt and Ott, 1996b ] Hunt and Ott, 1996a ] However, this conclusion does not hold generally and some applications may require long period orbits also [ Zoldi and Greenside, 1997a ] In this case it may be possible to apply the cycles expansion theory [ Arutso et al. 1990a ] Arutso et al. 1990b ] in order to manage all the periodic orbits set. The instability of periodic orbits of chaotic systems make the problem of their numerical search more complicated. They can not be found by a simple stabilisation method used generally to nd stable orbits. Therefore, ....

.... in [ Hunt and Ott, 1996b ] Hunt and Ott, 1996a ] However, this conclusion does not hold generally and some applications may require long period orbits also [ Zoldi and Greenside, 1997a ] In this case it may be possible to apply the cycles expansion theory [ Arutso et al. 1990a ] Arutso et al. 1990b ] in order to manage all the periodic orbits set. The instability of periodic orbits of chaotic systems make the problem of their numerical search more complicated. They can not be found by a simple stabilisation method used generally to nd stable orbits. Therefore, some special algorithm must ....

[Article contains additional citation context not shown here]

R. Arutso, E. Aurell, and P. Cvitanovic. Recycling of strange sets: 1. cycle expansion. Nonlinearity, 3:325, 1990.

.... This point of view is argued in [ Hunt and Ott, 1996b ] Hunt and Ott, 1996a ] However, this conclusion does not hold generally and some applications may require long period orbits also [ Zoldi and Greenside, 1997a ] In this case it may be possible to apply the cycles expansion theory [ Arutso et al. 1990a ] Arutso et al. 1990b ] in order to manage all the periodic orbits set. To treat periodic orbits of a geophysical system we need rst to detect some of them numerically. This requires an eOEcient algorithm of the unstable periodic orbit search, applicable to geophysical models. To develop ....

.... argued in [ Hunt and Ott, 1996b ] Hunt and Ott, 1996a ] However, this conclusion does not hold generally and some applications may require long period orbits also [ Zoldi and Greenside, 1997a ] In this case it may be possible to apply the cycles expansion theory [ Arutso et al. 1990a ] Arutso et al. 1990b ] in order to manage all the periodic orbits set. To treat periodic orbits of a geophysical system we need rst to detect some of them numerically. This requires an eOEcient algorithm of the unstable periodic orbit search, applicable to geophysical models. To develop such an algorithm and to ....

[Article contains additional citation context not shown here]

R. Arutso, E. Aurell, and P. Cvitanovic. Recycling of strange sets: 2. applications. Nonlinearity, 3:361, 1990.

.... This point of view is argued in [ Hunt and Ott, 1996b ] Hunt and Ott, 1996a ] However, this conclusion does not hold generally and some applications may require long period orbits also [ Zoldi and Greenside, 1997a ] In this case it may be possible to apply the cycles expansion theory [ Arutso et al. 1990a ] Arutso et al. 1990b ] in order to manage all the periodic orbits set. To treat periodic orbits of a geophysical system we need rst to detect some of them numerically. This requires an eOEcient algorithm of the unstable periodic orbit search, applicable to geophysical models. To develop ....

.... in [ Hunt and Ott, 1996b ] Hunt and Ott, 1996a ] However, this conclusion does not hold generally and some applications may require long period orbits also [ Zoldi and Greenside, 1997a ] In this case it may be possible to apply the cycles expansion theory [ Arutso et al. 1990a ] Arutso et al. 1990b ] in order to manage all the periodic orbits set. To treat periodic orbits of a geophysical system we need rst to detect some of them numerically. This requires an eOEcient algorithm of the unstable periodic orbit search, applicable to geophysical models. To develop such an algorithm and to ....

[Article contains additional citation context not shown here]

R. Arutso, E. Aurell, and P. Cvitanovic. Recycling of strange sets: 1. cycle expansion. Nonlinearity, 3:325, 1990.

.... dynamics of the map F , and can be a very useful way to classify the system; often the partition can be chosen so that the map between points and sequences is one to one, allowing us to enumerate its periodic points [17] and calculate quantities like entropies, escape rates and Liapunov exponents [1]. As another example, consider a cellular automaton (CA) in one dimension. This is a dynamical system on sequences where each site is updated according to some local rule, as a function of its state and those of its neighbors; for instance, suppose the state at each site is 0 or 1, and F (a) i = ....

R. Artuso, E. Aurell and P. Cvitanovi'c, "Recycling strange sets." Nonlinearity 3 (1990) 325--.

....attractor may lead to the fact that orbits are dense on some attractor subset. Hence it is not evident either the attractor set can be approximated by orbits, or this subset of the attractor only. The orbits encoding, application of symbolic dynamics and possible cycle expansion (see for example [Arutso et al. 1990]) allow to estimate easily many attractor properties and predictability characteristics for simple system like Lorenz one [Eckhard and Ott, 1994] Franceschini et al. 1993] In particular, encoding of periodic orbits allows to know whether all the solutions with periods T less than some T 0 have ....

R. Arutso, E. Aurell, and P. Cvitanovic. Recycling of strange sets: 1. cycle expansion. Nonlinearity, 3:325, 1990.

....For the present, we shall concentrate on establishing our method in the concrete case of computing the dimension of E 2 , and closely related sets. The approach we use is motivated by related calculations for quantities arising in the study of the linearized Feigenbaum renormalization operator [A A C], Pol] and the functional determinant of the Laplacian [Pol R] A common feature is an explicit expression as a power series, whose coefficients can readily be computed in terms of closed orbits for an underlying dynamical system and which tend to zero very rapidly, allowing very good ....

R. Artuso, E. Aurell, P. Cvitanovi'c, Recycling of strange sets. II. Applications., Nonlinearity 3 (1990), 361--386.

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R. Artuso, E. Aurell, and P. Cvitanovi'c, Recycling of strange sets: I. Cycle expansions, II. Applications, Nonlinearity 3 (1990), 325--359 and 361--386.

No context found.

R. Artuso, E. Aurell, and P. Cvitanovi'c, Recycling of strange sets I, Nonlinearity 3 (1990) 325.

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