Pseudo-Deterministic Chaotic Systems

Viana, Ricardo L.; Pinto, Sandro Ely de Souza; Barbosa, Jose Renato Ramos; Grebogi, Celso

doi:10.1142/s0218127403008636

Cited by 25 publications

(16 citation statements)

References 48 publications

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“…This figure is very similar to the corresponding diagram for the infinitetime Lyapunov exponents [16,23]. In fact, for Gaussian distributions it follows that [8] Figure 7 points out that the average time-n exponent crosses zero at a constant rate in the neighborhood of the hyper-chaos transition. Since the distributions depicted in Fig.…”

Section: Kicked Double Rotor Mapsupporting

confidence: 75%

“…In other words, the relative weight of local expansions or contractions is roughly the same, what is the worst situation when one tries to obtain a "true" chaotic trajectory which shadows a numerical one. This argument can be made more precise by assigning these relative contributions of contractions and expansions the weights of unstable periodic orbits with different unstable dimensions by computing the natural measure of the chaotic attractor [23,8].…”

Section: Kicked Double Rotor Mapmentioning

confidence: 99%

“…We will choose as a paradigm of a high-dimensional dynamical system a mechanical system consisting of a double rotor subjected to periodic impulsive excitation, in which we adress the most severe of those dynamical difficulties, the so-called unstable dimension variability [8]. The approach we choose to investigate the onset, evolution and effects of unstable dimension variability in dynamical system is to consider the behavior of their numerically computed finite-time Lyapunov exponents.…”

Section: Introductionmentioning

confidence: 99%

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Simulating a chaotic process

Viana¹,

Barbosa²,

Grebogi

et al. 2005

Braz. J. Phys.

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Computer simulations of partial differential equations of mathematical physics typically lead to some kind of high-dimensional dynamical system. When there is chaotic behavior we are faced with fundamental dynamical difficulties. We choose as a paradigm of such high-dimensional system a kicked double rotor. This system is investigated for parameter values at which it is strongly non-hyperbolic through a mechanism called unstable dimension variability, through which there are periodic orbits embedded in a chaotic attractor with different numbers of unstable directions. Our numerical investigation is primarily based on the analysis of the finite-time Lyapunov exponents, which gives us useful hints about the onset and evolution of unstable dimension variability for the double rotor map, as a system parameter (the forcing amplitude) is varied.

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Section: Kicked Double Rotor Mapsupporting

confidence: 75%

Section: Kicked Double Rotor Mapmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Simulating a chaotic process

Viana¹,

Barbosa²,

Grebogi

et al. 2005

Braz. J. Phys.

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“…Such Markov chains can be obtained by so-called Markov partitions that exist for hyperbolic dynamical systems (Sinai [82], Bowen [83], Ruelle [84]). For non-hyperbolic systems, no corresponding existence theorem is available, and the construction can be even more tedious than for hyperbolic systems (Viana et al [85]). For instance, both Markov and generating partitions for nonlinear systems are generally non-homogeneous, i.e., their cells are typically of different sizes and forms (see Remark A10).…”

Section: Generating Partitionsmentioning

confidence: 99%

On Macrostates in Complex Multi-Scale Systems

Atmanspacher

2016

Entropy

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A characteristic feature of complex systems is their deep structure, meaning that the definition of their states and observables depends on the level, or the scale, at which the system is considered. This scale dependence is reflected in the distinction of micro-and macro-states, referring to lower and higher levels of description. There are several conceptual and formal frameworks to address the relation between them. Here, we focus on an approach in which macrostates are contextually emergent from (rather than fully reducible to) microstates and can be constructed by contextual partitions of the space of microstates. We discuss criteria for the stability of such partitions, in particular under the microstate dynamics, and outline some examples. Finally, we address the question of how macrostates arising from stable partitions can be identified as relevant or meaningful.

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“…For non-hyperbolic systems, much less is known (cf. Viana et al 2003). Now let us introduce the notion of epistemic observables.…”

mentioning

confidence: 99%

Contextual emergence of mental states

Atmanspacher

2015

Cogn Process

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The emergence of mental states from neural states by partitioning the neural phase space is analyzed in terms of symbolic dynamics. Well-defined mental states provide contexts inducing a criterion of structural stability for the neurodynamics that can be implemented by particular partitions. This leads to distinguished subshifts of finite type that are either cyclic or irreducible. Cyclic shifts correspond to asymptotically stable fixed points or limit tori whereas irreducible shifts are obtained from generating partitions of mixing hyperbolic systems. These stability criteria are applied to the discussion of neural correlates of consiousness, to the definition of macroscopic neural states, and to aspects of the symbol grounding problem. In particular, it is shown that compatible mental descriptions, topologically equivalent to the neurodynamical description, emerge if the partition of the neural phase space is generating. If this is not the case, mental descriptions are incompatible or complementary. Consequences of this result for an integration or unification of cognitive science or psychology, respectively, will be indicated. Interlevel RelationsKnowledge of well-defined relations among different levels of descriptions of physical and other systems is inevitable if one wants to understand how (elements of) different descriptions depend on each other, give rise to each other, or even imply each other. The most ambitious program in this regard is physical reduction in the sense that higher-level descriptions of features of a system are determined by the description of features at the most fundamental level of physical theory, no matter how remote the higher level is from that most fundamental level. This program assumes that the description of all features which are not included at the fundamental level can be constructed or derived from this level without additional input.However, already physical examples pose serious difficulties for this program. It has recently been proposed that the concept of contextual emergence (Atmanspacher and Bishop, this issue; Bishop and Atmanspacher, preprint) addresses such situations more properly. Contextual emergence is characterized by the fact that lowerlevel descriptions provide necessary, but not sufficient conditions for higher-level descriptions. (Note that such a relation between descriptive levels does not necessarily entail the same relation between ontological levels.) The presence of necessary conditions indicates that lower-level descriptions provide a basis for higher-level descriptions, while the absence of sufficient conditions means that higher-level features are neither logical consequences of lower-level descriptions nor can they be rigorously derived from them alone. Hence, a full-blown reductive program is inapplicable in these cases. Sufficient conditions for a rigorous derivation of higher-level features can be introduced through specifying contexts reflecting the particular kinds of contingency in a given situation.A key ingredient of this proc...

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Pseudo-Deterministic Chaotic Systems

Cited by 25 publications

References 48 publications

Simulating a chaotic process

Simulating a chaotic process

On Macrostates in Complex Multi-Scale Systems

Contextual emergence of mental states

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