SHADOWING ON FRACTALS

Fractals ◽  
1994 ◽  
Vol 02 (02) ◽  
pp. 307-310
Author(s):  
FERNANDA BOTELHO ◽  
MAX GARZON
Keyword(s):  
Neural Networks ◽  
Computer Simulation ◽  
Dynamical Systems ◽  
Cellular Automata ◽  
Cantor Sets ◽  
Simulation And Modeling ◽  
Small Perturbations ◽  
Shadowing Property ◽  
Finite Approximations ◽  
Approximation Errors

We consider several families of continous dynamical systems on Cantor sets arising, in particular, from computer simulation and modeling of neural networks by discrete and/or finite approximations (such as cellular automata). It is shown that such approximations are always observable (have the shadowing property), in the sense that pseudo-orbits obtained by small perturbations of an orbit are approximated by actual orbits. It follows that the true behavior of locally defined dynamical systems can be observed exactly on computer simulations, despite unavoidable discretization and approximation errors.

BIFURCATIONS AND OSCILLATORY BEHAVIOR IN A CLASS OF COMPETITIVE CELLULAR NEURAL NETWORKS

2000 ◽  
Vol 10 (06) ◽  
pp. 1267-1293 ◽  
Author(s):  
M. DI MARCO ◽  
A. TESI ◽  
M. FORTI
Keyword(s):  
Neural Networks ◽  
Dynamical Systems ◽  
Opposite Sign ◽  
Symmetric Matrix ◽  
General Theorem ◽  
Dynamical Behavior ◽  
Cellular Neural Networks ◽  
Oscillatory Behavior ◽  
Third Order ◽  
Small Perturbations

When the neuron interconnection matrix is symmetric, the standard Cellular Neural Networks (CNN's) introduced by Chua and Yang [1988a] are known to be completely stable, that is, each trajectory converges towards some stationary state. In this paper it is shown that the interconnection symmetry, though ensuring complete stability, is not in the general case sufficient to guarantee that complete stability is robust with respect to sufficiently small perturbations of the interconnections. To this end, a class of third-order CNN's with competitive (inhibitory) interconnections between distinct neurons is introduced. The analysis of the dynamical behavior shows that such a class contains nonsymmetric CNN's exhibiting persistent oscillations, even if the interconnection matrix is arbitrarily close to some symmetric matrix. This result is of obvious relevance in view of CNN's implementation, since perfect interconnection symmetry in unattainable in hardware (e.g. VLSI) realizations. More insight on the behavior of the CNN's here introduced is gained by discussing the analogies with the dynamics of the May and Leonard model of the voting paradox, a special Volterra–Lotka model of three competing species. Finally, it is shown that the results in this paper can also be viewed as an extension of previous results by Zou and Nossek for a two-cell CNN with opposite-sign interconnections between distinct neurons. Such an extension has a significant interpretation in the framework of a general theorem by Smale for competitive dynamical systems.

scholarly journals On Smoother Attributions using Neural Stochastic Differential Equations

2021 ◽  
Author(s):  
Sumit Jha ◽  
Rickard Ewetz ◽  
Alvaro Velasquez ◽  
Susmit Jha
Keyword(s):  
Neural Networks ◽  
Dynamical Systems ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Systems Theory ◽  
Dynamical Systems Theory ◽  
Small Perturbations

Several methods have recently been developed for computing attributions of a neural network's prediction over the input features. However, these existing approaches for computing attributions are noisy and not robust to small perturbations of the input. This paper uses the recently identified connection between dynamical systems and residual neural networks to show that the attributions computed over neural stochastic differential equations (SDEs) are less noisy, visually sharper, and quantitatively more robust. Using dynamical systems theory, we theoretically analyze the robustness of these attributions. We also experimentally demonstrate the efficacy of our approach in providing smoother, visually sharper and quantitatively robust attributions by computing attributions for ImageNet images using ResNet-50, WideResNet-101 models and ResNeXt-101 models.

Lambda and the Edge of Chaos in Recurrent Neural Networks

2015 ◽  
Vol 21 (1) ◽  
pp. 55-71 ◽  
Author(s):  
Jared Seifter ◽  
James A. Reggia
Keyword(s):  
Neural Networks ◽  
Dynamical Systems ◽  
Cellular Automata ◽  
Recurrent Neural Networks ◽  
Artificial Life ◽  
Network Connectivity ◽  
Transitional Region ◽  
Edge Of Chaos ◽  
Dynamical Behaviors ◽  
Special Meaning

The idea that there is an edge of chaos, a region in the space of dynamical systems having special meaning for complex living entities, has a long history in artificial life. The significance of this region was first emphasized in cellular automata models when a single simple measure, λCA, identified it as a transitional region between order and chaos. Here we introduce a parameter λNN that is inspired by λCA but is defined for recurrent neural networks. We show through a series of systematic computational experiments that λNN generally orders the dynamical behaviors of randomly connected/weighted recurrent neural networks in the same way that λCA does for cellular automata. By extending this ordering to larger values of λNN than has typically been done with λCA and cellular automata, we find that a second edge-of-chaos region exists on the opposite side of the chaotic region. These basic results are found to hold under different assumptions about network connectivity, but vary substantially in their details. The results show that the basic concept underlying the lambda parameter can usefully be extended to other types of complex dynamical systems than just cellular automata.

SYNCHRONIZATION OF COUPLED EXTENDED DYNAMICAL SYSTEMS: A SHORT REVIEW

2003 ◽  
Vol 13 (04) ◽  
pp. 781-796 ◽  
Author(s):  
DAMIÁN H. ZANETTE ◽  
LUIS G. MORELLI
Keyword(s):  
Neural Networks ◽  
Dynamical Systems ◽  
Cellular Automata ◽  
Critical Behavior ◽  
Short Review ◽  
Ginzburg Landau ◽  
Spatially Extended ◽  
Ginzburg Landau Equations ◽  
Analytical Approaches ◽  
Joint Evolution

We briefly review the synchronization properties of cross-coupled spatially extended dynamical systems, with particular emphasis on elementary cellular automata and Kauffman networks subject to stochastic coupling. We also discuss the main results for the joint evolution of deterministically cross-coupled Ginzburg–Landau equations and neural networks. Both numerical and analytical approaches are addressed, and the main differences with the synchronization of zero-dimensional systems are highlighted. New results are presented characterizing the critical behavior at the synchronization transition of coupled Kauffman networks.

A STUDY OF BIFURCATIONS IN A CIRCULAR REAL CELLULAR AUTOMATON

1993 ◽  
Vol 03 (02) ◽  
pp. 293-321 ◽  
Author(s):  
JÜRGEN WEITKÄMPER
Keyword(s):  
Hopf Bifurcation ◽  
Dynamical Systems ◽  
Cellular Automata ◽  
Cellular Automaton ◽  
Parallel Computers ◽  
Discrete Dynamical Systems ◽  
Two Dimensional ◽  
Bifurcation Structure ◽  
Logistic Mapping ◽  
Henon Mapping

Real cellular automata (RCA) are time-discrete dynamical systems on ℝN. Like cellular automata they can be obtained from discretizing partial differential equations. Due to their structure RCA are ideally suited to implementation on parallel computers with a large number of processors. In a way similar to the Hénon mapping, the system we consider here embeds the logistic mapping in a system on ℝN, N>1. But in contrast to the Hénon system an RCA in general is not invertible. We present some results about the bifurcation structure of such systems, mostly restricting ourselves, due to the complexity of the problem, to the two-dimensional case. Among others we observe cascades of cusp bifurcations forming generalized crossroad areas and crossroad areas with the flip curves replaced by Hopf bifurcation curves.

HIGHER ORDER CELLULAR AUTOMATA

2005 ◽  
Vol 08 (02n03) ◽  
pp. 169-192 ◽  
Author(s):  
NILS A. BAAS ◽  
TORBJØRN HELVIK
Keyword(s):  
Dynamical Systems ◽  
Cellular Automata ◽  
Level Structure ◽  
Higher Order ◽  
New Concepts ◽  
Multi Level

We introduce a class of dynamical systems called Higher Order Cellular Automata (HOCA). These are based on ordinary CA, but have a hierarchical, or multi-level, structure and/or dynamics. We present a detailed formalism for HOCA and illustrate the concepts through four examples. Throughout the article we emphasize the principles and ideas behind the construction of HOCA, such that these easily can be applied to other types of dynamical systems. The article also presents new concepts and ideas for describing and studying hierarchial dynamics in general.

scholarly journals Numerically evaluated functional equivalence between chaotic dynamics in neural networks and cellular automata under totalistic rules

2006 ◽  
Vol 1 (3) ◽  
pp. 189-202 ◽  
Author(s):  
Ryu Takada ◽  
Daigo Munetaka ◽  
Shoji Kobayashi ◽  
Yoshikazu Suemitsu ◽  
Shigetoshi Nara
Keyword(s):  
Neural Networks ◽  
Cellular Automata ◽  
Chaotic Dynamics ◽  
Functional Equivalence
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