Computer dynamics and shadowing of chaotic orbits

Fryska, Slawomir T.; Zohdy, Mohamed

doi:10.1016/0375-9601(92)90719-3

Cited by 27 publications

(23 citation statements)

References 5 publications

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“…The comparison agrees with results obtained from the theory of random perturbation models [Blank, 1997;Diamond et al, 1994;Lasota & Mackey, 1997] and are consistent with the reported experiments [Blank, 1994;Fryska & Zohdy, 1992;Philip & Joseph, 2001;Pokrovskii et al, 1999;Sang et al, 1998a,b;Čermák, 1996;: (pseudo-)random perturbation is a better solution to dynamical degradation. Another feature about the perturbation algorithm is also found: perturbing system variables has better performance than perturbing control parameters, which is hardly observed from the theory of random perturbation models and experiments.…”

Section: Introductionsupporting

confidence: 90%

“…5 to Fig. 7 in [Fryska & Zohdy, 1992]). In [Liu & Chen, 2004], it was reported that the quantization errors in the chaotic evolution can generate a fake 4-scroll attractor, although the attractor should only has two scrolls in theory.…”

Section: Intractable Quantization Errorsmentioning

confidence: 93%

“…Actually, motivated by various "strange" phenomena of chaos observed on digital computers and in numerical simulations, pathologies of digital chaotic systems have been observed and extensively studied in the field of chaos theory [Arrowsmith & Vivaldi, 1994;Beck & Roepstorff, 1987;Benettin et al, 1978;Binder, 1992;Binder & Jensen, 1986;Blank, 1994Blank, , 1997Borcherds & McCauley, 1993;Bosioand & Vivaldi, 2000;Chambers, 1999;Chirkikov & Vivaldi, 1999;Diamond et al, 1994Diamond et al, , 1995Earn & Tremaine, 1992;Fryska & Zohdy, 1992;Góra & Boyarsku, 1988;Grebogi et al, 1988;Hogg & Huberman, 1985;Huberman, 1986;Kaneko, 1988;Karney, 1983;Keating, 1991;Levy, 1982;Li et al, 2001a;Lowenstein & Vivaldi, 1998;Masuda & Aihara, 2002b;McCauley & Palmore, 1986;Palmore & Herring, 1990;Palmore & McCauley, 1987;Percival & Vivaldi, 1987;Pokrovskii et al, 1999;Rannou, 1974;Thiran et al, 1989;Čermák, 1996;Vivaldi, 1994;Waelbroeck & Zertuche, 1999;Zhang & Vivaldi, 1998]. To show how such dynamical degradation occurs, assume that the discretized space has ...…”

Section: Theoretical Work: Dynamical Degradation Of Digital Chaotic Smentioning

confidence: 99%

“…Due to the sensitivity of chaotic systems to initial conditions and control parameters, the pseudo-orbits in finite precision can be entirely different from the theoretical ones even after a few number of iterations (a lower bound of this number can be calculated using the Kolmogorov entropy [Chen, 1992]). A good demonstration on this problem was given in [Fryska & Zohdy, 1992]: for a 3-D piecewise linear chaotic system, when the system is realized in 32-bit singleprecision floating-point arithmetic, a two-scroll attractor is obtained; when the system is realized in 80-bit extended double-precision floating-point arithmetic, the attractor collapses to be a non-chaotic periodic orbit; while the attractor theoretically solved from the chaotic equations is a one-scroll orbit (see Fig. 5 to Fig.…”

Section: Intractable Quantization Errorsmentioning

confidence: 99%

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On the Dynamical Degradation of Digital Piecewise Linear Chaotic Maps

Chen

Mou

2005

Int. J. Bifurcation Chaos

355

189

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When chaotic systems are realized with finite precisions in digital computers, their dynamical properties are often found to be entirely different from the original versions in the continuous setting. In the literature, there does not seem to be much work on quantitative analysis of such degradation of digitized chaos and how to reduce its negative influence on chaos-based digital systems. Focusing on 1D piecewise linear chaotic maps (PWLCM), this paper reports some findings on a new series of dynamical indicators, which can quantitatively reflect the degradation effects on a digital PWLCM realized with a fixed-point finite precision. On top of that, the paper introduces a new method for studying digital chaos from an algorithmic point of view. In addition, the theoretical results obtained in this paper should be very helpful for the consideration of reducing negative influence of dynamical degradation in real design of various digital chaotic systems. As typical examples, the proposed dynamical indicators are applied to the performance comparison of different remedies for improving dynamical degradation, cryptanalysis of digital chaotic ciphers based on 1D PWLCM, and the design of chaotic pseudo-random number generators with desired characteristics.

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Section: Introductionsupporting

confidence: 90%

Section: Intractable Quantization Errorsmentioning

confidence: 93%

Section: Theoretical Work: Dynamical Degradation Of Digital Chaotic Smentioning

confidence: 99%

Section: Intractable Quantization Errorsmentioning

confidence: 99%

See 2 more Smart Citations

On the Dynamical Degradation of Digital Piecewise Linear Chaotic Maps

Chen

Mou

2005

Int. J. Bifurcation Chaos

355

189

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“…Because the normalizing-rounding process in the floating point arithmetic is a deterministic process occurring at the least significant bit level, the shadowing lemma to the digital integration for a chaotic solution is not applicable. Fryska and Zohdy, 1992 injected controlled amounts of uniformly distributed random noise to a three-dimensional chaotic system (a piecewise linear system) during digital integration and thereby closely approximated the statistics of the invariant chaotic attractor. Due to simplicity of their system, an exact analytical solution was obtainable, which was used as a reference for comparing with numerical solutions from different precisions.…”

Section: Shadowing Of the Chaotic Solutionmentioning

confidence: 99%

Optimization of olfactory model in software to give 1/f power spectra reveals numerical instabilities in solutions governed by aperiodic (chaotic) attractors

1998

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TitleOptimization of olfactory model in software to give 1/f power spectra reveals numerical instabilities in solutions governed by aperiodic (chaotic) attractors. AbstractWe present a general connectionist model for an olfactory system. The dynamical behavior of each node (neural ensemble) of the model is governed by a second-order ordinary differential equation (ODE) followed by an asymmetric sigmoidal function, relating the aggregate activity of neurons to system parameters and stimuli from an outside environment. A digital implementation of the general connectionist model simulates the characteristics of a mammalian olfactory system having modifiable synaptic connections and spatio-temporal interactions among neural ensembles. Each of four distributed delay terms is represented by a second-order ODE. A parameter optimization algorithm is an integral component of the model. The parameter optimization discussed in this paper results in aperiodic oscillations having a near 1/f-type power spectrum with a peak in the gamma range, simulating the electro-encephalographic (EEG) potentials from the neural olfactory system. Random optimization is used for a rough search in a global parameter domain and the parameter self-adaptation rule serves for fine tuning within a local domain after the global search. By design the model is started from an unstable zero point by an external impulse input. It requires 650-850 ms to pass through an initializing transient before settling into a strange attractor. The attractor persists for at least 1500 ms, which is 7.5-20 times longer than the duration of the maximal stationary states observed in the EEGs and it is stable under perturbation by simulated sensory inputs giving response amplitudes less than 2 times the basal aperiodic activity. However, the optimization to 1/f-type activity reveals an inherent limit on digital simulation of chaotic states, owing to attractor crowding such that the size of basins decreases with increasing size of the model, until it approaches the size of digitizing step in computation, here a 64-bit word (ϳ10 ¹16). Outputs that are not optimized to approach 1/f-type power spectra transit earlier to limit cycle activity, usually well before 2000 ms. The duration of stationarity is increased by randomizing the terminal bit of the 64-bit words representing the state variables. The 1/f-type solutions are also exquisitely sensitive to parameter truncation; parameter values must be saved in their full binary form for re-starting. The implications in terms of numerical instability, chaos, attractor crowding and the shadowing theorem are discussed. ᭧

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Stochastic Differential Equations and Random Number Generators Minimize Numerical Instabilities in Digital Simulations

Freeman

2000

Neurodynamics: An Exploration in Mesoscopic Brain Dynamics

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Computer dynamics and shadowing of chaotic orbits

Cited by 27 publications

References 5 publications

On the Dynamical Degradation of Digital Piecewise Linear Chaotic Maps

On the Dynamical Degradation of Digital Piecewise Linear Chaotic Maps

Optimization of olfactory model in software to give 1/f power spectra reveals numerical instabilities in solutions governed by aperiodic (chaotic) attractors

Stochastic Differential Equations and Random Number Generators Minimize Numerical Instabilities in Digital Simulations

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