On the Dynamics of the q-Deformed Gaussian Map

2020 ◽  
Vol 30 (08) ◽  
pp. 2030021
Author(s):  
J. Cánovas ◽  
M. Muñoz-Guillermo
Keyword(s):  
Dynamical Systems ◽  
Topological Entropy ◽  
Chaotic Attractor ◽  
Schwarzian Derivative ◽  
Discrete Dynamical Systems ◽  
Coexistence Of Attractors ◽  
Wide Range ◽  
Gaussian Map ◽  
Parametric Region

Following the scheme inspired by Tsallis [Jagannathan & Sudeshna, 2005; Patidar & Sud, 2009], we study the Gaussian map and its [Formula: see text]-deformed version. We compute the topological entropies of the discrete dynamical systems which are obtained for both maps, the original Gaussian map and its [Formula: see text]-modification. In particular, we are able to obtain the parametric region in which the topological entropy is positive. The analysis of the sign of Schwarzian derivative and the topological entropy allow us a deeper analysis of the dynamics. We also highlight the coexistence of attractors, even if it is possible to determine a wide range of parameters in which one of them is a chaotic attractor.

ON THE STRUCTURE OF THE ω-LIMIT SETS FOR CONTINUOUS MAPS OF THE INTERVAL

1999 ◽  
Vol 09 (09) ◽  
pp. 1719-1729 ◽  
Author(s):  
LLUÍS ALSEDÀ ◽  
MOIRA CHAS ◽  
JAROSLAV SMÍTAL
Keyword(s):  
Dynamical Systems ◽  
Topological Entropy ◽  
Discrete Dynamical Systems ◽  
Limit Sets ◽  
Interval Maps ◽  
Positive Topological Entropy ◽  
Countable Ordinal ◽  
Continuous Maps

We introduce the notion of the center of a point for discrete dynamical systems and we study its properties for continuous interval maps. It is known that the Birkhoff center of any such map has depth at most 2. Contrary to this, we show that if a map has positive topological entropy then, for any countable ordinal α, there is a point xα∈I such that its center has depth at least α. This improves a result by [Sharkovskii, 1966].

scholarly journals On the origin and development of some notions of entropy

2015 ◽  
Vol 3 (1) ◽  
Author(s):  
Francisco Balibrea
Keyword(s):  
Applied Sciences ◽  
Dynamical Systems ◽  
Topological Entropy ◽  
Discrete Dynamical Systems ◽  
Point Of View ◽  
Topological Dynamics ◽  
Theoretical Point ◽  
Compact Metric Space ◽  
Continuous Maps ◽  
And Control

AbstractDiscrete dynamical systems are given by the pair (X, f ) where X is a compact metric space and f : X → X a continuous maps. During years, a long list of results have appeared to precise and understand what is the complexity of the systems. Among them, one of the most popular is that of topological entropy. In modern applications other conditions on X and f have been considered. For example X can be non-compact or f can be discontinuous (only in a finite number of points and with bounded jumps on the values of f or even non-bounded jumps). Such systems are interesting from theoretical point of view in Topological Dynamics and appear frequently in applied sciences such as Electronics and Control Theory. In this paper we are dealing mainly with the original ideas of entropy in Thermodinamics and their evolution until the appearing in the twenty century of the notions of Shannon and Kolmogorov-Sinai entropies and the subsequent topological entropy. In turn such notions have to evolve to other recent situations where it is necessary to give some extended versions of them adapted to the new problems.

scholarly journals Hidden Attractors in Discrete Dynamical Systems

Entropy ◽  
2021 ◽  
Vol 23 (5) ◽  
pp. 616
Author(s):  
Marek Berezowski ◽  
Marcin Lawnik
Keyword(s):  
Dynamical Systems ◽  
Chaos Theory ◽  
Scientific Literature ◽  
Discrete Systems ◽  
Discrete Dynamical Systems ◽  
Continuous Systems ◽  
Hidden Attractors ◽  
Negative Effects ◽  
Chemical Reactors ◽  
Points Of View

Research using chaos theory allows for a better understanding of many phenomena modeled by means of dynamical systems. The appearance of chaos in a given process can lead to very negative effects, e.g., in the construction of bridges or in systems based on chemical reactors. This problem is important, especially when in a given dynamic process there are so-called hidden attractors. In the scientific literature, we can find many works that deal with this issue from both the theoretical and practical points of view. The vast majority of these works concern multidimensional continuous systems. Our work shows these attractors in discrete systems. They can occur in Newton’s recursion and in numerical integration.

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.

scholarly journals Discovering governing equations from data by sparse identification of nonlinear dynamical systems

2016 ◽  
Vol 113 (15) ◽  
pp. 3932-3937 ◽  
Author(s):  
Steven L. Brunton ◽  
Joshua L. Proctor ◽  
J. Nathan Kutz
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Dynamical Systems ◽  
Measurement Data ◽  
Climate Science ◽  
Model Complexity ◽  
Governing Equations ◽  
Canonical Systems ◽  
Nonlinear Dynamical ◽  
Balance Accuracy ◽  
Wide Range

Extracting governing equations from data is a central challenge in many diverse areas of science and engineering. Data are abundant whereas models often remain elusive, as in climate science, neuroscience, ecology, finance, and epidemiology, to name only a few examples. In this work, we combine sparsity-promoting techniques and machine learning with nonlinear dynamical systems to discover governing equations from noisy measurement data. The only assumption about the structure of the model is that there are only a few important terms that govern the dynamics, so that the equations are sparse in the space of possible functions; this assumption holds for many physical systems in an appropriate basis. In particular, we use sparse regression to determine the fewest terms in the dynamic governing equations required to accurately represent the data. This results in parsimonious models that balance accuracy with model complexity to avoid overfitting. We demonstrate the algorithm on a wide range of problems, from simple canonical systems, including linear and nonlinear oscillators and the chaotic Lorenz system, to the fluid vortex shedding behind an obstacle. The fluid example illustrates the ability of this method to discover the underlying dynamics of a system that took experts in the community nearly 30 years to resolve. We also show that this method generalizes to parameterized systems and systems that are time-varying or have external forcing.

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