scholarly journals Asymptotic (statistical) periodicity in two-dimensional maps

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Fumihiko Nakamura ◽  
Michael C. Mackey
Keyword(s):  
Dynamical System ◽  
Bounded Variation ◽  
High Dimensional ◽  
Sufficient Condition ◽  
Two Dimensional ◽  
Function Of Bounded Variation ◽  
Dimensional Dynamical System ◽  
Asymptotic Periodicity ◽  
Parameter Values

<p style='text-indent:20px;'>In this paper we give a new sufficient condition for the existence of asymptotic periodicity of Frobenius–Perron operators corresponding to two–dimensional maps. Asymptotic periodicity for strictly expanding systems, that is, all eigenvalues of the system are greater than one, in a high-dimensional dynamical system was already known. Our new result enables one to deal with systems having an eigenvalue smaller than one. The key idea for the proof is to use a function of bounded variation defined by line integration. Finally, we introduce a new two-dimensional dynamical system numerically exhibiting asymptotic periodicity with different periods depending on parameter values, and discuss the application of our theorem to the example.</p>

Integrable two-dimensional dynamical systems and the characteristic Lie algebras

2007 ◽  
Vol 5 ◽  
pp. 195-200
Author(s):  
A.V. Zhiber ◽  
O.S. Kostrigina
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Second Order ◽  
System Of Equations ◽  
Two Dimensional ◽  
Finite Dimensional ◽  
Dimensional Dynamical System

In the paper it is shown that the two-dimensional dynamical system of equations is Darboux integrable if and only if its characteristic Lie algebra is finite-dimensional. The class of systems having a full set of fist and second order integrals is described.

scholarly journals Clustered Echo State Networks for Signal Observation and Frequency Filtering

2020 ◽  
Author(s):  
Laércio Oliveira Junior ◽  
Florian Stelzer ◽  
Liang Zhao
Keyword(s):  
Dynamical System ◽  
Neural Networks ◽  
Input Signal ◽  
Recurrent Neural Networks ◽  
High Dimensional ◽  
Frequency Filtering ◽  
Echo State Networks ◽  
Chaotic Signals ◽  
Network Topologies ◽  
Dimensional Dynamical System

Echo State Networks (ESNs) are recurrent neural networks that map an input signal to a high-dimensional dynamical system, called reservoir, and possess adaptive output weights. The output weights are trained such that the ESN’s output signal fits the desired target signal. Classical reservoirs are sparse and randomly connected networks. In this article, we investigate the effect of different network topologies on the performance of ESNs. Specifically, we use two types of networks to construct clustered reservoirs of ESN: the clustered Erdös–Rényi and the clustered Barabási-Albert network model. Moreover, we compare the performance of these clustered ESNs (CESNs) and classical ESNs with the random reservoir by employing them to two different tasks: frequency filtering and the reconstruction of chaotic signals. By using a clustered topology, one can achieve a significant increase in the ESN’s performance.

DYNAMICS AT INFINITY OF A CUBIC CHUA'S SYSTEM

2011 ◽  
Vol 21 (01) ◽  
pp. 333-340 ◽  
Author(s):  
MARCELO MESSIAS
Keyword(s):  
Dynamical System ◽  
Vector Field ◽  
Numerical Study ◽  
Three Dimensional ◽  
Polynomial Vector ◽  
Cubic Nonlinearity ◽  
Poincaré Compactification ◽  
Global Study ◽  
Dimensional Dynamical System ◽  
Parameter Values

We use the Poincaré compactification for a polynomial vector field in ℝ3 to study the dynamics near and at infinity of the classical Chua's system with a cubic nonlinearity. We give a complete description of the phase portrait of this system at infinity, which is identified with the sphere 𝕊2 in ℝ3 after compactification, and perform a numerical study on how the solutions reach infinity, depending on the parameter values. With this global study we intend to give a contribution in the understanding of this well known and extensively studied complex three-dimensional dynamical system.

scholarly journals ON A PLANAR DYNAMICAL SYSTEM ARISING IN THE NETWORK CONTROL THEORY

2016 ◽  
Vol 21 (3) ◽  
pp. 385-398 ◽  
Author(s):  
Svetlana Atslega ◽  
Dmitrijs Finaskins ◽  
Felix Sadyrbaev
Keyword(s):  
Dynamical System ◽  
Control Theory ◽  
Network Control ◽  
Two Dimensional ◽  
Planar Dynamical System ◽  
Attracting Set ◽  
Dimensional Dynamical System

We study the structure of attractors in the two-dimensional dynamical system that appears in the network control theory. We provide description of the attracting set and follow changes this set suffers under the changes of positive parameters µ and Θ.

scholarly journals The Topological Entropy of Cyclic Permutation Maps and Some Chaotic Properties on Their MPE sets

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Risong Li ◽  
Tianxiu Lu
Keyword(s):  
Dynamical System ◽  
Topological Entropy ◽  
Real Line ◽  
Cyclic Permutation ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
The Real ◽  
Compact Subinterval ◽  
Dimensional Dynamical System ◽  
Necessary And Sufficient

In this paper, we study some chaotic properties of s-dimensional dynamical system of the form Ψa1,a2,…,as=gsas,g1a1,…,gs−1as−1, where ak∈Hk for any k∈1,2,…,s, s≥2 is an integer, and Hk is a compact subinterval of the real line ℝ=−∞,+∞ for any k∈1,2,…,s. Particularly, a necessary and sufficient condition for a cyclic permutation map Ψa1,a2,…,as=gsas,g1a1,…,gs−1as−1 to be LY-chaotic or h-chaotic or RT-chaotic or D-chaotic is obtained. Moreover, the LY-chaoticity, h-chaoticity, RT-chaoticity, and D-chaoticity of such a cyclic permutation map is explored. Also, we proved that the topological entropy hΨ of such a cyclic permutation map is the same as the topological entropy of each of the following maps: gj∘gj−1∘⋯∘g1l∘gs∘gs−1∘⋯∘gj+1, if j=1,…,s−1and gs∘gs−1∘⋯∘g1, and that Ψ is sensitive if and only if at least one of the coordinates maps of Ψs is sensitive.

scholarly journals STABILITY ANALYSIS WITH APPLICATIONS OF A TWO-DIMENSIONAL DYNAMICAL SYSTEM ARISING FROM A STOCHASTIC MODEL FOR AN ASSET MARKET

2011 ◽  
Vol 11 (04) ◽  
pp. 715-752
Author(s):  
VLADIMIR BELITSKY ◽  
ANTONIO LUIZ PEREIRA ◽  
FERNANDO PIGEARD DE ALMEIDA PRADO
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Asset Price ◽  
Excess Demand ◽  
Market Model ◽  
Asset Market ◽  
Two Dimensional ◽  
Equilibrium Solutions ◽  
Dimensional Dynamical System ◽  
Demand Fluctuations

We analyze the stability properties of equilibrium solutions and periodicity of orbits in a two-dimensional dynamical system whose orbits mimic the evolution of the price of an asset and the excess demand for that asset. The construction of the system is grounded upon a heterogeneous interacting agent model for a single risky asset market. An advantage of this construction procedure is that the resulting dynamical system becomes a macroscopic market model which mirrors the market quantities and qualities that would typically be taken into account solely at the microscopic level of modeling. The system's parameters correspond to: (a) the proportion of speculators in a market; (b) the traders' speculative trend; (c) the degree of heterogeneity of idiosyncratic evaluations of the market agents with respect to the asset's fundamental value; and (d) the strength of the feedback of the population excess demand on the asset price update increment. This correspondence allows us to employ our results in order to infer plausible causes for the emergence of price and demand fluctuations in a real asset market. The employment of dynamical systems for studying evolution of stochastic models of socio-economic phenomena is quite usual in the area of heterogeneous interacting agent models. However, in the vast majority of the cases present in the literature, these dynamical systems are one-dimensional. Our work is among the few in the area that construct and study analytically a two-dimensional dynamical system and apply it for explanation of socio-economic phenomena.

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