Isochronous dynamical systems

2011 ◽  
Vol 369 (1939) ◽  
pp. 1118-1136 ◽  
Author(s):  
F. Calogero
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Initial Data ◽  
Large Fraction ◽  
Original System ◽  
Current Standard ◽  
Time Intervals ◽  
Open Set ◽  
Isochronous Systems ◽  
Autonomous Dynamical System

This is a terse review of recent results on isochronous dynamical systems, namely systems of (first-order, generally nonlinear) ordinary differential equations (ODEs) featuring an open set of initial data (which might coincide with the entire set of all initial data), from which emerge solutions all of which are completely periodic (i.e. periodic in all their components) with a fixed period (independent of the initial data, provided they are within the isochrony region). A leitmotif of this presentation is that ‘isochronous systems are not rare’. Indeed, it is shown how any (autonomous) dynamical system can be modified or extended so that the new (also autonomous) system thereby obtained is isochronous with an arbitrarily assigned period T , while its dynamics, over time intervals much shorter than the period T , mimics closely that of the original system, or even, over an arbitrarily large fraction of its period T , coincides exactly with that of the original system. It is pointed out that this fact raises the issue of developing criteria providing, for a dynamical system, some kind of measure associated with a finite time scale of the complexity of its behaviour (while the current, standard definitions of integrable versus chaotic dynamical systems are related to the behaviour of a system over infinite time).

scholarly journals Sensitivity Analysis and Stabilization for Two Dynamical Systems

2018 ◽  
pp. 33-40
Author(s):  
Frank Etin-Osa Bazuaye
Keyword(s):  
Dynamical System ◽  
Sensitivity Analysis ◽  
Dynamical Systems ◽  
Political Parties ◽  
Initial Data ◽  
Differential Equations ◽  
Democratic Process ◽  
Initial State ◽  
Mathematical Problems ◽  
The Impact

This paper focuses on the sensitivity analysis for two dominant political parties. In contrast to Misra, Bazuaye and Khan, who developed the model without investigating the impact of varying the initial state of political parties on the solution trajectory of each political parties, we have developed a sound numerical algorithm to analyze the impact of change on the initial data on the behavior of the democratic process which is a rare contribution to knowledge. Two Matlab standard solvers for ordinary differential equations, ode45 and ode23, have been utilized to handle these formidable mathematical problems. Our findings indicate that as the initial data varies, the dynamical system describing the interaction between two political parties is stabilized over a period of eight years. As duration increases, the systems get de-stabilized.

scholarly journals Neutrosophic Orbit Topological Spaces

2021 ◽  
Vol 18 (24) ◽  
pp. 1443
Author(s):  
T Madhumathi ◽  
F NirmalaIrudayam
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Topological Space ◽  
Necessary Conditions ◽  
Neutrosophic Set ◽  
Topological Spaces ◽  
Open Set ◽  
Evolution Function

Neutrosophy is a flourishing arena which conceptualizes the notion of true, falsity and indeterminancy attributes of an event. In the study of dynamical systems, an orbit is a collection of points related by the evolution function of the dynamical system. Hence in this paper we focus on introducing the concept of neutrosophic orbit topological space denoted as (X, tNO). Also, some of the important characteristics of neutrosophic orbit open sets are discussed with suitable examples. HIGHLIGHTS The orbit in mathematics has an important role in the study of dynamical systems Neutrosophy is a flourishing arena which conceptualizes the notion of true, falsity and indeterminancy attributes of an event. We combine the above two topics and create the following new concept The collection of all neutrosophic orbit open sets under the mapping . we introduce the necessary conditions on the mapping 𝒇 in order to obtain a fixed orbit of a neutrosophic set (i.e., 𝒇(𝝁) = 𝝁) for any neutrosophic orbit open set 𝝁 under the mapping 𝒇

Algorithmic complexity of points in dynamical systems

1993 ◽  
Vol 13 (4) ◽  
pp. 807-830 ◽  
Author(s):  
Homer S. White
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Topological Entropy ◽  
Invariant Probability Measure ◽  
Algorithmic Complexity ◽  
Maximal Entropy ◽  
Property A ◽  
Open Set ◽  
Unique Measure ◽  
Set Of Points

AbstractThis work is based on the author's dissertation. We examine the algorithmic complexity (in the sense of Kolmogorov and Chaitin) of the orbits of points in dynamical systems. Extending a theorem of A. A. Brudno, we note that for any ergodic invariant probability measure on a compact dynamical system, almost every trajectory has a limiting complexity equal to the entropy of the system. We use these results to show that for minimal dynamical systems, and for systems with the tracking property (a weaker version of specification), the set of points whose trajectories have upper complexity equal to the topological entropy is residual. We give an example of a topologically transitive system with positive entropy for which an uncountable open set of points has upper complexity equal to zero. We use techniques from universal data compression to prove a recurrence theorem: if a compact dynamical system has a unique measure of maximal entropy, then any point whose lower complexity is equal to the topological entropy is generic for that unique measure. Finally, we discuss algorithmic versions of the theorem of Kamae on preservation of the class of normal sequences under selection by sequences of zero Kamae-entropy.

Sensitivity and specification property of fuzzified dynamical systems

2018 ◽  
Vol 32 (23) ◽  
pp. 1850268
Author(s):  
Nan Li ◽  
Lidong Wang ◽  
Fengchun Lei
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Original System ◽  
Shadowing Property ◽  
Specification Property

The main purpose of this paper is to further explore the complexity of fuzzified dynamical systems. Especially, we study several kinds of specification properties of Zadeh’s extension. Among other things, we discuss the “stronger” sensitivity on product dynamical systems of g-fuzzification. There are two major ingredients. Firstly, it is proved that the specification (respectively almost specification) property of the original system and its Zadeh’s extension is equivalent, when the original system has the shadowing property. Moreover, we study the [Formula: see text]-sensitivity (respectively multi-sensitivity) of g-fuzzification and its induced product dynamical system.

Asymptotic Independence and Uniform Distribution of Quantization Errors for Spatially Discretized Dynamical Systems

1998 ◽  
Vol 08 (07) ◽  
pp. 1479-1490 ◽  
Author(s):  
P. Diamond ◽  
I. Vladimirov
Keyword(s):  
Dynamical System ◽  
Computer Simulation ◽  
Dynamical Systems ◽  
Computer Arithmetic ◽  
Original System ◽  
Unit Interval ◽  
Asymptotic Independence ◽  
System Study ◽  
Polynomial Mappings ◽  
Finite Set

Computer simulation of dynamical systems involves a state space which is the finite set of computer arithmetic. Restricting state values to this grid produces roundoff effects which can be studied by replacing the original system with a spatially discretized dynamical system. Study of the deviation of the discretized trajectories from those of the original system reduces to that of appropriately defined quantization errors. As the grid is refined, the asymptotic behavior of these quantization errors follows probabilistic laws. These results are applied to discretized polynomial mappings of the unit interval.

Random Bit Generator Based on Non-Autonomous Chaotic Systems

2015 ◽  
pp. 203-229
Author(s):  
Christos Volos ◽  
Ioannis Kyprianidis ◽  
Ioannis Stouboulos ◽  
Sundarapandian Vaidyanathan
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Chaos Theory ◽  
Chaotic Systems ◽  
Statistical Tests ◽  
Van Der Pol ◽  
Chaotic Dynamical Systems ◽  
Random Bit Generator ◽  
Close Relationship ◽  
Autonomous Dynamical System

In the last decade, a very interesting relationship between cryptography and chaos theory was developed. As a result of this close relationship, several chaos-based cryptosystems, especially using autonomous chaotic dynamical systems, have been put forward. However, this chapter presents a novel Chaotic Random Bit Generator (CRBG), which is based on the Poincaré map of a non-autonomous dynamical system. For this reason, the very-well known Duffing-van der Pol system has been used. The proposed CRBG also uses the X-OR function for improving the “randomness” of the produced bit streams, which are subjected to the most stringent statistical tests, the FIPS-140-2 suite tests, to detect the specific characteristics that are expected from random bit sequences.

scholarly journals Attractors for non-autonomous retarded lattice dynamical systems

2015 ◽  
Vol 2 (1) ◽  
Author(s):  
Tomás Caraballo ◽  
Francisco Morillas ◽  
José Valero
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Nonlinear Term ◽  
Pullback Attractor ◽  
System With Delay ◽  
General Growth ◽  
Lattice Dynamical System ◽  
Lattice Dynamical Systems ◽  
Autonomous Dynamical System

AbstractIn this paperwe study a non-autonomous lattice dynamical system with delay. Under rather general growth and dissipative conditions on the nonlinear term,we define a non-autonomous dynamical system and prove the existence of a pullback attractor for such system as well. Both multivalued and single-valued cases are considered.

Mapping of non-autonomous dynamical systems to autonomous ones

2019 ◽  
Vol 16 (06) ◽  
pp. 1950089
Author(s):  
Davood Momeni ◽  
Phongpichit Channuie ◽  
Mudhahir Al Ajmi
Keyword(s):  
Dynamical System ◽  
Partial Differential Equations ◽  
Dynamical Systems ◽  
Differential Equations ◽  
Coordinate Transformations ◽  
Partial Differential ◽  
First Order ◽  
Special Cases ◽  
Autonomous Dynamical Systems ◽  
Autonomous Dynamical System

Using a proper choice of the dynamical variables, we show that a non-autonomous dynamical system transforming to an autonomous dynamical system with a certain coordinate transformations can be obtained by solving a general nonlinear first-order partial differential equations. We examine some special cases and provide particular physical examples. Our framework constitutes general machineries in investigating other non-autonomous dynamical systems.

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 Potential Well in Poincaré Recurrence

Entropy ◽  
2021 ◽  
Vol 23 (3) ◽  
pp. 379
Author(s):  
Miguel Abadi ◽  
Vitor Amorim ◽  
Sandro Gallo
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Stochastic Processes ◽  
Recurrence Time ◽  
Potential Well ◽  
Exponential Approximation ◽  
Mixing Processes ◽  
System Perspective ◽  
Return Times ◽  
Error Terms

From a physical/dynamical system perspective, the potential well represents the proportional mass of points that escape the neighbourhood of a given point. In the last 20 years, several works have shown the importance of this quantity to obtain precise approximations for several recurrence time distributions in mixing stochastic processes and dynamical systems. Besides providing a review of the different scaling factors used in the literature in recurrence times, the present work contributes two new results: (1) For ϕ-mixing and ψ-mixing processes, we give a new exponential approximation for hitting and return times using the potential well as the scaling parameter. The error terms are explicit and sharp. (2) We analyse the uniform positivity of the potential well. Our results apply to processes on countable alphabets and do not assume a complete grammar.

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