scholarly journals Statistics of unstable periodic orbits of a chaotic dynamical system with a large number of degrees of freedom

2005 ◽  
Vol 72 (3) ◽  
Author(s):  
Mitsuhiro Kawasaki ◽  
Shin-ichi Sasa
Keyword(s):  
Dynamical System ◽  
Periodic Orbits ◽  
Degrees Of Freedom ◽  
Unstable Periodic Orbits ◽  
Chaotic Dynamical System

On the Potential for Entangled States Between Chaotic Systems

2014 ◽  
Vol 24 (06) ◽  
pp. 1450077 ◽  
Author(s):  
Matthew A. Morena ◽  
Kevin M. Short
Keyword(s):  
Dynamical System ◽  
Periodic Orbits ◽  
Chaotic Systems ◽  
Entangled States ◽  
Future Research ◽  
Unstable Periodic Orbits ◽  
Qualitative Information ◽  
Chaotic Dynamical System ◽  
Naturally Occurring ◽  
External Intervention

We report on the tendency of chaotic systems to be controlled onto their unstable periodic orbits in such a way that these orbits are stabilized. The resulting orbits are known as cupolets and collectively provide a rich source of qualitative information on the associated chaotic dynamical system. We show that pairs of interacting cupolets may be induced into a state of mutually sustained stabilization that requires no external intervention in order to be maintained and is thus considered bound or entangled. A number of properties of this sort of entanglement are discussed. For instance, should the interaction be disturbed, then the chaotic entanglement would be broken. Based on certain properties of chaotic systems and on examples which we present, there is further potential for chaotic entanglement to be naturally occurring. A discussion of this and of the implications of chaotic entanglement in future research investigations is also presented.

FEEDBACK CONTROL OF THE LIU CHAOTIC DYNAMICAL SYSTEM

2010 ◽  
Vol 24 (03) ◽  
pp. 397-404 ◽  
Author(s):  
XINGYUAN WANG ◽  
XINGUANG LI
Keyword(s):  
Dynamical System ◽  
Feedback Control ◽  
Asymptotic Stability ◽  
Numerical Simulations ◽  
Limit Cycles ◽  
Equilibrium Points ◽  
Unstable Periodic Orbits ◽  
Chaotic Dynamical System ◽  
Feedback Method ◽  
Liu System

Classical feedback method is used to control chaos in the Liu dynamical system. Based on the Routh–Hurwitz criteria, the conditions of the asymptotic stability of the steady states of the controlled Liu system are discussed, and they are also proved theoretically. Numerical simulations show that the method can suppress chaos to both unstable equilibrium points and unstable periodic orbits (limit cycles) successfully.

OPTIMAL NONLINEAR MODELS FROM EMPIRICAL TIME SERIES: AN APPLICATION TO CLIMATE

2004 ◽  
Vol 14 (06) ◽  
pp. 2041-2052 ◽  
Author(s):  
RAFAEL M. GUTIÉRREZ
Keyword(s):  
Dynamical System ◽  
Time Series ◽  
Surface Temperature ◽  
Degrees Of Freedom ◽  
Minimum Description Length ◽  
Lower Atmosphere ◽  
External Parameter ◽  
Optimal Size ◽  
Chaotic Dynamical System ◽  
Macroscopic System

In this work we propose a method that exploits the feedback between empirical and theoretical knowledge of a complex macroscopic system in order to build a nonlinear model. We apply the method to the monthly earth's mean surface temperature time series. The problems of contamination and stationarity are considered noting the importance of observation and modeling scales. We construct a dynamical system of ordinary differential equations where the vector field relating the relevant degrees of freedom and their variations in time is expressed in terms of a polynomial base orthonormal to the measure associated to the time series under study. The optimal size of the model and the values of its parameters are estimated with the principle of minimum description length and the Adams–Molton predictor–corrector method. This procedure is self-consistent because it does not use any external parameter or assumption. We then present a first approach to find the closest chaotic dynamical system corresponding to the earth's mean surface temperature and compare it with scale consistent theoretical or phenomenological models of the lower atmosphere. This comparison allows us to obtain an explicit functional form of the heat capacity of the earth's surface as a function of the earth's mean surface temperature.

THE PENDULUM WEAVES ALL KNOTS AND LINKS

2005 ◽  
Vol 14 (07) ◽  
pp. 869-881
Author(s):  
JOHN STARRETT
Keyword(s):  
Dynamical System ◽  
Strange Attractor ◽  
Complex Behavior ◽  
Unstable Periodic Orbits ◽  
Physical Pendulum ◽  
Chaotic Dynamical System ◽  
Manifold With Boundary ◽  
Infinite Set ◽  
Local Stable Manifolds ◽  
Knots And Links

The behavior of a dissipative chaotic dynamical system is determined by the skeleton of its strange attractor, which consists of an uncountably infinite set of unstable periodic orbits. Each orbit is a topological knot, and the set is an infinite link. The types of knots and links supported by the system may be determined by collapsing the attractor along its local stable manifolds to form a template, a branched two manifold with boundary that supports the same set of knots and links as the original attractor. We show that the strange attractor of a chaotic, vertically forced physical pendulum can be collapsed to a template that supports all knots and links. Thus, one of the simplest and most well known dynamical systems is capable of the most complex behavior possible.

scholarly journals Locating and Stabilizing Unstable Periodic Orbits Embedded in the Horseshoe Map

2021 ◽  
Vol 31 (04) ◽  
pp. 2150110
Author(s):  
Yuu Miino ◽  
Daisuke Ito ◽  
Tetsushi Ueta ◽  
Hiroshi Kawakami
Keyword(s):  
Dynamical System ◽  
Periodic Orbits ◽  
Ad Hoc ◽  
Computation Method ◽  
Control Input ◽  
Unstable Periodic Orbits ◽  
Stable And Unstable Manifolds ◽  
Controlling Chaos ◽  
Dimensional Dynamical System ◽  
The Given

Based on the theory of symbolic dynamical systems, we propose a novel computation method to locate and stabilize the unstable periodic points (UPPs) in a two-dimensional dynamical system with a Smale horseshoe. This method directly implies a new framework for controlling chaos. By introducing the subset based correspondence between a planar dynamical system and a symbolic dynamical system, we locate regions sectioned by stable and unstable manifolds comprehensively and identify the specified region containing a UPP with the particular period. Then Newton’s method compensates the accurate location of the UPP with the regional information as an initial estimation. On the other hand, the external force control (EFC) is known as an effective method to stabilize the UPPs. By applying the EFC to the located UPPs, robust controlling chaos is realized. In this framework, we never use ad hoc approaches to find target UPPs in the given chaotic set. Moreover, the method can stabilize UPPs with the specified period regardless of the situation where the targeted chaotic set is attractive. As illustrative numerical experiments, we locate and stabilize UPPs and the corresponding unstable periodic orbits in a horseshoe structure of the Duffing equation. In spite of the strong instability of UPPs, the controlled orbit is robust and the control input retains being tiny in magnitude.

scholarly journals Characteristics of Periodic Orbits in Elliptical Galaxies

1983 ◽  
Vol 74 ◽  
pp. 271-274
Author(s):  
N. Caranicolas
Keyword(s):  
Dynamical System ◽  
Periodic Orbits ◽  
Degrees Of Freedom ◽  
Basic Characteristic ◽  
Elliptical Galaxies ◽  
Resonance Case ◽  
Two Degrees Of Freedom ◽  
Exact Resonance ◽  
Near Resonance ◽  
Families Of Periodic Orbits

AbstractThe properties of the characteristic curves of several families of periodic orbits, in a conservative dynamical system of two degrees of freedom, symmetric with respect to both axes, are reviewed. The two main types of families are presented. One sees that the pattern of the characteristics in the exact resonance case is similar to that of the near resonance case except for the basic characteristic . The form of the characteristics can be found theoretically by means of the second integral.

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