scholarly journals Non-existence of wandering intervals and structure of topological attractors of one dimensional dynamical systems: 1. The case of negative Schwarzian derivative

1989 ◽  
Vol 9 (4) ◽  
pp. 737-749 ◽  
Author(s):  
M. Yu. Lyubich
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Limit Cycle ◽  
Critical Points ◽  
Schwarzian Derivative ◽  
Dimensional Manifold ◽  
Generic Point ◽  
One Dimensional ◽  
Manifold With Boundary ◽  
Dimensional Dynamical System

AbstractIt is proved that an arbitrary one dimensional dynamical system with negative Schwarzian derivative and non-degenerate critical points has no wandering intervals. This result implies a rather complete view of the dynamics of such a system. In particular, every minimal topological attractor is either a limit cycle, or a one dimensional manifold with boundary, or a solenoid. The orbit of a generic point tends to some minimal attractor.

scholarly journals Non-existence of wandering intervals and structure of topological attractors of one dimensional dynamical systems 2. The smooth case

1989 ◽  
Vol 9 (4) ◽  
pp. 751-758 ◽  
Author(s):  
A. M. Blokh ◽  
M. Yu. Lyubich
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Critical Points ◽  
One Dimensional ◽  
Smooth Case ◽  
Smooth Dynamical System

AbstractWe prove that an arbitrary one dimensional smooth dynamical system with non-degenerate critical points has no wandering intervals.

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.

PERIOD-DOUBLING SCENARIO WITHOUT FLIP BIFURCATIONS IN A ONE-DIMENSIONAL MAP

2005 ◽  
Vol 15 (04) ◽  
pp. 1267-1284 ◽  
Author(s):  
V. AVRUTIN ◽  
M. SCHANZ
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Symmetry Breaking ◽  
Period Doubling ◽  
Pitchfork Bifurcation ◽  
Period Doubling Bifurcation ◽  
Coexisting Attractors ◽  
One Dimensional ◽  
Poincaré Return Map ◽  
Smooth Dynamical System

In this work a one-dimensional piecewise-smooth dynamical system, representing a Poincaré return map for dynamical systems of the Lorenz type, is investigated. The system shows a bifurcation scenario similar to the classical period-doubling one, but which is influenced by so-called border collision phenomena and denoted as border collision period-doubling bifurcation scenario. This scenario is formed by a sequence of pairs of bifurcations, whereby each pair consists of a border collision bifurcation and a pitchfork bifurcation. The mechanism leading to this scenario and its characteristic properties, like symmetry-breaking and symmetry-recovering as well as emergence of coexisting attractors, are investigated.

Zero-dimensional covers of finite dimensional dynamical systems

1995 ◽  
Vol 15 (5) ◽  
pp. 939-950 ◽  
Author(s):  
John Kulesza
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Periodic Points ◽  
Finite Dimensional ◽  
Dimensional Dynamical System

AbstractIf (X, f) is a compact metric, finite-dimensional dynamical system with a zero-dimensional set of periodic points, then there is a zero-dimensional compact metric dynamical system (C, g) and a finite-to-one (in fact, at most (n + l)n-to-one) surjection h: C → X such that h o g = f o h. An example shows that the requirement on the set of periodic points is necessary.

scholarly journals BISTABILITY, BIFURCATION AND CHAOS IN A LASER SYSTEM

1995 ◽  
Vol 05 (04) ◽  
pp. 1021-1031 ◽  
Author(s):  
F. O'CAIRBRE ◽  
A. G. O'FARRELL ◽  
A. O'REILLY
Keyword(s):  
Dynamical System ◽  
Laser Physics ◽  
Laser System ◽  
Bifurcation And Chaos ◽  
One Dimensional ◽  
Fabry Perot ◽  
Dimensional Dynamical System ◽  
Parameter Ranges

In this paper we study a one-dimensional dynamical system that provides a model for the evolution of a Fabry–Perot cavity in Laser Physics. We determine the parameter ranges for bistability and chaos in the system. We also examine the bifurcations of the system and the occurrence of various types of cycles.

scholarly journals Markov covers and finiteness of periodic attractors for diffeomorphisms with a dominated splitting

2000 ◽  
Vol 20 (1) ◽  
pp. 1-14
Author(s):  
MASAYUKI ASAOKA
Keyword(s):  
Dynamical System ◽  
Dominated Splitting ◽  
One Dimensional ◽  
Surface Diffeomorphisms ◽  
Periodic Attractors ◽  
Dimensional Dynamical System

In this paper, we show the existence of Markov covers for $C^2$ surface diffeomorphisms with a dominated splitting under some assumptions. Using a Markov cover, we can reduce the dynamics to a one-dimensional dynamical system having a good metric property. As an application, we show finiteness of periodic attractors for the above diffeomorphisms.

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.

The Dynamical Systems Safety Analysis by SQMD Method

2014 ◽  
Vol 693 ◽  
pp. 92-97
Author(s):  
Pavol Tanuska ◽  
Milan Strbo ◽  
Augustin Gese ◽  
Barbora Zahradnikova
Keyword(s):  
Dynamical System ◽  
Control System ◽  
Dynamical Systems ◽  
Critical Points ◽  
Safety Analysis ◽  
Critical System ◽  
Safety Critical ◽  
Qualitative And Quantitative ◽  
Safety Critical System

The objective of the article is to demonstrate the principle of the SQMD method concept for performing safety analysis on the example of a dynamical system. The safety analysis is performed in the process of designing a control system for safety-critical system processes. The safety analysis is aimed at using the models to monitor different critical points of the system. For the purpose of modelling, we suggest using the SQMD method combining qualitative and quantitative procedures of modelling and taking both methods advantages.

scholarly journals Forced Oscillations of a Damped Korteweg-de Vries Equation in a Quarter Plane

2003 ◽  
Vol 05 (03) ◽  
pp. 369-400 ◽  
Author(s):  
Jerry L. Bona ◽  
S. M. Sun ◽  
Bing-Yu Zhang
Keyword(s):  
Dynamical System ◽  
Limit Cycle ◽  
Small Amplitude ◽  
Vries Equation ◽  
Initial Condition ◽  
Infinite Dimensional ◽  
De Vries ◽  
Exponentially Stable ◽  
Dimensional Dynamical System ◽  
Infinite Dimensional Dynamical System

Laboratory experiments have shown that when nonlinear, dispersive waves are forced periodically from one end of an undisturbed stretch of the medium of propagation, the signal eventually becomes temporally periodic at each spatial point. It is our purpose here to establish this as a fact at least in the context of a damped Korteweg-de Vries equation. Thus, consideration is given to the initial-boundary-value problem [Formula: see text] For this problem, it is shown that if the small amplitude, boundary forcing h is periodic of period T, say, then the solution u of (*) is eventually periodic of period T. More precisely, we show for each x > 0, that u(x, t + T) - u(x, t) converges to zero exponentially as t → ∞. Viewing (*) (without the initial condition) as an infinite dimensional dynamical system in the Hilbert space Hs(R+) for suitable values of s, we also demonstrate that for a given, small amplitude time-periodic boundary forcing, the system (*) admits a unique limit cycle, or forced oscillation (a solution of (*) without the initial condition that is exactly periodic of period T). Furthermore, we show that this limit cycle is globally exponentially stable. In other words, it comprises an exponentially stable attractor for the infinite-dimensional dynamical system described by (*).

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