Global dynamics of a planar mapping exhibiting orbits of periods 1, 2, 3 and no chaos

1990 ◽  
Vol 333 (1630) ◽  
pp. 209-259
Keyword(s):  
Invariant Manifolds ◽  
Periodic Points ◽  
Chaotic Behaviour ◽  
Global Dynamics ◽  
Geometrical Analysis ◽  
Small Oscillations ◽  
Self Similar ◽  
Planar Mapping ◽  
Exponentially Small

A geometrical analysis of the planar mapping A : ( x,y ) -> ( y+xy,x ) is presented. A complete global portrait of the invariant manifolds of A is found, primarily by deductive methods. The behaviour of some manifolds was initially investigated numerically, but theoretical explanations for the observations are given in every case. The most significant features of the mapping A are : that it has periodic points of periods 1, 2 and 3 only; that it possesses no chaotic behaviour; that it has sequences of abutting regions of self-similar structure, and that it exhibits heteroclinic behaviour manifesting itself as exponentially small oscillations in some of the invariant manifolds.

scholarly journals Classification of rough transformations of a circle from a modern point of view

2018 ◽  
Vol 20 (4) ◽  
pp. 408-418
Author(s):  
A. E. Kolobyanina ◽  
E. V. Nozdrinova ◽  
O. V. Pochinka
Keyword(s):  
Invariant Manifolds ◽  
Sufficient Conditions ◽  
Periodic Points ◽  
Point Of View ◽  
Topological Classification ◽  
Topological Invariants ◽  
Modern Theory ◽  
Ambient Manifold ◽  
Periodic Data

In this paper the authors use modern methods and approaches to present a solution to the problem of the topological classification of circle’s rough transformations in canonical formulation. In the modern theory of dynamical systems such problems are understood as the complete topological classification: finding topological invariants, proving the completeness of the set of invariants found and constructing a standard representative from a given set of topological invariants. Namely, in the first theorem of this paper the type of periodic data of circle’s rough transformations is established. In the second theorem necessary and sufficient conditions of their conjugacy are proved. These conditions mean coincidence of periodic data and rotation numbers. In the third theorem the admissible set of parameters is implemented by a rough transformation of a circle. While proving the theorems, we assume that the results on the local topological classification of hyperbolic periodic points, as well as the results on the global representation of the ambient manifold as a union of invariant manifolds of periodic points, are known.

An Intrinsically Three-Dimensional Fractal

2014 ◽  
Vol 24 (06) ◽  
pp. 1430017 ◽  
Author(s):  
M. Fernández-Guasti
Keyword(s):  
Bifurcation Diagram ◽  
Chaotic Behavior ◽  
Logistic Map ◽  
Three Dimensional ◽  
Periodic Points ◽  
Three Dimensions ◽  
Two Dimensional ◽  
Hyperbolic Numbers ◽  
Self Similar ◽  
One To One

The quadratic iteration is mapped using a nondistributive real scator algebra in three dimensions. The bound set S has a rich fractal-like boundary. Periodic points on the scalar axis are necessarily surrounded by off axis divergent magnitude points. There is a one-to-one correspondence of this set with the bifurcation diagram of the logistic map. The three-dimensional S set exhibits self-similar 3D copies of the elementary fractal along the negative scalar axis. These 3D copies correspond to the windows amid the chaotic behavior of the logistic map. Nonetheless, the two-dimensional projection becomes identical to the nonfractal quadratic iteration produced with hyperbolic numbers. Two- and three-dimensional renderings are presented to explore some of the features of this set.

scholarly journals On Embedding of the Morse-Smale Diffeomorphisms in a Topological Flow

2020 ◽  
Vol 66 (2) ◽  
pp. 160-181
Author(s):  
V. Z. Grines ◽  
E. Ya. Gurevich ◽  
O. V. Pochinka
Keyword(s):  
Invariant Manifolds ◽  
Three Dimensional ◽  
Sufficient Conditions ◽  
Periodic Points ◽  
Topological Classification ◽  
Higher Dimension ◽  
Topological Information ◽  
Topology Of Manifolds ◽  
Necessary And Sufficient

This review presents the results of recent years on solving of the Palis problem on finding necessary and sufficient conditions for the embedding of Morse-Smale cascades in topological flows. To date, the problem has been solved by Palis for Morse-Smale diffeomorphisms given on manifolds of dimension two. The result for the circle is a trivial exercise. In dimensions three and higher new effects arise related to the possibility of wild embeddings of closures of invariant manifolds of saddle periodic points that leads to additional obstacles for Morse-Smale diffeomorphisms to embed in topological flows. The progress achieved in solving of Paliss problem in dimension three is associated with the recently obtained complete topological classification of Morse-Smale diffeomorphisms on three-dimensional manifolds and the introduction of new invariants describing the embedding of separatrices of saddle periodic points in a supporting manifold. The transition to a higher dimension requires the latest results from the topology of manifolds. The necessary topological information, which plays key roles in the proofs, is also presented in the survey.

scholarly journals Exponentially Small Lower Bounds for the Splitting of Separatrices to Whiskered Tori with Frequencies of Constant Type

2014 ◽  
Vol 24 (08) ◽  
pp. 1440011 ◽  
Author(s):  
Amadeu Delshams ◽  
Marina Gonchenko ◽  
Pere Gutiérrez
Keyword(s):  
Lower Bounds ◽  
Invariant Manifolds ◽  
Irrational Number ◽  
Melnikov Method ◽  
Arithmetic Properties ◽  
Constant Type ◽  
Whiskered Tori ◽  
Vector Formula ◽  
Splitting Of Invariant Manifolds ◽  
Exponentially Small

We study the splitting of invariant manifolds of whiskered tori with two frequencies in nearly-integrable Hamiltonian systems, such that the hyperbolic part is given by a pendulum. We consider a two-dimensional torus with a fast frequency vector [Formula: see text], with ω = (1, Ω) where Ω is an irrational number of constant type, i.e. a number whose continued fraction has bounded entries. Applying the Poincaré–Melnikov method, we find exponentially small lower bounds for the maximal splitting distance between the stable and unstable invariant manifolds associated to the invariant torus, and we show that these bounds depend strongly on the arithmetic properties of the frequencies.

scholarly journals The splitting of separatrices for analytic diffeomorphisms

1990 ◽  
Vol 10 (2) ◽  
pp. 295-318 ◽  
Author(s):  
E. Fontich ◽  
C. Simó
Keyword(s):  
Invariant Manifolds ◽  
Complex Singularities ◽  
Homoclinic Points ◽  
Unperturbed Problem ◽  
Splitting Of Separatrices ◽  
Maximum Separation ◽  
Exponentially Small

AbstractWe study families of diffeomorphisms close to the identity, which tend to it when the parameter goes to zero, and having homoclinic points. We consider the analytical case and we find that the maximum separation between the invariant manifolds, in a given region, is exponentially small with respect to the parameter. The exponent is related to the complex singularities of a flow which is taken as an unperturbed problem. Finally several examples are given.

Birkhoff signature change: a criterion for the instability of chaotic resonance

1992 ◽  
Vol 338 (1651) ◽  
pp. 557-568 ◽  
Keyword(s):  
Homoclinic Orbit ◽  
Invariant Manifolds ◽  
Complex Dynamics ◽  
Period Doubling ◽  
Nonlinear Oscillators ◽  
Global Dynamics ◽  
Significant Information ◽  
Signature Change ◽  
Numerical Effort ◽  
Initial Change

For periodically forced nonlinear oscillators permitting escape from a potential well a relation is observed between two well-known phenomena, the period-doubling cascade leading to the chaotic escape of the resonant attractor and the complex dynamics associated with the creation of a structurally unstable homoclinic orbit. The particular homoclinic orbit is identified as that created at the initial change of the period one Birkhoff signature of the invariant manifolds of the hilltop saddle. The primary resonant attractor may thus be viewed as the period one simple Newhouse orbit. Significant subharmonic and superharmonic escape events may likewise be associated with nearby Birkhoff signature changes. Significant information about the global dynamics may thus be obtained with little numerical effort by inspection of the signatures of the invariant manifolds of the hilltop saddle.

scholarly journals On topological conjugacy of some chaotic dynamical systems on the Sierpinski gasket

Filomat ◽  
2021 ◽  
Vol 35 (7) ◽  
pp. 2317-2331
Author(s):  
Nisa Aslan ◽  
Mustafa Saltan ◽  
Bünyamin Demir
Keyword(s):  
Dynamical Systems ◽  
Sierpinski Gasket ◽  
Iterated Function Systems ◽  
Time Algorithm ◽  
Periodic Points ◽  
Topological Conjugacy ◽  
Escape Time ◽  
Sierpiński Gasket ◽  
Iterated Function ◽  
Self Similar

The dynamical systems on the classical fractals can naturally be obtained with the help of their iterated function systems. In the recent years, different ways have been developed to define dynamical systems on the self similar sets. In this paper, we give composition functions by using expanding and folding mappings which generate the classical Sierpinski Gasket via the escape time algorithm. These functions also indicate dynamical systems on this fractal. We express the dynamical systems by using the code representations of the points. Then, we investigate whether these dynamical systems are topologically conjugate (equivalent) or not. Finally, we show that the dynamical systems are chaotic in the sense of Devaney and then we also compute and compare the periodic points.

BIFURCATION STRUCTURE AT 1/3-SUBHARMONIC RESONANCE IN AN ASYMMETRIC NONLINEAR ELASTIC OSCILLATOR

1996 ◽  
Vol 06 (08) ◽  
pp. 1529-1546 ◽  
Author(s):  
G. REGA ◽  
A. SALVATORI
Keyword(s):  
Invariant Manifolds ◽  
Initial Conditions ◽  
Basins Of Attraction ◽  
Subharmonic Resonance ◽  
Phase Portraits ◽  
Global Dynamics ◽  
Bifurcation Structure ◽  
Response Curves ◽  
Nonlinear Elastic ◽  
Insight Into

The attractor-basin bifurcation structure in an asymmetric nonlinear oscillator representative of the planar finite forced dynamics of elastic structural systems with initial curvature is studied at the 1/3-subharmonic resonance regime. Local and global analyses are made by means of different computational tools to obtain frequency-response curves of coexisting regular solutions, bifurcation diagrams ensuing from different sets of initial conditions, manifolds structure of direct and inverse saddles corresponding to unstable periodic solutions, basins of attraction at different values of the control parameter. Deep insight into the global dynamics of the system and its evolution is achieved through the analysis of synthetic attractor-basin-manifold phase portraits. The topological mechanisms which entail onset and disappearance of various attractors, and the main and secondary evolutions to chaos, are identified. Special attention is devoted to the analysis of sudden bifurcational events characterizing the system global dynamics, associated with the topological behavior of the invariant manifolds of several direct and inverse saddles. Features of basin metamorphosis, attractor-basin accessibility, and window occurrence are examined. The approach followed, consisting in combined bifurcation analysis of the attractor-basin structure and of the manifold structure, is thought to be useful for a variety of dynamical systems.

scholarly journals Chaotic Behaviour and Bifurcation in Real Dynamics of Two-Parameter Family of Functions including Logarithmic Map

2020 ◽  
Vol 2020 ◽  
pp. 1-13
Author(s):  
Mohammad Sajid
Keyword(s):  
Fixed Points ◽  
Research Work ◽  
Period Doubling ◽  
Periodic Points ◽  
Real Parameter ◽  
Chaotic Behaviour ◽  
Positive Real ◽  
The Real ◽  
Positive Real Parameter ◽  
Two Parameters

The focus of this research work is to obtain the chaotic behaviour and bifurcation in the real dynamics of a newly proposed family of functions fλ,ax=x+1−λxlnax;x>0, depending on two parameters in one dimension, where assume that λ is a continuous positive real parameter and a is a discrete positive real parameter. This proposed family of functions is different from the existing families of functions in previous works which exhibits chaotic behaviour. Further, the dynamical properties of this family are analyzed theoretically and numerically as well as graphically. The real fixed points of functions fλ,ax are theoretically simulated, and the real periodic points are numerically computed. The stability of these fixed points and periodic points is discussed. By varying parameter values, the plots of bifurcation diagrams for the real dynamics of fλ,ax are shown. The existence of chaos in the dynamics of fλ,ax is explored by looking period-doubling in the bifurcation diagram, and chaos is to be quantified by determining positive Lyapunov exponents.

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