Chaotic scattering on a double well: Periodic orbits, symbolic dynamics, and scaling

1993 ◽  
Vol 3 (4) ◽  
pp. 475-485 ◽  
Author(s):  
Vincent Daniels ◽  
Michel Vallières ◽  
Jian‐Min Yuan
Keyword(s):  
Periodic Orbits ◽  
Symbolic Dynamics ◽  
Chaotic Scattering

Topological classification of periodic orbits in the Kuramoto–Sivashinsky equation

2018 ◽  
Vol 32 (15) ◽  
pp. 1850155 ◽  
Author(s):  
Chengwei Dong
Keyword(s):  
Nonlinear System ◽  
Variational Method ◽  
Periodic Orbits ◽  
Symbolic Dynamics ◽  
Chaotic Systems ◽  
Building Blocks ◽  
Topological Classification ◽  
One Dimensional ◽  
Sivashinsky Equation

In this paper, we systematically research periodic orbits of the Kuramoto–Sivashinsky equation (KSe). In order to overcome the difficulties in the establishment of one-dimensional symbolic dynamics in the nonlinear system, two basic periodic orbits can be used as basic building blocks to initialize cycle searching, and we use the variational method to numerically determine all the periodic orbits under parameter [Formula: see text] = 0.02991. The symbolic dynamics based on trajectory topology are very successful for classifying all short periodic orbits in the KSe. The current research can be conveniently adapted to the identification and classification of periodic orbits in other chaotic systems.

Map Dynamics Study of the Lorenz Equations

1997 ◽  
Vol 07 (02) ◽  
pp. 373-382 ◽  
Author(s):  
Olivier Michielin ◽  
Paul E. Phillipson
Keyword(s):  
Periodic Orbits ◽  
Symbolic Dynamics ◽  
Strange Attractor ◽  
Good Accuracy ◽  
Simple Solution ◽  
Lorenz Equations ◽  
Computer Solution ◽  
One Dimensional

The Lorenz equations [Lorenz, 1963], in addition to a strange attractor, display sequences of periodic and aperiodic orbits. Approximate one-dimensional map solutions are heuristically constructed, supplementing previous symbolic dynamics studies, which closely reproduce these sequences. A relatively simple solution reproduces the sequence topology to good accuracy. A second more refined solution reproduces to higher accuracy both the topology and scale of the attractor. The second solution is sufficiently accurate to predict periodic orbits not previously observed and difficult to extract directly from computer solution of the Lorenz equations.

scholarly journals Effects of Chaos in the Galactic Halo

1996 ◽  
Vol 169 ◽  
pp. 419-421
Author(s):  
W.J. Schuster ◽  
C. Allen
Keyword(s):  
Periodic Orbits ◽  
Vertical Structure ◽  
Galactic Halo ◽  
Galactic Center ◽  
Galactic Plane ◽  
Orbital Energy ◽  
Chaotic Scattering ◽  
Halo Stars ◽  
Chaotic Orbits ◽  
Families Of Periodic Orbits

To study the way in which the principal periodic orbits in a Galactic potential determine orbital structure, horizontal and vertical surfaces of section, i.e. (dR/dt, R) and (dz/dt, z), are being used to explore the potential of Allen & Santillán (1991) and to investigate possible vertical structure in the Galactic halo. The chaotic “scattering” process due to the nearly spherical mass distribution close to the Galactic center in conjunction with the confinement of the chaotic orbits produces a vertical segregation of both chaotic and non-chaotic orbits in the halo. Certain zmax, zmin are preferred by the chaotic orbits over others as a result of the conservation of the total orbital energy and of the interaction and confinement of the chaotic orbits by the principal families of periodic orbits (Figure 1). Some of these periodic orbits have been identified. Correlations between the structure found in the observed W distribution and that of the numerically determined zmax, zmin histograms are shown for our sample of 280 halo stars (Schuster et al. 1993). W is the star's velocity perpendicular to the Galactic plane and zmax, zmin the maximum distances above or below the Galactic plane, respectively, reached by the star in the course of its orbit. This vertical structure may explain certain puzzling observations of the galactic halo, such as conflicting c/a values for the shape of the halo, and unusual velocity dispersions and/or distributions near the Galactic poles. These results are in good agreement with Hartwick's (1987) two component model for the halo.

Periodic orbits analysis for the Zhou system via variational approach

2019 ◽  
Vol 33 (19) ◽  
pp. 1950212 ◽  
Author(s):  
Chengwei Dong ◽  
Lian Jia
Keyword(s):  
Fixed Points ◽  
Periodic Orbits ◽  
Symbolic Dynamics ◽  
Variational Approach ◽  
Topological Classification ◽  
Dissipative Systems ◽  
Systematic Calculation ◽  
Low Dimensional ◽  
General Method

We proposed a general method for the systematic calculation of unstable cycles in the Zhou system. The variational approach is employed for the cycle search, and we establish interesting symbolic dynamics successfully based on the orbits circuiting property with respect to different fixed points. Upon the defined symbolic rule, cycles with topological length up to five are sought and ordered. Further, upon parameter changes, the homotopy evolution of certain selected cycles are investigated. The topological classification methodology could be widely utilized in other low-dimensional dissipative systems.

VALIDATED STUDY OF THE EXISTENCE OF SHORT CYCLES FOR CHAOTIC SYSTEMS USING SYMBOLIC DYNAMICS AND INTERVAL TOOLS

2011 ◽  
Vol 21 (02) ◽  
pp. 551-563 ◽  
Author(s):  
ZBIGNIEW GALIAS ◽  
WARWICK TUCKER
Keyword(s):  
Periodic Orbits ◽  
Symbolic Dynamics ◽  
Poincaré Map ◽  
Lorenz System ◽  
Interval Methods ◽  
Theoretical Argument ◽  
Poincare Map ◽  
Distinct Symbol ◽  
The Lorenz System ◽  
Symbol Sequences

We show that, for a certain class of systems, the problem of establishing the existence of periodic orbits can be successfully studied by a symbolic dynamics approach combined with interval methods. Symbolic dynamics is used to find approximate positions of periodic points, and the existence of periodic orbits in a neighborhood of these approximations is proved using an interval operator. As an example, the Lorenz system is studied; a theoretical argument is used to prove that each periodic orbit has a distinct symbol sequence. All periodic orbits with the period p ≤ 16 of the Poincaré map associated with the Lorenz system are found. Estimates of the topological entropy of the Poincaré map and the flow, based on the number and flow-times of short periodic orbits, are calculated. Finally, we establish the existence of several long periodic orbits with specific symbol sequences.

TOPOLOGICAL WINDING NUMBERS FOR PERIOD-DOUBLING CASCADES

1996 ◽  
Vol 06 (01) ◽  
pp. 185-187
Author(s):  
CARSTEN KNUDSEN
Keyword(s):  
Periodic Orbits ◽  
Symbolic Dynamics ◽  
Period Doubling ◽  
Accumulation Point ◽  
Winding Number ◽  
Unimodal Maps ◽  
Forced Oscillators ◽  
Essential Properties ◽  
Limiting Value

We define the topological winding number for unimodal maps that share the essential properties of that of winding numbers for forced oscillators exhibiting period-doubling cascades. It is demonstrated how this number can be computed for any of the periodic orbits in the first period-doubling cascade. The limiting winding number at the accumulation point of the first period-doubling cascade is also derived. It is shown that the limiting value for the winding number ω∞ can be computed as the Farey sum of any two neighbouring topological winding numbers in the period-doubling cascade. The derivations are all based on symbolic dynamics and simple combinatorics.

A GEOMETRIC MODEL FOR THE DUFFING OSCILLATOR

1993 ◽  
Vol 03 (03) ◽  
pp. 685-691 ◽  
Author(s):  
J.W.L. McCALLUM ◽  
R. GILMORE
Keyword(s):  
Periodic Orbits ◽  
Symbolic Dynamics ◽  
Chaotic Attractor ◽  
Geometric Model ◽  
Duffing Oscillator ◽  
Chaotic Attractors ◽  
Unstable Periodic Orbits ◽  
Linking Numbers ◽  
Full Period ◽  
Parameter Values

A geometric model for the Duffing oscillator is constructed by analyzing the unstable periodic orbits underlying the chaotic attractors present at particular parameter values. A template is constructed from observations of the motion of the chaotic attractor in a Poincaré section as the section is swept for one full period. The periodic orbits underlying the chaotic attractor are found and their linking numbers are computed. These are compared with the linking numbers from the template and the symbolic dynamics of the orbits are identified. This comparison is used to validate the template identification and label the orbits by their symbolic dynamics.

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