Rigorous study of short periodic orbits for the Lorenz system

Abstract. An infinite number of unstable periodic orbits (UPOs) are embedded in a chaotic system which models some complex phenomenon. Several algorithms which extract UPOs numerically from continuous-time chaotic systems have been proposed. In this article the damped Newton-Raphson-Mees algorithm is reviewed, and some important techniques and remarks concerning the practical numerical computations are exemplified by employing the Lorenz system.

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

International Journal of Bifurcation and Chaos ◽

10.1142/s021812741102857x ◽

2011 ◽

Vol 21 (02) ◽

pp. 551-563 ◽

Cited By ~ 4

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.

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PERIODIC ORBITS OF THE LORENZ SYSTEM THROUGH PERTURBATION THEORY

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127401003632 ◽

2001 ◽

Vol 11 (10) ◽

pp. 2559-2566 ◽

Cited By ~ 3

Author(s):

J. PALACIÁN ◽

P. YANGUAS

Keyword(s):

Periodic Orbits ◽

Lorenz System ◽

Linear Part ◽

Three Dimensional ◽

Invariant Sets ◽

Dimensional System ◽

The Matrix ◽

Lie Transformations ◽

The Lorenz System ◽

The Stability

Different transformations are applied to the Lorenz system with the aim of reducing the initial three-dimensional system into others of dimension two. The symmetries of the linear part of the system are determined by calculating the matrices which commute with the matrix associated to the linear part. These symmetries are extended to the whole system up to an adequate order by using Lie transformations. After the reduction, we formulate the resulting systems using the invariants associated to each reduction. At this step, we calculate for each reduced system the equilibria and their stability. They are in correspondence with the periodic orbits and invariant sets of the initial system, the stability being the same.

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Periodic Orbits, Lyapunov Vectors, and Singular Vectors in the Lorenz System

Journal of the Atmospheric Sciences ◽

10.1175/1520-0469(1998)055<0390:polvas>2.0.co;2 ◽

1998 ◽

Vol 55 (3) ◽

pp. 390-398 ◽

Cited By ~ 47

Author(s):

Anna Trevisan ◽

Francesco Pancotti

Keyword(s):

Periodic Orbits ◽

Lorenz System ◽

Singular Vectors ◽

The Lorenz System

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Templates of Two Foliated Attractors — Lorenz and Chen Systems

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127416500371 ◽

2016 ◽

Vol 26 (02) ◽

pp. 1650037

Author(s):

Martin Rosalie

Keyword(s):

Periodic Orbits ◽

Chaotic Attractor ◽

Lorenz System ◽

Chen System ◽

Attractor Solution ◽

The Lorenz System ◽

Foliated Structure

A chaotic attractor solution of the Lorenz system [Lorenz, 1963] with foliated structure is topologically characterized. Its template permits to both summarize the organization of its periodic orbits and detail the topology of the solution as a branched manifold. A template of an attractor solution of the Chen system [Chen & Ueta, 1999] with a similar foliated structure is also established.

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