Localization Method of Compact Invariant Sets with Application to the Chua System

2016 ◽  
Vol 26 (05) ◽  
pp. 1650073 ◽  
Author(s):  
A. F. Gribov ◽  
A. N. Kanatnikov ◽  
A. P. Krishchenko
Keyword(s):  
Phase Space ◽  
Nonlinear System ◽  
Bifurcation Analysis ◽  
Invariant Set ◽  
Nonempty Intersection ◽  
Invariant Sets ◽  
Localization Method ◽  
Poincaré Sections ◽  
Systems Of Inequalities ◽  
Parametric Families

In this paper, we consider the problem of compact invariant sets localization for the Chua system. To obtain our results we develop and apply a localization method. This method allows us to find two types of subsets in the phase space of a nonlinear system. The first type consists of Poincaré sections having a nonempty intersection with any compact invariant set of the system. The second type consists of localizing sets containing all compact invariant sets of the system. The considered localization method produces systems of inequalities describing the localizing sets and specifies the equations of the appropriate global sections. These inequalities and equations depend on parameters of the system and, therefore, the obtained localization results can be used in the bifurcation analysis. We find one-parametric families of both compact global sections and nontrivial localizing sets for the Chua system. These localizing sets are compact or unbounded. The intersection of unbounded localizing sets in some cases is a compact localizing set. We indicate the domains where trajectories of the Chua system go to infinity.

CHAOTIC SCATTERING IN TIME-DEPENDENT HAMILTONIAN SYSTEMS

1991 ◽  
Vol 01 (03) ◽  
pp. 667-679 ◽  
Author(s):  
YING-CHENG LAI ◽  
CELSO GREBOGI
Keyword(s):  
Phase Space ◽  
Measure Zero ◽  
Invariant Set ◽  
Time Dependent ◽  
Invariant Sets ◽  
Physical Origin ◽  
Chaotic Scattering ◽  
Periodic Oscillations ◽  
Surface Of Section ◽  
Oscillating Potential

We consider the classical scattering of particles in a one-degree-of-freedom, time-dependent Hamiltonian system. We demonstrate that chaotic scattering can be induced by periodic oscillations in the position of the potential. We study the invariant sets on a surface of section for different amplitudes of the oscillating potential. It is found that for small amplitudes, the phase space consists of nonescaping KAM islands and an escaping set. The escaping set is made up of a nonhyperbolic set that gives rise to chaotic scattering and remains of KAM islands. For large amplitudes, the phase space contains a Lebesgue measure zero invariant set that gives rise to chaotic scattering. In this regime, we also discuss the physical origin of the Cantor set responsible for the chaotic scattering and calculate its fractal dimension.

Quasi-Min-Max MPC for Nonlinear System via Embedding Approach

2011 ◽  
Vol 320 ◽  
pp. 481-486
Author(s):  
Min Zhao ◽  
Ping Ping Song
Keyword(s):  
Computational Complexity ◽  
Nonlinear System ◽  
Predictive Control ◽  
Time Instant ◽  
Invariant Set ◽  
Invariant Sets ◽  
Control Law ◽  
Linear Parameter ◽  
Linear Parameter Varying ◽  
One Step

A quasi-min-max model predictive control (MPC) algorithm is proposed for constrained nonlinear system via an embedding approach. The nonlinear system can be approximated by a linear parameter varying (LPV) model. And a method based on invariant set is proposed for the embedding model to reduce the computational complexity. The proposed method constructs a one-step invariant set comprises an interpolation between several pre-computed invariant sets at each time instant. Then control law is obtained by solving a constrained QP problem, which is also useful for the nonlinear system. The performances of the approach are presented via an example.

Bi-invariant sets and measures have integer Hausdorff dimension

1999 ◽  
Vol 19 (2) ◽  
pp. 523-534 ◽  
Author(s):  
DAVID MEIRI ◽  
YUVAL PERES
Keyword(s):  
Hausdorff Dimension ◽  
Invariant Measure ◽  
Ergodic Measure ◽  
Invariant Set ◽  
Invariant Sets ◽  
Dimensional Torus ◽  
Uniformly Distributed Sequence ◽  
Invariant Ergodic Measure

Let $A,B$ be two diagonal endomorphisms of the $d$-dimensional torus with corresponding eigenvalues relatively prime. We show that for any $A$-invariant ergodic measure $\mu$, there exists a projection onto a torus ${\mathbb T}^r$ of dimension $r\ge\dim\mu$, that maps $\mu$-almost every $B$-orbit to a uniformly distributed sequence in ${\mathbb T}^r$. As a corollary we obtain that the Hausdorff dimension of any bi-invariant measure, as well as any closed bi-invariant set, is an integer.

ESSENTIAL ENTROPY-CARRYING HORSESHOES AS SET LIMITS

2012 ◽  
Vol 22 (08) ◽  
pp. 1250195 ◽  
Author(s):  
STEVEN M. PEDERSON
Keyword(s):  
Topological Entropy ◽  
Convergent Sequence ◽  
Invariant Set ◽  
Invariant Sets ◽  
Piecewise Monotone ◽  
Interval Maps ◽  
Monotone Map ◽  
Monotone Maps

This paper studies the set limit of a sequence of invariant sets corresponding to a convergent sequence of piecewise monotone interval maps. To do this, the notion of essential entropy-carrying set is introduced. A piecewise monotone map f with an essential entropy-carrying horseshoe S(f) and a sequence of piecewise monotone maps [Formula: see text] converging to f is considered. It is proven that if each gi has an invariant set T(gi) with at least as much topological entropy as f, then the set limit of [Formula: see text] contains S(f).

scholarly journals Some compact invariant sets for hyperbolic linear automorphisms of torii

1988 ◽  
Vol 8 (2) ◽  
pp. 191-204 ◽  
Author(s):  
Albert Fathi
Keyword(s):  
Hausdorff Dimension ◽  
Homology Group ◽  
Invariant Set ◽  
Finite Union ◽  
Invariant Sets ◽  
Pisot Number ◽  
Toral Automorphism ◽  
Lower Capacity

AbstractIf the action induced by a pseudo-Anosov map on the first homology group is hyperbolic, it is possible, by a theorem of Franks, to find a compact invariant set for the toral automorphism associated with this action. If the stable and unstable foliations of the Pseudo-Anosov map are orientable, we show that the invariant set is a finite union of topological 2-discs. Using some ideas of Urbański, it is possible to prove that the lower capacity of the associated compact invariant set is >2; in particular, the invariant set is fractal. When the dilatation coefficient is a Pisot number, we can compute the Hausdorff dimension of the compact invariant set.

Dynamics of Thermoacoustic Oscillations Leading to Lean Flame Blowout

2012 ◽  
Author(s):  
Lipika Kabiraj ◽  
R. I. Sujith
Keyword(s):  
Time Series ◽  
Steady State ◽  
Phase Space ◽  
Dynamical Systems ◽  
Nonlinear Oscillation ◽  
Bifurcation Analysis ◽  
Nonlinear Time Series Analysis ◽  
Pressure Oscillations ◽  
Reconstructed Phase Space ◽  
Thermoacoustic Oscillations

Lean flame blowout induced by thermoacoustic oscillations is a serious problem faced by the power and propulsion industry. We analyze a prototypical thermoacoustic system through systematic bifurcation analysis and find that starting from a steady state, this system exhibits successive bifurcations resulting in complex nonlinear oscillation states, eventually leading to flame blowout. To understand the observed bifurcations, we analyze the oscillation states using nonlinear time series analysis, particularly through the representation of pressure oscillations on a reconstructed phase space. Prior to flame blowout, a bursting phenomenon is observed in pressure oscillations. These burst oscillations are found to exhibit similarities with the phenomenon known as intermittency in the dynamical systems theory. This investigation based on nonlinear analysis of experimentally acquired data from a thermoacoustic system sheds light on how thermoacoustic oscillations lead to flame blowout.

scholarly journals A refinement of the Conley index and an application to the stability of hyperbolic invariant sets

1987 ◽  
Vol 7 (1) ◽  
pp. 93-103 ◽  
Author(s):  
Andreas Floer
Keyword(s):  
Continuous Flow ◽  
Index Theory ◽  
Conley Index ◽  
Invariant Set ◽  
Continuation Theorem ◽  
Invariant Sets ◽  
Additional Structure ◽  
Isolated Invariant Set ◽  
Continuation Property ◽  
The Stability

AbstractA compact and isolated invariant set of a continuous flow possesses a so called Conley index, which is the homotopy type of a pointed compact space. For this index a well known continuation property holds true. Our aim is to prove in this context a continuation theorem for the invariant set itself, using an additional structure. This refinement of Conley's index theory will then be used to prove a global and topological continuation-theorem for normally hyperbolic invariant sets.

scholarly journals Invariant Sets of Hybrid Autonomous Systems with Disturbance

2010 ◽  
Vol 2010 ◽  
pp. 1-12
Author(s):  
Li Jian-Qiang ◽  
Zhu Zexuan ◽  
Ji Zhen ◽  
Pei Hai-Long
Keyword(s):  
Lyapunov Function ◽  
Hybrid Systems ◽  
Autonomous Systems ◽  
Invariant Set ◽  
Invariant Sets ◽  
Common Lyapunov Function ◽  
Transition Mode ◽  
Convex Problem

The concept and model of hybrid systems are introduced. Invariant sets introduced by LaSalle are proposed, and the concept is extended to invariant sets in hybrid systems which include disturbance. It is shown that the existence of invariant sets by arbitrary transition in hybrid systems is determined by the existence of common Lyapunov function in the systems. Based on the Lyapunov function, an efficient transition method is proposed to ensure the existence of invariant sets. An algorithm is concluded to compute the transition mode, and the invariant set can also be computed as a convex problem. The efficiency and correctness of the transition algorithm are demonstrated by an example of hybrid systems.

GLOBALLY EXPONENTIALLY ATTRACTIVE SETS AND POSITIVE INVARIANT SETS OF THREE-DIMENSIONAL AUTONOMOUS SYSTEMS WITH ONLY CROSS-PRODUCT NONLINEARITIES

2013 ◽  
Vol 23 (01) ◽  
pp. 1350007 ◽  
Author(s):  
XINQUAN ZHAO ◽  
FENG JIANG ◽  
JUNHAO HU
Keyword(s):  
Chaotic Systems ◽  
Autonomous Systems ◽  
Three Dimensional ◽  
Sufficient Conditions ◽  
Cross Product ◽  
Invariant Set ◽  
Invariant Sets ◽  
Special Cases ◽  
Attractive Sets ◽  
Positive Invariant Set

In this paper, the existence of globally exponentially attractive sets and positive invariant sets of three-dimensional autonomous systems with only cross-product nonlinearities are considered. Sufficient conditions, which guarantee the existence of globally exponentially attractive set and positive invariant set of the system, are obtained. The results of this paper comprise some existing relative results as in special cases. The approach presented in this paper can be applied to study other chaotic systems.

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