Strange Nonchaotic Trajectories on Torus

1997 ◽  
Vol 07 (02) ◽  
pp. 423-429 ◽  
Author(s):  
T. Kapitaniak ◽  
L. O. Chua
Keyword(s):  
Lebesgue Measure ◽  
Parameter Space ◽  
Positive Lebesgue Measure ◽  
Strange Nonchaotic Attractors

In this letter we have shown that aperiodic nonchaotic trajectories characteristic of strange nonchaotic attractors can occur on a two-frequency torus. We found that these trajectories are robust as they exist on a positive Lebesgue measure set in the parameter space.

A STRANGE ATTRACTOR WITH LARGE ENTROPY IN THE UNFOLDING OF A LOW RESONANT DEGENERATE HOMOCLINIC ORBIT

2006 ◽  
Vol 16 (12) ◽  
pp. 3509-3522 ◽  
Author(s):  
M. MARTENS ◽  
V. NAUDOT ◽  
J. YANG
Keyword(s):  
Vector Field ◽  
Lebesgue Measure ◽  
Parameter Space ◽  
Homoclinic Orbit ◽  
Strange Attractor ◽  
Linear Part ◽  
Unperturbed System ◽  
Positive Lebesgue Measure ◽  
Open Set ◽  
Inclination Flip

The unfolding of a vector field exhibiting a degenerate homoclinic orbit of inclination-flip type is studied. The linear part of the unperturbed system possesses a resonance but the coefficient of the corresponding monomial vanishes. We show that for an open set in the parameter space, the system possesses a suspended cubic Hénon-like map. As a consequence, strange attractors with entropy close to log 3 persist in a positive Lebesgue measure set.

scholarly journals Torsion Discriminance for Stability of Linear Time-Invariant Systems

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 386
Author(s):  
Yuxin Wang ◽  
Huafei Sun ◽  
Yueqi Cao ◽  
Shiqiang Zhang
Keyword(s):  
Lebesgue Measure ◽  
Linear Time ◽  
Zero Solution ◽  
Positive Lebesgue Measure ◽  
Asymptotically Stable ◽  
Linear Time Invariant ◽  
Measurable Set ◽  
Linear Time Invariant Systems ◽  
The Stability ◽  
Time Invariant

This paper extends the former approaches to describe the stability of n-dimensional linear time-invariant systems via the torsion τ ( t ) of the state trajectory. For a system r ˙ ( t ) = A r ( t ) where A is invertible, we show that (1) if there exists a measurable set E 1 with positive Lebesgue measure, such that r ( 0 ) ∈ E 1 implies that lim t → + ∞ τ ( t ) ≠ 0 or lim t → + ∞ τ ( t ) does not exist, then the zero solution of the system is stable; (2) if there exists a measurable set E 2 with positive Lebesgue measure, such that r ( 0 ) ∈ E 2 implies that lim t → + ∞ τ ( t ) = + ∞ , then the zero solution of the system is asymptotically stable. Furthermore, we establish a relationship between the ith curvature ( i = 1 , 2 , ⋯ ) of the trajectory and the stability of the zero solution when A is similar to a real diagonal matrix.

On a Minimization Problem Involving the Critical Sobolev Exponent

2007 ◽  
Vol 7 (4) ◽  
Author(s):  
Francesca Prinari ◽  
Nicola Visciglia
Keyword(s):  
Lebesgue Measure ◽  
Minimization Problem ◽  
Lorentz Space ◽  
Critical Sobolev Exponent ◽  
Positive Lebesgue Measure ◽  
Sobolev Exponent ◽  
Interior Part

AbstractFollowing [3] we study the following minimization problem:in any dimension n ≥ 4 and under suitable assumptions on a(x). Mainly we assume that a(x) belongs to the Lorentz space LN ≡ {x ∈ Ω|a(x) < 0}has positive Lebesgue measure. Notice that this last condition is satisfied when the set N has a nontrivial interior part (in fact this is the typical assumption imposed in the literature on the set N).

Transitive maps which are not ergodic with respect to Lebesgue measure

1996 ◽  
Vol 16 (4) ◽  
pp. 833-848 ◽  
Author(s):  
Sebastian Van Strien
Keyword(s):  
Lebesgue Measure ◽  
Riemann Sphere ◽  
Julia Set ◽  
Iteration Scheme ◽  
Rational Maps ◽  
Positive Lebesgue Measure ◽  
Rational Map ◽  
Interval Maps ◽  
Open Set ◽  
Totally Disconnected

AbstractIn this paper we shall give examples of rational maps on the Riemann sphere and also of polynomial interval maps which are transitive but not ergodic with respect to Lebesgue measure. In fact, these maps have two disjoint compact attractors whose attractive basins are ‘intermingled’, each having a positive Lebesgue measure in every open set. In addition, we show that there exists a real bimodal polynomial with Fibonacci dynamics (of the type considered by Branner and Hubbard), whose Julia set is totally disconnected and has positive Lebesgue measure. Finally, we show that there exists a rational map associated to the Newton iteration scheme corresponding to a polynomial whose Julia set has positive Lebesgue measure.

scholarly journals Attracting Cantor set of positive measure for a C∞ map of an interval

1982 ◽  
Vol 2 (3-4) ◽  
pp. 405-415 ◽  
Author(s):  
Michał Misiurewicz
Keyword(s):  
Lebesgue Measure ◽  
Positive Measure ◽  
Cantor Set ◽  
Positive Lebesgue Measure ◽  
Smooth Map

AbstractWe give an example of a smooth map of an interval into itself, conjugate to the Feigenbaum map, for which the attracting Cantor set has positive Lebesgue measure.

scholarly journals Lebesgue measure of escaping sets of entire functions

2018 ◽  
Vol 40 (1) ◽  
pp. 89-116 ◽  
Author(s):  
WEIWEI CUI
Keyword(s):  
Entire Function ◽  
Recent Work ◽  
Lebesgue Measure ◽  
Finite Order ◽  
Infinite Order ◽  
Entire Functions ◽  
Transcendental Entire Function ◽  
Positive Lebesgue Measure

For a transcendental entire function $f$ of finite order in the Eremenko–Lyubich class ${\mathcal{B}}$, we give conditions under which the Lebesgue measure of the escaping set ${\mathcal{I}}(f)$ of $f$ is zero. This complements the recent work of Aspenberg and Bergweiler [Math. Ann. 352(1) (2012), 27–54], in which they give conditions on entire functions in the same class with escaping sets of positive Lebesgue measure. We will construct an entire function in the Eremenko–Lyubich class to show that the condition given by Aspenberg and Bergweiler is essentially sharp. Furthermore, we adapt our idea of proof to certain infinite-order entire functions. Under some restrictions to the growth of these entire functions, we show that the escaping sets have zero Lebesgue measure. This generalizes a result of Eremenko and Lyubich.

On some properties of sticky Brownian motion

2020 ◽  
pp. 2150037
Author(s):  
Haoyan Zhang ◽  
Pingping Jiang
Keyword(s):  
Brownian Motion ◽  
Laplace Transform ◽  
Explicit Expression ◽  
Lebesgue Measure ◽  
Transition Density ◽  
Positive Lebesgue Measure ◽  
First Hitting Time ◽  
Skew Brownian Motion ◽  
Basic Probability ◽  
Random Jump

In this paper, we investigate a generalization of Brownian motion, called sticky skew Brownian motion, which has two interesting characteristics: stickiness and skewness. This kind of processes spends a lot more time at its sticky points so that the time they spend at the sticky points has positive Lebesgue measure. By using time change, we obtain an SDE for the sticky skew Brownian motion. Then, we present the explicit relationship between symmetric local time and occupation time. Some basic probability properties, such as transition density, are studied and we derive the explicit expression of Laplace transform of transition density for the sticky skew Brownian motion. We also consider the first hitting time problems over a constant boundary and a random jump boundary, respectively, and give some corollaries based on the results above.

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