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.

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.

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.

Intermingled basins for the triangle map

1996 ◽  
Vol 16 (4) ◽  
pp. 651-662 ◽  
Author(s):  
James C. Alexander ◽  
Brian R. Hunt ◽  
Ittai Kan ◽  
James A. Yorke
Keyword(s):  
Lebesgue Measure ◽  
Basin Of Attraction ◽  
The Other ◽  
Positive Lebesgue Measure ◽  
Quadratic Maps ◽  
Open Set ◽  
Parameter Values ◽  
The Basin Of Attraction

AbstractA family of quadratic maps of the plane has been found numerically for certain parameter values to have three attractors, in a triangular pattern, with ‘intermingled’ basins. This means that for every open set S, if the basin of attraction of one of the attractors intersects S in a set of positive Lebesgue measure, then so do the other two basins. In this paper we mathematically verify this observation for a particular parameter, and prove that our results hold for a set of parameters with positive Lebesgue measure.

Strange attractor in the unfolding of an inclination-flip homoclinic orbit

1996 ◽  
Vol 16 (5) ◽  
pp. 1071-1086 ◽  
Author(s):  
Vincent Naudot
Keyword(s):  
Equilibrium Point ◽  
Unstable Manifold ◽  
Homoclinic Orbit ◽  
Strange Attractor ◽  
Stable Manifold ◽  
Weak Type ◽  
Trivial Intersection ◽  
Inclination Flip ◽  
Hyperbolic Equilibrium ◽  
Dimension One

AbstractWe study the unfolding of a smooth vector-fieldXon ℝ3having a homoclinic orbit to a hyperbolic equilibrium point with three real eigenvalues satisfying − λss< λs< 0 < λuWe say that Γ is an inclination-flip homoclinic orbit if the extended unstable manifold at the equilibrium point is, along Γ, non-transverse to the stable manifold and that Γ is of weak type if the unstable manifold has a non-trivial intersection with a specialC2weak stable manifold of dimension one. In this paper, we show the existence of a strange attractor in the unfolding of an inclination-flip homoclinic orbit (of weak type) in the case where the divergence at the equilibrium point is negative. The crucial idea is to compare the Poincaré return map with the Hénon family:being close to 0.

Bifurcation Complexity from Orbit-Flip Homoclinic Orbit of Weak Type

2016 ◽  
Vol 26 (04) ◽  
pp. 1650059 ◽  
Author(s):  
Qiuying Lu ◽  
Vincent Naudot
Keyword(s):  
Homoclinic Orbit ◽  
Weak Type ◽  
Blow Up ◽  
Three Dimensional ◽  
Homoclinic Orbits ◽  
Unperturbed System ◽  
Orbit Flip ◽  
Inclination Flip ◽  
Three Dimensional Vector Field ◽  
Degenerate Version

In this paper, we study the unfolding of a three-dimensional vector field having an orbit-flip homoclinic orbit of weak type. Such a homoclinic orbit is a degenerate version of the so-called orbit-flip homoclinic orbit. We show the existence of inclination-flip homoclinic orbits of arbitrary higher order bifurcating from the unperturbed system. Our strategy consists of using the local moving coordinates method and blow up in the parameter space. In addition, the numerical existence of the orbit-flip homoclinic orbit of weak type is presented based on the truncated Taylor expansion and the numerical computation for both the strong stable manifold and unstable manifold.

The cusp horseshoe and its bifurcations in the unfolding of an inclination-flip homoclinic orbit

1994 ◽  
Vol 14 (4) ◽  
pp. 667-693 ◽  
Author(s):  
Ale Jan Homburg ◽  
Hiroshi Kokubu ◽  
Martin Krupa
Keyword(s):  
Vector Field ◽  
Homoclinic Orbit ◽  
Homoclinic Orbits ◽  
Small Perturbations ◽  
Smale Horseshoe ◽  
Inclination Flip

AbstractDeng has demonstrated a mechanism through which a perturbation of a vector field having an inclination-flip homoclinic orbit would have a Smale horseshoe. In this article we prove that if the eigenvalues of the saddle to which the homoclinic orbit is asymptotic satisfy the condition 2λu > min{−λs, λuu} then there are arbitrarily small perturbations of the vector field which possess a Smale horseshoe. Moreover we analyze a sequence of bifurcations leading to the annihilation of the horseshoe. This sequence contains, in particular, the points of existence of n-homoclinic orbits with arbitrary n.

Maximum-likelihood estimation in non-standard conditions

1971 ◽  
Vol 70 (3) ◽  
pp. 441-450 ◽  
Author(s):  
P. A. P. Moran
Keyword(s):  
Maximum Likelihood ◽  
Parameter Space ◽  
Null Hypothesis ◽  
Maximum Likelihood Estimators ◽  
Likelihood Estimation ◽  
Standard Theory ◽  
True Parameter ◽  
Open Set ◽  
Estimation Problems ◽  
Parameter Point

The origin of the present paper is the desire to study the asymptotic behaviour of certain tests of significance which can be based on maximum-likelihood estimators. The standard theory of such problems (e.g. Wald(4)) assumes, sometimes tacitly, that the parameter point corresponding to the null hypothesis lies inside an open set in the parameter space. Here we wish to study what happens when the true parameter point, in estimation problems, lies on the boundary of the parameter space.

Osculatory Packings by Spheres

1970 ◽  
Vol 13 (1) ◽  
pp. 59-64 ◽  
Author(s):  
David W. Boyd
Keyword(s):  
Hausdorff Dimension ◽  
Lebesgue Measure ◽  
Simple Proof ◽  
Pairwise Disjoint ◽  
Exponent Of Convergence ◽  
Residual Set ◽  
Open Set ◽  
Finite Lebesgue Measure

If U is an open set in Euclidean N-space EN which has finite Lebesgue measure |U| then a complete packing of U by open spheres is a collection C={Sn} of pairwise disjoint open spheres contained in U and such that Σ∞n=1|Sn| = |U|. Such packings exist by Vitali's theorem. An osculatory packing is one in which the spheres Sn are chosen recursively so that from a certain point on Sn+1 is the largest possible sphere contained in (Here S- will denote the closure of a set S). We give here a simple proof of the "well-known" fact that an osculatory packing is a complete packing. Our method of proof shows also that for osculatory packings, the Hausdorff dimension of the residual set is dominated by the exponent of convergence of the radii of the Sn.

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