scholarly journals Two-parameter unfolding of a parabolic point of a vector field in ℂ fixing the origin

2021 ◽  
Vol 30 (1) ◽  
pp. 139-169
Author(s):  
Christiane Rousseau
Keyword(s):  
Vector Field ◽  
Parabolic Point ◽  
Two Parameter

scholarly journals Lorenz attractors through Šil'nikov-type bifurcation. Part I

1990 ◽  
Vol 10 (4) ◽  
pp. 793-821 ◽  
Author(s):  
Marek Ryszard Rychlik
Keyword(s):  
Fixed Point ◽  
Vector Field ◽  
Differential Equations ◽  
Unstable Manifold ◽  
Parameter Family ◽  
Double Loop ◽  
Hyperbolic Fixed Point ◽  
Lorenz Attractors ◽  
Two Parameter

AbstractThe main result of this paper is a construction of geometric Lorenz attractors (as axiomatically defined by J. Guckenheimer) by means of an Ω-explosion. The unperturbed vector field on ℝ3is assumed to have a hyperbolic fixed point, whose eigenvalues satisfy the inequalities λ1> 0, λ2> 0, λ3> 0 and |λ2|>|λ1|>|λ3|. Moreover, the unstable manifold of the fixed point is supposed to form a double loop. Under some other natural assumptions a generic two-parameter family containing the unperturbed vector field contains geometric Lorenz attractors.A possible application of this result is a method of proving the existence of geometric Lorenz attractors in concrete families of differential equations. A detailed discussion of the method is in preparation and will be published as Part II.

scholarly journals On the Universal Unfolding of Vector Fields in One Variable: A Proof of Kostov’s Theorem

2020 ◽  
Vol 19 (3) ◽  
Author(s):  
Martin Klimeš ◽  
Christiane Rousseau
Keyword(s):  
Vector Field ◽  
Vector Fields ◽  
Versal Deformation ◽  
Dimensional Vector ◽  
Universal Unfolding ◽  
Parabolic Point ◽  
The Real ◽  
Real Analytic

AbstractIn this note we present variants of Kostov’s theorem on a versal deformation of a parabolic point of a complex analytic 1-dimensional vector field. First we provide a self-contained proof of Kostov’s theorem, together with a proof that this versal deformation is indeed universal. We then generalize to the real analytic and formal cases, where we show universality, and to the $${\mathcal {C}}^\infty $$ C ∞ case, where we show that only versality is possible.

scholarly journals Torus-Breakdown Near a Heteroclinic Attractor: A Case Study

2021 ◽  
Vol 31 (10) ◽  
pp. 2130029
Author(s):  
Luísa Castro ◽  
Alexandre Rodrigues
Keyword(s):  
Vector Field ◽  
Vector Fields ◽  
Strange Attractors ◽  
Complete Understanding ◽  
Arnold Tongue ◽  
The Family ◽  
Organizing Center ◽  
Torus Breakdown ◽  
Two Parameter

There are few explicit examples in the literature of vector fields exhibiting observable chaos that may be proved analytically. This paper reports numerical experiments performed for an explicit two-parameter family of [Formula: see text]-symmetric vector fields whose organizing center exhibits an attracting heteroclinic network linking two saddle-foci. Each vector field in the family is the restriction to [Formula: see text] of a polynomial vector field in [Formula: see text]. We investigate global bifurcations due to symmetry-breaking and we detect strange attractors via a mechanism called Torus-Breakdown. We explain how an attracting torus gets destroyed by following the changes in the unstable manifold of a saddle-focus. Although a complete understanding of the corresponding bifurcation diagram and the mechanisms underlying the dynamical changes is out of reach, we uncover complex patterns for the symmetric family under analysis, using a combination of theoretical tools and computer simulations. This article suggests a route to obtain rotational horseshoes and strange attractors; additionally, we make an attempt to elucidate some of the bifurcations involved in an Arnold tongue.

Some results of the Barbier-Chalonge-Divan system of stellar classification

1966 ◽  
Vol 24 ◽  
pp. 77-90 ◽  
Author(s):  
D. Chalonge
Keyword(s):  
Early Type ◽  
Parameter System ◽  
Early Type Stars ◽  
Two Parameter

Several years ago a three-parameter system of stellar classification has been proposed (1, 2), for the early-type stars (O-G): it was an improvement on the two-parameter system described by Barbier and Chalonge (3).

Subjective Time Preference and Willingness to Pay for an Energy-Saving Durable Good

2001 ◽  
Vol 32 (3) ◽  
pp. 133-141 ◽  
Author(s):  
Gerrit Antonides ◽  
Sophia R. Wunderink
Keyword(s):  
Willingness To Pay ◽  
Energy Saving ◽  
Subjective Time ◽  
Durable Good ◽  
Discount Function ◽  
Hyperbolic Discount ◽  
The One ◽  
Two Parameter ◽  
Difference Effect ◽  
Better Than

Summary: Different shapes of individual subjective discount functions were compared using real measures of willingness to accept future monetary outcomes in an experiment. The two-parameter hyperbolic discount function described the data better than three alternative one-parameter discount functions. However, the hyperbolic discount functions did not explain the common difference effect better than the classical discount function. Discount functions were also estimated from survey data of Dutch households who reported their willingness to postpone positive and negative amounts. Future positive amounts were discounted more than future negative amounts and smaller amounts were discounted more than larger amounts. Furthermore, younger people discounted more than older people. Finally, discount functions were used in explaining consumers' willingness to pay for an energy-saving durable good. In this case, the two-parameter discount model could not be estimated and the one-parameter models did not differ significantly in explaining the data.

Hydrodynamics of polymer solutions via two-parameter scaling

1994 ◽  
Vol 4 (8) ◽  
pp. 1299-1310 ◽  
Author(s):  
Ralph H. Colby ◽  
Michael Rubinstein ◽  
Mohamed Daoud
Keyword(s):  
Polymer Solutions ◽  
Two Parameter
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