scholarly journals On nondegeneracy conditions for the Levi map in higher codimension: a survey

2020 ◽  
Vol 6 (2) ◽  
Author(s):  
Léa Blanc-Centi ◽  
Francine Meylan
Keyword(s):  
Nondegeneracy Conditions

EXISTENCE OF STRONG TRACES FOR GENERALIZED SOLUTIONS OF MULTIDIMENSIONAL SCALAR CONSERVATION LAWS

2005 ◽  
Vol 02 (04) ◽  
pp. 885-908 ◽  
Author(s):  
E. YU. PANOV
Keyword(s):  
Wide Class ◽  
Conservation Law ◽  
Generalized Solutions ◽  
Flux Vector ◽  
Scalar Conservation Laws ◽  
Generalized Entropy ◽  
Scalar Conservation Law ◽  
Nondegeneracy Conditions ◽  
Scalar Conservation ◽  
Class Of Functions

In the half-space t > 0 a multidimensional scalar conservation law with only continuous flux vector is considered. For the wide class of functions including generalized entropy sub- and super-solutions to this equation, we prove existence of the strong trace on the initial hyperspace t = 0. No nondegeneracy conditions on the flux are required.

scholarly journals Partial Preservation of Frequencies and Floquet Exponents of Invariant Tori in the Reversible KAM Context 2

2017 ◽  
Vol 63 (3) ◽  
pp. 516-541
Author(s):  
M B Sevryuk
Keyword(s):  
Fixed Point ◽  
Invariant Torus ◽  
Variational Equation ◽  
Invariant Tori ◽  
Kam Theory ◽  
Coefficient Matrix ◽  
Nondegeneracy Conditions

We consider the persistence of smooth families of invariant tori in the reversible context 2 of KAM theory under various weak nondegeneracy conditions via Herman’s method. The reversible KAM context 2 refers to the situation where the dimension of the fixed point manifold of the reversing involution is less than half the codimension of the invariant torus in question. The nondegeneracy conditions we employ ensure the preservation of any prescribed subsets of the frequencies of the unperturbed tori and of their Floquet exponents (the eigenvalues of the coefficient matrix of the variational equation along the torus).

scholarly journals Formation and construction of a shock wave for 3-D compressible Euler equations with the spherical initial data

2004 ◽  
Vol 175 ◽  
pp. 125-164 ◽  
Author(s):  
Huicheng Yin
Keyword(s):  
Shock Wave ◽  
Initial Data ◽  
Euler Equations ◽  
Blow Up ◽  
Three Dimensional ◽  
Compressible Euler Equations ◽  
Lipschitz Continuous ◽  
Compressible Euler ◽  
Nondegeneracy Conditions ◽  
Weak Entropy Solution

AbstractIn this paper, the problem on formation and construction of a shock wave for three dimensional compressible Euler equations with the small perturbed spherical initial data is studied. If the given smooth initial data satisfy certain nondegeneracy conditions, then from the results in [22], we know that there exists a unique blowup point at the blowup time such that the first order derivatives of a smooth solution blow up, while the solution itself is still continuous at the blowup point. From the blowup point, we construct a weak entropy solution which is not uniformly Lipschitz continuous on two sides of a shock curve. Moreover the strength of the constructed shock is zero at the blowup point and then gradually increases. Additionally, some detailed and precise estimates on the solution are obtained in a neighbourhood of the blowup point.

ON THE COMPUTATION OF LOCAL BIFURCATION DIAGRAMS NEAR DEGENERATE HOPF BIFURCATIONS OF CERTAIN TYPES

1993 ◽  
Vol 03 (05) ◽  
pp. 1103-1122 ◽  
Author(s):  
JORGE LUIS MOIOLA
Keyword(s):  
System Theory ◽  
Phase Portrait ◽  
Harmonic Balance ◽  
Graphical Method ◽  
Feedback System ◽  
Hopf Bifurcations ◽  
Local Bifurcation ◽  
Bifurcation Diagrams ◽  
Nondegeneracy Conditions ◽  
Analytical Expressions

The computation of local bifurcation diagrams near degenerate Hopf bifurcations of certain types using feedback system theory and harmonic balance techniques is presented. This approach also provides the analytical expressions for the defining and the nondegeneracy conditions in the so-called frequency domain counterpart. A classical graphical method is easily adapted to carry on the continuation of the oscillatory branches to depict the local bifurcation diagrams. Moreover, several higher-order harmonic balance approximations are implemented to compare the accuracy of the computed solutions. The results are presented using local bifurcation diagrams, phase portrait plots and period diagrams, with similar ones obtained by using AUTO.

Bifurcations of a Bouncing Ball Dynamical System

2019 ◽  
Vol 29 (14) ◽  
pp. 1950191
Author(s):  
Shengfu Deng
Keyword(s):  
Melnikov Method ◽  
Flip Bifurcation ◽  
Difference System ◽  
Form Analysis ◽  
Center Manifold Theorem ◽  
Bouncing Ball ◽  
Nondegeneracy Conditions ◽  
The Difference ◽  
The Stability ◽  
Simple Difference

A deceptively simple difference system of a bouncing ball is investigated. Applying the center manifold theorem and the normal form analysis, we first give the parameter conditions for the flip bifurcation. Then we discuss the 1:2 resonance. Because the nondegeneracy conditions are not satisfied, we use the approximation of mappings by a flow and change this difference system into an ordinary differential system with dimension two. With the aid of the Melnikov method, the homoclinic bifurcation and the Poincaré bifurcation are analyzed, which imply the existence of the invariant circles for the difference system. Furthermore, we compute the normal forms to provide the parameter conditions for the Chenciner bifurcation and also obtain the stability of the fixed point. Finally, some numerical simulations are presented to verify the obtained results.

scholarly journals Topological normal forms of high degree for the simplest bifurcations

2001 ◽  
Vol 2 (2) ◽  
pp. 155 ◽  
Author(s):  
Francisco Balibrea ◽  
Jose C. Valverde Fajardo
Keyword(s):  
Normal Forms ◽  
Nondegeneracy Conditions ◽  
High Degree

This paper is devoted to study the topological normal forms of families of maps on R which, under nondegeneracy conditions of high degree, also present the simplest bifurcations.

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