Singularities of Solutions of Hamilton–Jacobi Equations

Dynamical Systems ◽

Differential Equations ◽

Riemannian Geometry ◽

Survey Paper ◽

The Past ◽

Propagation Of Singularities ◽

Hamiltonian Dynamical Systems ◽

Jacobi Equations

AbstractThis is a survey paper on the quantitative analysis of the propagation of singularities for the viscosity solutions to Hamilton–Jacobi equations in the past decades. We also review further applications of the theory to various fields such as Riemannian geometry, Hamiltonian dynamical systems and partial differential equations.

PARTIAL DIFFERENTIAL EQUATIONS AND DYNAMICAL SYSTEMS

Demonstratio Mathematica ◽

10.1515/dema-1978-0219 ◽

1978 ◽

Vol 11 (2) ◽

Author(s):

Jan Rogulski

Keyword(s):

Dynamical Systems ◽

Differential Equations ◽

II.1 Quantitative Analysis of Partial Differential Equations

More Progresses in Analysis ◽

10.1142/9789812835635_others05 ◽

2009 ◽

Keyword(s):

Quantitative Analysis ◽

Differential Equations ◽

Invariant measures for the flow of a first order partial differential equation

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385700003059 ◽

1985 ◽

Vol 5 (3) ◽

pp. 437-443 ◽

Cited By ~ 20

Author(s):

R. Rudnicki

Keyword(s):

Differential Equation ◽

Partial Differential Equation ◽

Dynamical Systems ◽

Differential Equations ◽

Invariant Measures ◽

Order Partial Differential Equation ◽

Partial Differential ◽

First Order

AbstractWe prove that the dynamical systems generated by first order partial differential equations are K-flows and chaotic in the sense of Auslander & Yorke.

Advances in Numerical Analysis, Vol. I: Nonlinear Partial Differential Equations and Dynamical Systems.

Mathematics of Computation ◽

10.2307/2153128 ◽

1993 ◽

Vol 60 (202) ◽

pp. 849

Author(s):

V. T. ◽

Will Light

Keyword(s):

Numerical Analysis ◽

Dynamical Systems ◽

Differential Equations ◽

Nonlinear Partial Differential Equations ◽

FINITE DIMENSIONALITY OF ATTRACTORS FOR NON-AUTONOMOUS DYNAMICAL SYSTEMS GIVEN BY PARTIAL DIFFERENTIAL EQUATIONS

Stochastics and Dynamics ◽

10.1142/s0219493704001127 ◽

2004 ◽

Vol 04 (03) ◽

pp. 385-404 ◽

Cited By ~ 28

Author(s):

J. A. LANGA ◽

B. SCHMALFUSS

Keyword(s):

Dynamical Systems ◽

Differential Equations ◽

Sufficient Conditions ◽

Fractal Dimensionality ◽

Natural Generalization ◽

Pullback Attractor ◽

Compact Sets ◽

Qualitative Properties ◽

In the last decade, the concept of pullback attractor has become one of the usual tools to describe some qualitative properties of non-autonomous partial differential equations. A pullback attractor is a family of compact sets, invariant for the corresponding process related to the equation, and attracting from the past, and it assumes a natural generalization of the now classical concept of global attractor for autonomous partial differential equations. In this work we give sufficient conditions in order to prove the finite Hausdorff and fractal dimensionality of pullback attractors for non-autonomous infinite dimensional dynamical systems, and we apply our results to a generalized non-autonomous partial differential equation of Navier–Stokes type.

A Modular Dynamical Cryptosystem Based on Continuous-Interval Cellular Automata

Cognitive Informatics for Revealing Human Cognition ◽

10.4018/978-1-4666-2476-4.ch017 ◽

2012 ◽

pp. 261-283

Author(s):

Jesus D. Terrazas Gonzalez ◽

Witold Kinsner

Keyword(s):

Fractal Dimension ◽

Dynamical Systems ◽

Cellular Automata ◽

Differential Equations ◽

Data Protection ◽

Research Group ◽

The Past ◽

Chaotic Sequences ◽

Encryption And Decryption ◽

Complex Sequences

This paper presents a new cryptosystem based on chaotic continuous-interval cellular automata (CCA) to increase data protection as demonstrated by their flexibility to encrypt and decrypt information from distinct sources. Enhancements to cryptosystems are also presented including (i) a model based on a new chaotic CCA attractor, (ii) the dynamical integration of modules containing dynamical systems to generate complex sequences, and (iii) an enhancement for symmetric cryptosystems by allowing them to generate an unlimited number of keys. This paper also presents a process of mixing chaotic sequences obtained from cellular automata, instead of using differential equations, as a basis to achieve higher security and higher speed for the encryption and decryption processes, as compared to other recent approaches. The complexity of the mixed sequences is measured using the variance fractal dimension trajectory to compare them to the unmixed chaotic sequences to verify that the former are more complex. This type of polyscale measure and evaluation has never been done in the past outside this research group.