Dynamical systems generated by sobolev class vector fields in finite and infinite dimensions

1999 ◽  
Vol 94 (3) ◽  
pp. 1394-1445
Author(s):  
V. Bogachev ◽  
E. Mayer-Wolf
Keyword(s):  
Dynamical Systems ◽  
Vector Fields ◽  
Sobolev Class ◽  
Infinite Dimensions

scholarly journals A Twenty-First Century Guidebook for Applied Dynamical Systems

2006 ◽  
Vol 1 (4) ◽  
pp. 279-282 ◽  
Author(s):  
A. R. Champneys
Keyword(s):  
Dynamical Systems ◽  
21St Century ◽  
Vector Fields ◽  
Nonlinear Oscillations ◽  
First Century ◽  
Twenty First Century ◽  
The Impact ◽  
Young Researchers

This paper represents the author’s view on the impact of the book Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields by John Guckenheimer and Philip Holmes, first published in 1983 (Springer-Verlag, Berlin). In particular, the questions addressed are: if one were to write a similar book for the 21st century, which topics should be contained and what form should the book take in order to have a similar impact on the modern generation of young researchers in applied dynamical systems?

SINGULAR FOLIATION GENERATED BY AN ORBIT OF FAMILY OF VECTOR FIELDS

2021 ◽  
Vol 10 (4) ◽  
pp. 2141-2147
Author(s):  
X.F. Sharipov ◽  
B. Boymatov ◽  
N. Abriyev
Keyword(s):  
Dynamical Systems ◽  
Euclidean Space ◽  
Control Theory ◽  
Vector Fields ◽  
Singular Foliation ◽  
Completely Integrable ◽  
Conformal Vector ◽  
Conformal Vector Fields ◽  
Systems Control

Geometry of orbit is a subject of many investigations because it has important role in many branches of mathematics such as dynamical systems, control theory. In this paper it is studied geometry of orbits of conformal vector fields. It is shown that orbits of conformal vector fields are integral submanifolds of completely integrable distributions. Also for Euclidean space it is proven that if all orbits have the same dimension they are closed subsets.

Stable Index Pairs for Discrete Dynamical Systems

1997 ◽  
Vol 40 (4) ◽  
pp. 448-455 ◽  
Author(s):  
Tomasz Kaczynski ◽  
Marian Mrozek
Keyword(s):  
Dynamical Systems ◽  
Vector Fields ◽  
Discrete Dynamical Systems ◽  
Discrete Case ◽  
Smooth Vector ◽  
Small Perturbations ◽  
Index Pairs ◽  
Open Question ◽  
Stable Index

AbstractA new shorter proof of the existence of index pairs for discrete dynamical systems is given. Moreover, the index pairs defined in that proof are stable with respect to small perturbations of the generating map. The existence of stable index pairs was previously known in the case of diffeomorphisms and flows generated by smooth vector fields but it was an open question in the general discrete case.

scholarly journals Finite-time Lagrangian transport analysis: stable and unstable manifolds of hyperbolic trajectories and finite-time Lyapunov exponents

2010 ◽  
Vol 17 (1) ◽  
pp. 1-36 ◽  
Author(s):  
M. Branicki ◽  
S. Wiggins
Keyword(s):  
Dynamical Systems ◽  
Finite Time ◽  
Vector Fields ◽  
Transient Flow ◽  
Time Interval ◽  
Stable And Unstable Manifolds ◽  
Lagrangian Transport ◽  
Transport Barriers ◽  
Unstable Manifolds ◽  
Hyperbolic Trajectories

Abstract. We consider issues associated with the Lagrangian characterisation of flow structures arising in aperiodically time-dependent vector fields that are only known on a finite time interval. A major motivation for the consideration of this problem arises from the desire to study transport and mixing problems in geophysical flows where the flow is obtained from a numerical solution, on a finite space-time grid, of an appropriate partial differential equation model for the velocity field. Of particular interest is the characterisation, location, and evolution of transport barriers in the flow, i.e. material curves and surfaces. We argue that a general theory of Lagrangian transport has to account for the effects of transient flow phenomena which are not captured by the infinite-time notions of hyperbolicity even for flows defined for all time. Notions of finite-time hyperbolic trajectories, their finite time stable and unstable manifolds, as well as finite-time Lyapunov exponent (FTLE) fields and associated Lagrangian coherent structures have been the main tools for characterising transport barriers in the time-aperiodic situation. In this paper we consider a variety of examples, some with explicit solutions, that illustrate in a concrete manner the issues and phenomena that arise in the setting of finite-time dynamical systems. Of particular significance for geophysical applications is the notion of flow transition which occurs when finite-time hyperbolicity is lost or gained. The phenomena discovered and analysed in our examples point the way to a variety of directions for rigorous mathematical research in this rapidly developing and important area of dynamical systems theory.

scholarly journals On diagrammatic technique for nonlinear dynamical systems

2014 ◽  
Vol 29 (35) ◽  
pp. 1430039
Author(s):  
Mykola Semenyakin
Keyword(s):  
Vector Field ◽  
Dynamical Systems ◽  
Fixed Points ◽  
Vector Fields ◽  
Nonlinear Dynamical Systems ◽  
Radius Of Convergence ◽  
Evolution Operators ◽  
Finite Degree ◽  
Nonlinear Dynamical ◽  
Diagrammatic Technique

In this paper, we investigate phase flows over ℂn and ℝn generated by vector fields V = ∑ Pi∂i where Pi are finite degree polynomials. With the convenient diagrammatic technique, we get expressions for evolution operators ev {V|t} : x(0) ↦ x(t) through the series in powers of x(0) and t, represented as sum over all trees of a particular type. Estimates are made for the radius of convergence in some particular cases. The phase flows behavior in the neighborhood of vector field fixed points are examined. Resonance cases are considered separately.

BIFURCATIONS OF LIMIT CYCLES IN A Z2-EQUIVARIANT PLANAR POLYNOMIAL VECTOR FIELD OF DEGREE 7

2006 ◽  
Vol 16 (04) ◽  
pp. 925-943 ◽  
Author(s):  
JIBIN LI ◽  
MINGJI ZHANG ◽  
SHUMIN LI
Keyword(s):  
Vector Field ◽  
Dynamical Systems ◽  
Limit Cycles ◽  
Bifurcation Theory ◽  
Vector Fields ◽  
Compound Eyes ◽  
Polynomial Vector ◽  
Polynomial Vector Fields ◽  
Planar Dynamical Systems ◽  
Equivariant Vector Field

By using the bifurcation theory of planar dynamical systems and the method of detection functions, the bifurcations of limit cycles in a Z2-equivariant planar perturbed Hamiltonian polynomial vector fields of degree 7 are studied. An example of a special Z2-equivariant vector field having 50 limit cycles with a configuration of compound eyes are given.

ON DYNAMICAL SYSTEMS GENERATED BY TWO ALTERNATING VECTOR FIELDS

1996 ◽  
Vol 06 (11) ◽  
pp. 2015-2030 ◽  
Author(s):  
A. KLÍČ ◽  
P. POKORNÝ
Keyword(s):  
Dynamical Systems ◽  
Fixed Points ◽  
Periodic Solutions ◽  
Time Evolution ◽  
Autonomous System ◽  
Vector Fields ◽  
Numerical Study ◽  
Method Of Averaging ◽  
Controlling Of Chaos

Dynamical systems with time evolution determined by two alternating vector fields are investigated both analytically and numerically. When the two vector fields are related by an involutory diffeomorphism G then the fixed points of G (either isolated or non-isolated) are shown to give rise to branches of periodic solutions of the resulting non-autonomous system. The method of averaging is used for small switching periods. Detailed numerical study of both conservative (“blinking vortex”) and dissipative (“blinking nodes”, “blinking cycles” and “blinking Lorenz”) systems shows that the technique of blinking can be used to initiating and controlling of chaos.

Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields

Physics Today ◽  
1985 ◽  
Vol 38 (11) ◽  
pp. 102-105 ◽  
Author(s):  
J. Gluckheimer ◽  
P. Holmes ◽  
K. K. Lee
Keyword(s):  
Dynamical Systems ◽  
Vector Fields ◽  
Nonlinear Oscillations
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