PARAMETER DEPENDENCE OF STABLE MANIFOLDS FOR IMPULSIVE EQUATIONS

2014 ◽  
Vol 12 (02) ◽  
pp. 131-160
Author(s):  
LUIS BARREIRA ◽  
CLAUDIA VALLS
Keyword(s):  
Vector Fields ◽  
Invariant Manifolds ◽  
Linear Part ◽  
Small Perturbations ◽  
Stable Manifolds ◽  
Class C ◽  
Nonautonomous Equations ◽  
Parameter Dependent ◽  
Nonuniform Exponential Dichotomy ◽  
Fiber Contraction

We establish the existence of stable manifolds under sufficiently small perturbations of a linear impulsive equation. Our results are optimal, in the sense that for vector fields of class C1 outside the jumping times, the invariant manifolds are also of class C1 outside these times. We also consider the case of C1 parameter-dependent perturbations and we establish the C1 dependence of the stable manifolds on the parameter. The proof uses the fiber contraction principle. We emphasize that we consider the general case of nonautonomous equations for which the linear part has a nonuniform exponential dichotomy.

scholarly journals ON POLYNOMIAL DIFFERENTIAL EQUATIONS OF THE SECOND ORDER ON A CIRCLE WITHOUT SINGULAR POINTS

2020 ◽  
Vol 12 (4) ◽  
pp. 33-40
Author(s):  
V.Sh. Roitenberg ◽  
Keyword(s):  
Differential Equations ◽  
Limit Cycles ◽  
Vector Fields ◽  
Second Order ◽  
Singular Points ◽  
Free Term ◽  
Small Perturbations ◽  
Cylindrical Phase ◽  
Autonomous Differential Equations ◽  
The Right

In this paper, autonomous differential equations of the second order are considered, the right-hand sides of which are polynomials of degree n with respect to the first derivative with periodic continuously differentiable coefficients, and the corresponding vector fields on the cylindrical phase space. The free term and the leading coefficient of the polynomial is assumed not to vanish, which is equivalent to the absence of singular points of the vector field. Rough equations are considered for which the topological structure of the phase portrait does not change under small perturbations in the class of equations under consideration. It is proved that the equation is rough if and only if all its closed trajectories are hyperbolic. Rough equations form an open and everywhere dense set in the space of the equations under consideration. It is shown that for n > 4 an equation of degree n can have arbitrarily many limit cycles. For n = 4, the possible number of limit cycles is determined in the case when the free term and the leading coefficient of the equation have opposite signs.

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.

Random invariant manifolds for ill-posed stochastic evolution equations

2019 ◽  
Vol 20 (02) ◽  
pp. 2050013
Author(s):  
Alexandra Neamţu
Keyword(s):  
Invariant Manifolds ◽  
Evolution Equations ◽  
Linear Part ◽  
Time Behavior ◽  
Stochastic Evolution ◽  
Stochastic Evolution Equations ◽  
Stable And Unstable Manifolds ◽  
Long Time ◽  
Ill Posed ◽  
Unstable Manifolds

We establish the existence of random stable and unstable manifolds for ill-posed stochastic partial differential equations (SPDEs). Namely, we assume that the linear part does not generate a [Formula: see text]-semigroup. Using the theory of integrated semigroups, we are able to analyze the long-time behavior of random dynamical systems generated by such SPDEs.

scholarly journals GENERATION OF SYMMETRIC PERIODIC ORBITS BY A HETEROCLINIC LOOP FORMED BY TWO SINGULAR POINTS AND THEIR INVARIANT MANIFOLDS OF DIMENSIONS 1 AND 2 IN ℝ3

2007 ◽  
Vol 17 (09) ◽  
pp. 3295-3302 ◽  
Author(s):  
MONTSERRAT CORBERA ◽  
JAUME LLIBRE
Keyword(s):  
Periodic Orbits ◽  
Invariant Manifold ◽  
Vector Fields ◽  
Invariant Manifolds ◽  
Main Tool ◽  
Singular Points ◽  
Dimensional Sphere ◽  
Polynomial Vector ◽  
Polynomial Vector Fields ◽  
Heteroclinic Loop

In this paper we will find continuous periodic orbits passing near infinity for a class of polynomial vector fields in ℝ3. We consider polynomial vector fields that are invariant under a symmetry with respect to a plane Σ and that possess a "generalized heteroclinic loop" formed by two singular points e+ and e- at infinity and their invariant manifolds Γ and Λ. Γ is an invariant manifold of dimension 1 formed by an orbit going from e- to e+, Γ is contained in ℝ3 and is transversal to Σ. Λ is an invariant manifold of dimension 2 at infinity. In fact, Λ is the two-dimensional sphere at infinity in the Poincaré compactification minus the singular points e+ and e-. The main tool for proving the existence of such periodic orbits is the construction of a Poincaré map along the generalized heteroclinic loop together with the symmetry with respect to Σ.

scholarly journals A SURVEY OF METHODS FOR COMPUTING (UN)STABLE MANIFOLDS OF VECTOR FIELDS

2006 ◽  
pp. 67-95 ◽  
Author(s):  
B. KRAUSKOPF ◽  
H. M. OSINGA ◽  
E. J. DOEDEL ◽  
M. E. HENDERSON ◽  
J. GUCKENHEIMER ◽  
...  
Keyword(s):  
Vector Fields ◽  
Stable Manifolds

scholarly journals A note on continuation problems

1986 ◽  
Vol 28 (1) ◽  
pp. 55-61 ◽  
Author(s):  
Rita Nugari
Keyword(s):  
Banach Spaces ◽  
Topological Structure ◽  
Vector Fields ◽  
Existence Of Solution ◽  
Index Zero ◽  
Bounded Linear ◽  
Solution Sets ◽  
Connected Sets ◽  
Continuation Principle ◽  
Parameter Dependent

Recently M. Martelli [6] and M. Furi and M. P. Pera [1] proved some interesting results about the existence and the global topological structure of connected sets of solutions to problems of the form:Lx = N(λ, x)with L:E → F a bounded linear Fredholm operator of index zero (where E, F are real Banach spaces), and N:ℝ × E → F a nonlinear map satisfying suitable conditions.While the existence of solution sets for this kind of problem follows from the Leray–Schauder continuation principle, it is our aim to show in this note that their global topological structure can be obtained as a consequence of the theory developed by J. Ize, I. Massabò, J. Pejsachowicz and A. Vignoli in [3, 4] about parameter dependent compact vector fields in Banach spaces.

scholarly journals A SURVEY OF METHODS FOR COMPUTING (UN)STABLE MANIFOLDS OF VECTOR FIELDS

2005 ◽  
Vol 15 (03) ◽  
pp. 763-791 ◽  
Author(s):  
B. KRAUSKOPF ◽  
H. M. OSINGA ◽  
E. J. DOEDEL ◽  
M. E. HENDERSON ◽  
J. GUCKENHEIMER ◽  
...  
Keyword(s):  
Vector Field ◽  
Unstable Manifold ◽  
Stable Manifold ◽  
Vector Fields ◽  
Invariant Manifolds ◽  
Lorenz System ◽  
Dimensional Manifold ◽  
Two Dimensional ◽  
The Lorenz System ◽  
Global Invariant

The computation of global invariant manifolds has seen renewed interest in recent years. We survey different approaches for computing a global stable or unstable manifold of a vector field, where we concentrate on the case of a two-dimensional manifold. All methods are illustrated with the same example — the two-dimensional stable manifold of the origin in the Lorenz system.

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