Smoothness of Asymptotic Phase Revisited

2011 ◽  
Vol 11 (4) ◽  
Author(s):  
Flaviano Battelli ◽  
Kenneth J. Palmer
Keyword(s):  
Periodic Solution ◽  
Vector Field ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Stable Manifold ◽  
Autonomous System ◽  
Asymptotic Phase

AbstractIt is well-known that solutions on the stable manifold of a hyperbolic periodic solution of an autonomous system of ordinary differential equations have an asymptotic phase which has the same order of smoothness as the vector field. In this paper we show if the system depends on a parameter that, in general, the asymptotic phase loses one order of smoothness in the parameter.

Pathwise convergent higher order numerical schemes for random ordinary differential equations

2007 ◽  
Vol 463 (2087) ◽  
pp. 2929-2944 ◽  
Author(s):  
Peter E Kloeden ◽  
Arnulf Jentzen
Keyword(s):  
Stochastic Process ◽  
Brownian Motion ◽  
Vector Field ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Higher Order ◽  
Time Variable ◽  
Numerical Schemes ◽  
Hölder Continuous ◽  
Usual Order

Random ordinary differential equations (RODEs) are ordinary differential equations (ODEs) with a stochastic process in their vector field. They can be analysed pathwise using deterministic calculus, but since the driving stochastic process is usually only Hölder continuous in time, the vector field is not differentiable in the time variable, so traditional numerical schemes for ODEs do not achieve their usual order of convergence when applied to RODEs. Nevertheless deterministic calculus can still be used to derive higher order numerical schemes for RODEs via integral versions of implicit Taylor-like expansions. The theory is developed systematically here and applied to illustrative examples involving Brownian motion and fractional Brownian motion as the driving processes.

Submersive second order ordinary differential equations

1991 ◽  
Vol 110 (1) ◽  
pp. 207-224 ◽  
Author(s):  
Marek Kossowski ◽  
Gerard Thompson
Keyword(s):  
Vector Field ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Tangent Bundle ◽  
Second Order

The objectives of this paper are to define and to characterize submersive second order ordinary differential equations (ODE) and to examine several situations in which such ODE occur. This definition and characterization is in terms of tangent bundle geometry as developed in [4, 6, 7, 10, 11, 14]. From this viewpoint second order ODE are identified with a special vector field on the tangent bundle. The ODE are said to be submersive when this vector field and the canonical vertical endomorphism [14] define a foliation, relative to which the vector field passes to the local quotient.

Theory of Index for Dynamical Systems of Order Higher Than Two

1980 ◽  
Vol 47 (2) ◽  
pp. 421-427 ◽  
Author(s):  
C. S. Hsu
Keyword(s):  
Vector Field ◽  
Dynamical Systems ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Higher Order ◽  
Abstract Concept ◽  
Basic Tool ◽  
Degree Of A Map

This paper is concerned with the generalization of Poincare´’s theory of index to systems of order higher than two. The basic tool used in the generalization is the concept of the degree of a map. In topology this concept has been used to discuss the index of a vector field. In this paper we shall use the degree of a map concept to present a theory of index for higher-order systems in a form which might make it more accessible to engineers for applications. The theory utilizes the notion of the index of a hypersurface with respect to a given vector field. After presenting the theory, it is applied to dynamical systems governed by ordinary differential equations and also to dynamical systems governed by point mappings. Finally, in order to show how the abstract concept of the degree of a map, hence the index of a surface, may actually be evaluated, illustrative procedures of evaluation for two kinds of hypersurfaces are discussed in detail and an example of application is given.

scholarly journals Chaos in an anharmonic oscillator

1995 ◽  
Vol 37 (2) ◽  
pp. 186-207 ◽  
Author(s):  
J. R. Christie ◽  
K. Gopalsamy
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Autonomous System ◽  
Anharmonic Oscillator ◽  
Homoclinic Orbits ◽  
Chaotic Behaviour ◽  
Differential Systems ◽  
Melnikov's Method ◽  
One Dimensional ◽  
Melnikov’S Method

AbstractUsing Melnikov's method, the existence of chaotic behaviour in the sense of Smale in a particular time-periodically perturbed planar autonomous system of ordinary differential equations is established. Examples of planar autonomous differential systems with homoclinic orbits are provided, and an application to the dynamics of a one-dimensional anharmonic oscillator is given.

scholarly journals On constructing single-input non-autonomous systems of full rank

2018 ◽  
Keyword(s):  
Vector Field ◽  
Nonlinear System ◽  
Differential Equations ◽  
Autonomous System ◽  
Vector Fields ◽  
Autonomous Systems ◽  
Full Rank ◽  
System Of Differential Equations ◽  
Single Input ◽  
Class C

For a nonlinear system of differential equations $\dot x=f(x)$, a method of constructing a system of full rank $\dot x=f(x)+g(x)u$ is studied for vector fields of the class $C^k$, $1\le k<\infty$, in the case when $f(x)\not=0$. A method for constructing a non-autonomous system of full rank is proposed in the case when the vector field $f(x)$ can vanish.

scholarly journals Fourth-Order Four-Point Boundary Value Problem: A Solutions Funnel Approach

2012 ◽  
Vol 2012 ◽  
pp. 1-18
Author(s):  
Panos K. Palamides ◽  
Alex P. Palamides
Keyword(s):  
Vector Field ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Ordinary Differential Equations ◽  
Fourth Order ◽  
Boundary Value ◽  
Point Boundary ◽  
Negative Solution ◽  
Order Four ◽  
The Continuum

We investigate the existence of positive or a negative solution of several classes of four-point boundary-value problems for fourth-order ordinary differential equations. Although these problems do not always admit a (positive) Green's function, the obtained solution is still of definite sign. Furthermore, we prove the existence of an entire continuum of solutions. Our technique relies on the continuum property (connectedness and compactness) of the solutions funnel (Kneser's Theorem), combined with the corresponding vector field.

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