Differential-Difference Equations. By Richard Bellman and Kenneth L. Cooke. Pp. xvi, 465. 1963. (Academic Press)

Introduction to Ordinary Differential Equations * Linear Systems and Stability of Nonlinear Systems * Applications * Hyperbolic Theory * Continuation of Periodic Solutions * Homoclinic Orbits, Melnikov?s Method, and Chaos * Averaging * Local Bifurcation * References * Index.

http://www.math.ecnu.edu.cn/~zmwang/teaching/Advanced_ODE-2014-2015/1-Textbook_and_Reference_book/2-Ordinary%20Differential%20Equations%20with%20Applications.pdf

Ordinary Differential Equations with Applications

Integrals, Conditional Expectations and Martingales of Multivalued Functions

The original Hilbert's 16th problem can be split into four parts consisting of Problems A–D. In this paper, the progress of study on Hilbert's 16th problem is presented, and the relationship between Hilbert's 16th problem and bifurcations of planar vector fields is discussed. The material is presented in eight sections. Section 1: Introduction: what is Hilbert's 16th problem? Section 2: The first part of Hilbert's 16th problem. Section 3: The second part of Hilbert's 16th problem: introduction. Section 4: Focal values, saddle values and finite cyclicity in a fine focus, closed orbit and homoclinic loop. Section 5: Finiteness problem. Section 6: The weakened Hilbert's 16th problem. Section 7: Global and local bifurcations of Zq–equivariant vector fields. Section 8: The rate of growth of Hilbert number H(n) with n.

Hilbert's 16th problem and bifurcations of planar polynomial vector fields

An approximation scheme involving approximation of linear functional differential equations by systems of high order ordinary differential equations is formulated and convergence is established in the context of known results from linear semigroup theory. Applications to optimal control problems are discussed and a summary of numerical results is given. The paper is concluded with a brief survey of previous literature on this class of approximations for systems with delays.

Hereditary Control Problems: Numerical Methods Based on Averaging Approximations

On etudie un probleme de bifurcation dans des familles de champs de vecteurs analytiques plans qui ont un centre non degenere a l'origine pour toutes les valeurs d'un parametre λ∈R N . On definit la fonction periode: (ξ,λ)→P(ξ,λ). On determine les points critiques de cette fonction pres du centre avec λ comme parametre de bifurcation

/pdf/bifurcation-of-critical-periods-for-plane-vector-fields-2dvziqyxfr.pdf

Bifurcation of critical periods for plane vector fields

Bifurcation of limit cycles from quadratic isochrones

An attainable sets approach to optimal control of functional differential equations with function space terminal conditions

https://www.ams.org/tran/1968-134-03/S0002-9947-1968-0236868-8/S0002-9947-1968-0236868-8.pdf

Measurable multivalued mappings and Lusin’s theorem

Publisher Summary This chapter provides an overview of an optimization problem for an nth order scalar neutral functional differential equation with functional side conditions. It presents the study of optimization problems of this type. However, when the results are specialized to the present problem, the maximum principle may be degenerate, whereas the results which assure a normal variational problem require assumptions that are not satisfied when L(x, u) = 0, is converted to a first order n-dimensional system. If the optimal control turns out to be sufficiently smooth, so that appropriate derivatives for λ exist, then one can show that the necessary conditions of the theorem agree and in this case the sufficiency theory obtained there can be applied.

Marc Q. Jacobs

Papers

Bifurcation of critical periods for plane vector fields

Bifurcation of limit cycles from quadratic isochrones

An attainable sets approach to optimal control of functional differential equations with function space terminal conditions

Measurable multivalued mappings and Lusin’s theorem

AN OPTIMIZATION PROBLEM FOR AN nth ORDER SCALAR NEUTRAL FUNCTIONAL DIFFERENTIAL EQUATION WITH FUNCTIONAL SIDE CONDITIONS