A BRIEF SURVEY ON THE NUMERICAL DYNAMICS FOR FUNCTIONAL DIFFERENTIAL EQUATIONS
Copyright policy )A BRIEF SURVEY ON THE NUMERICAL DYNAMICS FOR FUNCTIONAL DIFFERENTIAL EQUATIONS
This is a survey on discretizing delay equations from a geometric-qualitative view-point. Concepts like compact attractors, hyperbolic periodic orbits, the saddle structure around hyperbolic equilibria, center-unstable manifolds of equilibria, inertial manifolds, structural stability, and Kamke monotonicity are considered. Error estimates for smooth and nonsmooth initial data in various Cj topologies are provided. The emphasis is put on Runge–Kutta methods with natural interpolants. The paper ends with a collection of the related results on retarded functional differential equations with bounded delay.
invariant manifolds, hyperbolic periodic orbits, Attractors and repellers of smooth dynamical systems and their topological structure, saddle structure, Kamke monotonicity, Generic properties, structural stability of dynamical systems, Numerical methods for Hamiltonian systems including symplectic integrators, Numerical approximation of solutions of functional-differential equations, Stability theory for smooth dynamical systems, Runge-Kutta discretizations, survey paper, Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations, compact attractors, hyperbolic equilibria, retarded functional differential equations, error estimates, Research exposition (monographs, survey articles) pertaining to numerical analysis, structural stability, Periodic orbits of vector fields and flows, inertial manifolds, delay equations, center-unstable manifolds, Error bounds for numerical methods for ordinary differential equations
invariant manifolds, hyperbolic periodic orbits, Attractors and repellers of smooth dynamical systems and their topological structure, saddle structure, Kamke monotonicity, Generic properties, structural stability of dynamical systems, Numerical methods for Hamiltonian systems including symplectic integrators, Numerical approximation of solutions of functional-differential equations, Stability theory for smooth dynamical systems, Runge-Kutta discretizations, survey paper, Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations, compact attractors, hyperbolic equilibria, retarded functional differential equations, error estimates, Research exposition (monographs, survey articles) pertaining to numerical analysis, structural stability, Periodic orbits of vector fields and flows, inertial manifolds, delay equations, center-unstable manifolds, Error bounds for numerical methods for ordinary differential equations
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