Abstract
This work is concerned with efficient numerical methods for computing high order Taylor and Fourier-Taylor approximations of unstable manifolds attached to equilibrium and periodic solutions of delay differential equations. In our approach we first reformulate the delay differential equation as an ordinary differential equation on an appropriate Banach space. Then we extend the Parameterization Method for ordinary differential equations so that we can define operator equations whose solutions are charts or covering maps for the desired invariant manifolds of the delay system. Finally we develop formal series solutions of the operator equations. Order-by-order calculations lead to linear recurrence equations for the coefficients of the formal series solutions. These recurrence equations are solved numerically to any desired degree. The method lends itself to a-posteriori error analysis, and recovers the dynamics on the manifold in addition to the embedding. Moreover, the manifold is not required to be a graph, hence the method is able to follow folds in the embedding. In order to demonstrate the utility of our approach we numerically implement the method for some 1, 2, 3 and 4 dimensional unstable manifolds in problems with constant, and (briefly) state dependent delays.
Highlights
Numerical methods for computing stable/unstable manifolds occupy a central position in the field of computational dynamics
This sections recalls the dynamical systems approach to delay differential equations, i.e. we describe how such problems lead to ODEs on a function space
In this paper we developed parameterization methods for unstable manifolds of both equilibrium and periodic solutions of delay differential equations (DDEs)
Summary
Numerical methods for computing stable/unstable manifolds occupy a central position in the field of computational dynamics. Even though the original references [4, 5, 6] framed the parameterization method in the general setting of infinite dimensional Banach spaces, the results there require invertibility of certain linear operators, and do not apply directly to the unbounded linear operators which appear in the context of DDEs. Many subsequent works on the parameterization method provide extensions of the parameterization method to problems involving unbounded operators. For example the works of [55, 60, 21] develop/discuss computational methods for studying unstable manifolds attached to equilibrium and periodic orbits for some infinite dimensional systems. The advantage of carefully developing the parameterization method in the classical context of retarded functional differential equations is that the correct form of the series expansion for the unstable manifold appears quite naturally as the results of a certain formal computation. More precisely we compute some high order Taylor and Fourier-Taylor approximations of one, two, three, and four dimensional unstable manifolds for three different example systems: two with constant and one with a state dependent delay
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