9 research outputs found
Computational Complexity of Smooth Differential Equations
The computational complexity of the solutions to the ordinary
differential equation , under various assumptions
on the function has been investigated. Kawamura showed in 2010 that the
solution can be PSPACE-hard even if is assumed to be Lipschitz
continuous and polynomial-time computable. We place further requirements on the
smoothness of and obtain the following results: the solution can still
be PSPACE-hard if is assumed to be of class ; for each , the
solution can be hard for the counting hierarchy even if is of class
.Comment: 15 pages, 3 figure
Computability of differential equations
In this chapter, we provide a survey of results concerning the computability and computational complexity of differential equations. In particular, we study the conditions which ensure computability of the solution to an initial value problem for an ordinary differential equation (ODE) and analyze the computational complexity of a computable solution. We also present computability results concerning the asymptotic behaviors of ODEs as well as several classically important partial differential equations.info:eu-repo/semantics/acceptedVersio
Computing the exact number of periodic orbits for planar flows
In this paper, we consider the problem of determining the \emph{exact} number
of periodic orbits for polynomial planar flows. This problem is a variant of
Hilbert's 16th problem. Using a natural definition of computability, we show
that the problem is noncomputable on the one hand and, on the other hand,
computable uniformly on the set of all structurally stable systems defined on
the unit disk. We also prove that there is a family of polynomial planar
systems which does not have a computable sharp upper bound on the number of its
periodic orbits
Computing the exact number of periodic orbits for planar flows
In this paper, we consider the problem of determining the exact number of periodic orbits for polynomial planar flows. This problem is a variant of Hilbert's 16th problem. Using a natural definition of computability, we show that the problem is noncomputable on the one hand and, on the other hand, computable uniformly on the set of all structurally stable systems defined on the unit disk. We also prove that there is a family of polynomial planar systems which does not have a computable sharp upper bound on the number of its periodic orbits.info:eu-repo/semantics/publishedVersio
Continuous-time computation with restricted integration capabilities
AbstractRecursion theory on the reals, the analog counterpart of recursive function theory, is an approach to continuous-time computation inspired by the models of Classical Physics. In recursion theory on the reals, the discrete operations of standard recursion theory are replaced by operations on continuous functions such as composition and various forms of differential equations like indefinite integrals, linear differential equations and more general Cauchy problems. We define classes of real recursive functions in a manner similar to the standard recursion theory and we study their complexity. We prove both upper and lower bounds for several classes of real recursive functions, which lie inside the elementary functions, and can be characterized in terms of space complexity. In particular, we show that hierarchies of real recursive classes closed under restricted integration operations are related to the exponential space hierarchy. The results in this paper, combined with earlier results, suggest that there is a close connection between analog complexity classes and subrecursive classes, at least in the region between FLINSPACE and the primitive recursive functions