Asymptotic Approximations of Integral Manifolds

• Published in: SIAM journal on applied mathematics 1987-10, Vol.47 (5), p.929-940
• Main Author: Gilsinn, David E.
• Format: Article
• Language: English
• Subjects: Approximation; Differential equations; Eigenvalues; Exact sciences and technology; Function theory, analysis; Mathematical integrals; Mathematical manifolds; Mathematical methods in physics; Mathematical vectors; Partial differential equations; Physics; Sine function
• ISSN: 0036-1399; 1095-712X
• DOI: 10.1137/0147061

Multidegree of freedom nonlinear differential equations can often be transformed by means of the method of averaging into equivalent systems with only high-order terms. Under appropriate small-order perturbation conditions these systems have unique surfaces of solutions called integral manifolds. They generalize the notions of periodic and almost periodic solutions for single degree of freedom systems. In parametric form the integral manifolds satisfy a certain system of partial differential equations. Conversely, an Nth-order asymptotic integral manifold is defined as a formal solution of that system of partial differential equations, up to order N in the perturbation parameter. In the main result a system of integral equations is written for the remainder terms. By a contraction argument the system of integral equations has a fixed point, which, added to the Nth-order asymptotic integral manifold, forms an integral manifold for the normal system. By uniqueness this is the integral manifold sought. This implies that the unique integral manifold can be written as a formal series plus high order error terms. As an example a second order asymptotic representation for the periodic solution of a van der Pol oscillator is then developed.