A General Method for Computer-Assisted Proofs of Periodic Solutions in Delay Differential Problems

Abstract

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A novel technique to obtain a time-delayed vibration control analytical solution with simulation of He’s formula

“…In recent years, differential equations, especially those containing delayed impacts that lead to the establishment of an important branch of nonlinear analysis, have gained great importance due to their useful role in modeling various events in many applications related to different areas of science such as physics, geography, chemistry, biology, nuclear reactors theory, economics, natural sciences, and engineering. [1][2][3] The governing time delay differential equations for dynamical systems are called an infinite-dimensional system. So, this kind of equation has greater complicated dynamics than the other non-delayed traditional ones.…”

Rotation invariant patterns for a nonlinear Laplace-Beltrami equation: A Taylor-Chebyshev series approach

“…These results have all been obtained using the weights µ = 1.1 and ν = 1.2. Other computational parameters can be found in Table 1 for the four particular solutions, and in the code available at [27] for all other solutions. Both solutions on the n = 1 branch are depicted in Figure 3, while the ones on the n = 16 branch are shown in Figure 4.…”

Verified Integration of Differential Equations with Discrete Delay

“…is obtained, for which we assume a constant initialization function x(t) = x 0 for t ≤ 0 with the consistent initial condition x 0 = 2 at the point t = 0 in the remainder of this subsection. The change of coordinates (36) helps to avoid that solutions cross the value y = 0 if initialized with non-negative functions y(t) > 0 for t ≤ 0 and positive parameters p > 0. The advantage of the exclusion of the solution y = 0 from the solution set is that a singularity in the iterations of Theorems 2 and 3 as well as Corollary 2 can be avoided.…”

“…In contrast to the existing techniques with result verification for solving delay differential equations [10,35,36] (that employ Taylor methods or radii polynomial approaches), we do not focus on obtaining especially tight enclosures, which is necessary for a computational proof of such properties as periodicity of solutions. Instead, we aim at computing guaranteed outer solution enclosures by an approach that represents state trajectories by simple (exponential) functions in a computationally cheap way.…”