Use of Shrink Wrapping for Interval Taylor Models in Algorithms of Computer-Assisted Proof of the Existence of Periodic Trajectories in Systems of Ordinary Differential Equations

Abstract

Using interval Taylor models (TM), we construct algorithms for the computerassisted proof of the existence of periodic trajectories in systems of ordinary differential equations (ODEs). Although TMs allow one to construct guaranteed estimates for families of solutions of systems of ODEs when integrating ODEs over large time intervals, the interval residual included in the TMs begins to grow exponentially and becomes the dominant part of the estimate of the solution pencil, making it practically unusable. To eliminate this deficiency, the creators of the TM K. Makino and M. Berz proposed the idea of so-called "shrink wrapping." We formalize the original algorithm within the framework of the TM definitions we have adopted and propose our own version of the "shrink wrapping," more accurately adapted to the problem of the computer-aided proof of the existence of periodic trajectories.

Reduction in Degrees of Freedom for Large-Scale Nonlinear Systems in Computer-Assisted Proofs

“…This ensures a unified convergence of the time stepping method towards a stable periodic orbit and minimizes possible numerical issues related with the rapidly changing time step values. In addition, it allows us the use of these time steps in Picard iterations in proving algorithms; see [26]. Such process is executed for both full and reduced systems.…”

“…We could observe that not only were we able to approximate the periodic orbit well, but we were also able to represent the tangent space to the periodic orbit relatively well, at least for the chosen set of leading eigenvalues. In order to verify this fact, we performed rigorous computations of the reduced systems using TMs of the second order for the original POD approximation and POD-mixed approximation using the same system size of 20; see Figure 9 and [25,26] for more information. All calculations were carried out on GPUs.…”

“…In this case, one wishes to obtain as few remainders in TMs as possible on a relatively large time integration span. Since the degree of polynomials resides in the exponent in computational complexity, there are two ways to obtain reasonable computational times: reduce n and reduce m. In order to achieve the first reduction in n, we use shrink wrapping for TMs; see [26]. This allows us a decrease in n from four to two for a simple Van der Pol oscillator example to prove a periodic orbit using Poincare section approach.…”

Universal Bifurcation Chaos Theory and Its New Applications

“…It is proved in papers by this author and in other papers that the FShM bifurcation scenario of transition to dynamical chaos is realized in classical dissipative two-dimensional non-autonomous systems with periodic coefficients, such as as Mathieu, Croquette and Duffing-Holmes systems, and in three-dimensional dissipative autonomous systems, such as Lorenz, Chua, Sprott, Ressler, Chen, Rabinovich-Fabricant, Vallis, Magnitskii, Anishchenko-Astakhov, Volterra-Gause, Pikovskii-Rabinovich-Trakhtengertz, Sviregev, Rucklidge, Genezio-Tesi, Wiedlich-Trubetskov systems [1][2][3][6][7][8][9][10][11], and many others. This scenario of transition to chaos also takes place in many and infinitely dimensional systems of nonlinear ordinary differential equations, such as Rikitaki system, Lorenz complex five-dimensional system, Mackey-Glass equation with delay argument [1][2][3], and many others.…”