The fractionally damped Van der Pol oscillator

Hilfer-derivative existence and uniqueness, structure, chaos and a Bernstein-splines approach

Master Thesis (2025)
Author(s)

N.J. Goedegebure (TU Delft - Electrical Engineering, Mathematics and Computer Science)

Contributor(s)

Kateryna Marynets – Mentor (TU Delft - Mathematical Physics)

Wim T. van Horssen – Graduation committee member (TU Delft - Mathematical Physics)

D. Toshniwal – Graduation committee member (TU Delft - Numerical Analysis)

Faculty
Electrical Engineering, Mathematics and Computer Science
More Info
expand_more
Publication Year
2025
Language
English
Graduation Date
24-01-2025
Awarding Institution
Delft University of Technology
Programme
['Applied Mathematics']
Faculty
Electrical Engineering, Mathematics and Computer Science
Reuse Rights

Other than for strictly personal use, it is not permitted to download, forward or distribute the text or part of it, without the consent of the author(s) and/or copyright holder(s), unless the work is under an open content license such as Creative Commons.

Abstract

In this thesis, we study fractional differential equations with Hilfer derivative operators. Solutions are approximated using a newly developed Bernstein-splines approach and subsequently applied to the Van der Pol oscillator with fractional damping. Fractional derivatives generalize differentiation to the order α ∈ (0,∞), resulting in order α differential operators. The Hilfer-derivative of order α ∈ (0,1) and type β ∈ [0,1] is one of such operators and combines two of the most commonly used operators through parametrization: the Riemmann-Liouville-derivative (β = 0) and Caputo-derivative (β=1). The choice of β results in different kernel behavior of the operator, commonly yielding singular behavior of solutions of initial value problems (IVP's) and boundary value problems (BVP's) for β ∈ [0,1). Based on existing results for IVP's, we develop a new proof for existence and uniqueness of solutions to BVP's for Hilfer-fractional derivatives. To obtain solution approximations numerically for IVP's and BVP's, a Bernstein splines solution approach is developed and implemented, providing accurate convergence results for nonlinear IVP's and BVP's in an efficient vectorized approach. Implementation difficulties for the singular behavior of solutions for β ∈ [0,1) are successfully resolved through time-domain transformation approximation techniques. Finally, the methods are applied to numerically study the behavior of the fractionally damped Van der Pol oscillator, a nonlinear equation of interest in electrical engineering and control theory. We study the approximate limit cycle, of which behavior corresponds with existing analytical results for the Caputo-derivative (β=1). Convergence of solutions can also be obtained for Hilfer type values of β ∈ [0,1), appearing to be of little influence on the long-term limit cycle. Furthermore, when forcing is applied, we observe periodic, quasiperiodic and chaotic behavior.

Files

License info not available