Tihamér A. Kocsis

56705224300

Publications - 4

Exactly Solvable Quadratic Differential Equation Systems Through Generalized Inversion

Ádám Bácsi Tihamér A. Kocsis

Publication Name: Qualitative Theory of Dynamical Systems

Publication Date: 2023-03-01

Volume: 22

Issue: 1

Page Range: Unknown

Description:

We study the autonomous systems of quadratic differential equations of the form x˙i(t)=x(t)TAix(t)+viTx(t) with x(t) = (x1(t) , x2(t) , … , xi(t) , ⋯) which, in general, cannot be solved exactly. In the present paper, we introduce a subclass of analytically solvable quadratic systems, whose solution is realized through a multi-dimensional generalization of the inversion which transforms a quadratic system into a linear one. We provide a constructive algorithm which, on one hand, decides whether the system of differential equations is analytically solvable with the inversion transformation and, on the other hand, provides the solution. The presented results apply for arbitrary, finite number of variables.

Open Access: Yes

DOI: 10.1007/s12346-023-00738-7

Optimal Monotonicity-Preserving Perturbations of a Given Runge–Kutta Method

Tihamér A. Kocsis Inmaculada Higueras David Ketcheson

Publication Name: Journal of Scientific Computing

Publication Date: 2018-09-01

Volume: 76

Issue: 3

Page Range: 1337-1369

Description:

Perturbed Runge–Kutta methods (also referred to as downwind Runge–Kutta methods) can guarantee monotonicity preservation under larger step sizes relative to their traditional Runge–Kutta counterparts. In this paper we study the question of how to optimally perturb a given method in order to increase the radius of absolute monotonicity (a.m.). We prove that for methods with zero radius of a.m., it is always possible to give a perturbation with positive radius. We first study methods for linear problems and then methods for nonlinear problems. In each case, we prove upper bounds on the radius of a.m., and provide algorithms to compute optimal perturbations. We also provide optimal perturbations for many known methods.

Open Access: Yes

DOI: 10.1007/s10915-018-0664-3

On the absolute stability regions corresponding to partial sums of the exponential function

Tihamér A. Kocsis David Ketcheson Lajos Lóczi

Publication Name: IMA Journal of Numerical Analysis

Publication Date: 2015-07-01

Volume: 35

Issue: 3

Page Range: 1426-1455

Description:

Certain numerical methods for initial value problems have as stability function the nth partial sum of the exponential function. We study the stability region, that is, the set in the complex plane over which the nth partial sum has at most unit modulus. It is known that the asymptotic shape of the part of the stability region in the left half-plane is a semidisc. We quantify this by providing discs that enclose or are enclosed by the stability region or its left half-plane part. The radius of the smallest disc centred at the origin that contains the stability region (or its portion in the left half-plane) is determined for 1 ≤ n ≤ 20. Bounds on such radii are proved for n ≥ 2; these bounds are shown to be optimal in the limit n → +∞. We prove that the stability region and its complement, restricted to the imaginary axis, consist of alternating intervals of length tending to π, as n → ∞. Finally, we prove that a semidisc in the left half-plane centred at the origin and with vertical boundary lying on the imaginary axis is included in the stability region if and only if n ≡ 0 mod 4 or n ≡ 3 mod 4. The maximal radii of such semidiscs are exactly determined for 1 ≤ n ≤ 20.

Open Access: Yes

DOI: 10.1093/imanum/dru039

On the control of a vehicle dynamics problem

Tihamér A. Kocsis Adrián Németh Zoltán Horváth

Publication Name: 2014 IEEE International Electric Vehicle Conference Ievc 2014

Publication Date: 2014-01-01

Volume: Unknown

Issue: Unknown

Page Range: Unknown

Description:

In this paper we present an approach to find the controlled invariant sets of a nonlinear dynamical system. Our method explores the state space and determines the set of those initial values where the system can be stabilized with a bounded control function. A continuous closed-loop feedback rule can also be obtained from this procedure. We illustrate the numerical results of our method on a problem describing the behaviour of actuators controlling the lateral dynamics of a vehicle.

Open Access: Yes

DOI: 10.1109/IEVC.2014.7056192