The Dynamics of Runge–kutta Methods

Abstract

The first step in investigating the dynamics of a continuous-time system described by an ordinary differential equation is to integrate to obtain trajectories. In this paper, we attempt to elucidate the dynamics of the most commonly used family of numerical integration schemes, Runge–Kutta methods, by the application of the techniques of dynamical systems theory to the maps produced in the numerical analysis.

A mathematical model with control to analyse the dynamics of dengue disease transmission in urban Colombo

“…Finally the entire system is numerically solved to investigate the consequences of these control strategies. The fourth order Runge Kutta scheme is used to obtain the solutions (Cartwright & Piro, 1992 …”

Construcción y estudio de códigos adaptativos de linealización local para ecuaciones diferenciales ordinarias

“…Es conocido [4,23,22] que los integradores convencionales como los méto-dos de Runge-Kutta y predictor-corrector de Adams-Bashforth-Moulton pueden producir comportamientos dinámicos errados en la integración de sistemas de ecuaciones diferenciales. Los principales problemas se presentan, por ejemplo, en la convergencia a estados estables falsos, cambios en las bases de atracción y aparición de bifurcaciones falsas.…”

Hardware Design of Embedded Systems for Security Applications

“…Among these, we can cite both Euler's and Runge-Kutta's numerical resolution methods [Yang et al, 2005;Cartwright & Piro, 1992]. Unlike the Euler method -a numerical procedure for solving the simplest approximation by the first-order differential equations with initial conditions -the Runge-Kutta method allows for the most accurate solutions.…”

“…Unlike the Euler method -a numerical procedure for solving the simplest approximation by the first-order differential equations with initial conditions -the Runge-Kutta method allows for the most accurate solutions. Indeed, in numerical analysis, the Runge-Kutta method characterises an important family of implicit and explicit iterative methods for the approximation of solutions for Ordinary Differential Equations (ODEs) [Cartwright & Piro, 1992]. These numerical methods are based on the principle of iteration, which is to say that the first estimate of the solution is used to calculate a second estimate, more precisely, and so on.…”

“…In this context, and in considering our embedded ciphering application, hardware implementation is designed and coded in VHDL with a structural description logic. This low level form of design seeks for resolving the Genesio-Tesi differential equation (18) through the RK-4 numerical resolution method in order to produce a more accurate estimate of the solution [Cartwright & Piro, 1992].…”