Kinetic differential equations, being nonlinear, are capable of producing many kinds of exotic phenomena. However, the existence of multistationarity, oscillation or chaos is usually proved by numerical methods. Here we investigate a relatively simple reaction among two species consisting of five reaction steps, one of the third order. About this reaction we show the following facts (using symbolic methods): the kinetic differential equation of the reaction has two limit cycles surrounding the stationary point of focus type in the positive quadrant. The outer limit cycle is always stable and the inner one is always unstable. We also performed the search for partial integrals of the system and have found one such integral. Application of the methods needs computer help (Wolfram language and Singular) because the symbolic calculations to carry out are too complicated to do by hand. Even to make characteristic drawings is very far from trivial: the inner limit cycle is very small. Some of the methods we use are relatively new, and all of them have never been used in reaction kinetics, although they can be used to have similar exact results on other models.
Multistep methods are important tools for solving ordinary differential equations with initial conditions. In order for these methods to be efficient they must be adaptive, that is, they must allow the choice of an appropriate step-size for each integration step. We present a comprehensive way of formulating multistep methods that is adaptive by construction and show how this methodology can be applied to particular situations. We also show the application to strong stability preserving methods, used to solve ODEs arising from the semi-discretization of time-dependent partial differential equations (PDEs), especially hyperbolic PDEs with shocks. We finally demonstrate how we apply the formulation to differential algebraic equations (DAEs), an ODE system coupled with algebraic constraints.
In the second talk on time step adaptivity, we focus on the special needs of conservative dynamical systems. This includes Hamiltonian problems, and weakly dissipative systems. In integrable Hamiltonian problems, the mathematical solution is time reversible, which precludes the use of classical controllers, which adapt the step size to manage the error observed in previous steps. Instead, a time reversible tracking algorithm is developed, which allows full reversibility of the adaptive computational process. This is shown to preserve first integrals over long times, and even improves the accuracy over constant step size symplectic integrators. We demonstrate the procedure in two examples from celestial mechanics, and then proceed to demonstrate how a similar approach can be combined with splitting methods in weakly dissipative systems. The latter approach has been put to effective use in rolling bearing dynamic simulation.
(Professors Gustaf Söderlind and Carmen Arévalo will present three talks during their visit to Budapest. All three talks deal with adaptive methods of ODEs. Professor Söderlind’s first talk will be held at the Seminar on Applied Analysis of the Department of Applied Analysis at ELTE university (details at the Miklós Farkas seminar), while his second talk together with Professor Arévalo’s talk will be presented at the Miklós Farkas Seminar. Although professor Söderlind’s talks form a complete presentation together, yet they can be followed separately.)