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Journal of Computational Dynamics

January 2021 , Volume 8 , Issue 1

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Homogeneous darboux polynomials and generalising integrable ODE systems
Peter H. van der Kamp, D. I. McLaren and G. R. W. Quispel
2021, 8(1): 1-8 doi: 10.3934/jcd.2021001 +[Abstract](485) +[HTML](239) +[PDF](305.11KB)
Abstract:

We show that any system of ODEs can be modified whilst preserving its homogeneous Darboux polynomials. We employ the result to generalise a hierarchy of integrable Lotka-Volterra systems.

A self-consistent dynamical system with multiple absolutely continuous invariant measures
Fanni M. Sélley
2021, 8(1): 9-32 doi: 10.3934/jcd.2021002 +[Abstract](473) +[HTML](232) +[PDF](6197.65KB)
Abstract:

In this paper we study a class of self-consistent dynamical systems, self-consistent in the sense that the discrete time dynamics is different in each step depending on current statistics. The general framework admits popular examples such as coupled map systems. Motivated by an example of [9], we concentrate on a special case where the dynamics in each step is a \begin{document}$ \beta $\end{document}-map with some \begin{document}$ \beta \geq 2 $\end{document}. Included in the definition of \begin{document}$ \beta $\end{document} is a parameter \begin{document}$ \varepsilon > 0 $\end{document} controlling the strength of self-consistency. We show such a self-consistent system which has a unique absolutely continuous invariant measure (acim) for \begin{document}$ \varepsilon = 0 $\end{document}, but at least two for any \begin{document}$ \varepsilon > 0 $\end{document}. With a slight modification, we transform this system into one which produces a phase transition-like behavior: it has a unique acim for \begin{document}$ 0< \varepsilon < \varepsilon^* $\end{document}, and multiple for sufficiently large values of \begin{document}$ \varepsilon $\end{document}. We discuss the stability of the invariant measures by the help of computer simulations employing the numerical representation of the self-consistent transfer operator.

The geometry of convergence in numerical analysis
George W. Patrick
2021, 8(1): 33-58 doi: 10.3934/jcd.2021003 +[Abstract](646) +[HTML](274) +[PDF](596.6KB)
Abstract:

The domains of mesh functions are strict subsets of the underlying space of continuous independent variables. Spaces of partial maps between topological spaces admit topologies which do not depend on any metric. Such topologies geometrically generalize the usual numerical analysis definitions of convergence.

A general framework for validated continuation of periodic orbits in systems of polynomial ODEs
Jan Bouwe van den Berg and Elena Queirolo
2021, 8(1): 59-97 doi: 10.3934/jcd.2021004 +[Abstract](321) +[HTML](138) +[PDF](948.32KB)
Abstract:

In this paper a parametrized Newton-Kantorovich approach is applied to continuation of periodic orbits in arbitrary polynomial vector fields. This allows us to rigorously validate numerically computed branches of periodic solutions. We derive the estimates in full generality and present sample continuation proofs obtained using an implementation in Matlab. The presented approach is applicable to any polynomial vector field of any order and dimension. A variety of examples is presented to illustrate the efficacy of the method.

Optimization-based subdivision algorithm for reachable sets
Wolfgang Riedl, Robert Baier and Matthias Gerdts
2021, 8(1): 99-130 doi: 10.3934/jcd.2021005 +[Abstract](417) +[HTML](165) +[PDF](5980.46KB)
Abstract:

Reachable sets for nonlinear control systems can be computed via the use of solvers for optimal control problems. The paper presents a new improved variant which applies adaptive concepts similar to the framework of known subdivision techniques by Dellnitz/Hohmann. Using set properties of the nearest point projection, the convergence and rigorousness of the algorithm can be proved without the assumption of diffeomorphism on a nonlinear mapping. The adaptive method is demonstrated by two nonlinear academic examples and for a more complex robot model with box constraints for four states, two controls and five boundary conditions. In these examples adaptive and non-adaptive techniques as well as various discretization methods and optimization solvers are compared.

The method also offers interesting features, like zooming into details of the reachable set, self-determination of the needed bounding box, easy parallelization and the use of different grid geometries. With the calculation of a 3d funnel in one of the examples, it is shown that the algorithm can also be used to approximate higher dimensional reachable sets and the resulting box collection may serve as a starting point for more sophisticated visualizations or algorithms.

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