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Cezary Olszowiec

Publications and source records attributed to Cezary Olszowiec.

2 recordsLinked to original sources
arXiv2018

On the non-transverse homoclinic channel of a center manifold

We consider a scenario when a stable and unstable manifolds of compact center manifold of a saddle-center coincide. The normal form of the ODE governing the system near the center manifold is derived and so is the normal form of the return map to the neighbourhood of the center manifold. The limit dynamics of the return map is investigated by showing that it might take the form of a Henon-like map possessing a Lorenz-like attractor or satisfy 'cone-field condition' resulting in partial hyperbolicity. We consider also motivating example from game theory.

math.DS↗
arXiv2016

Complex behaviour in cyclic competition bimatrix games

We consider an example of cyclic competition bimatrix game which is a Rock-Scissors-Paper game with assumption about perfect memory of the playing agents. At first we investigate the dynamics in the neighbourhood of the Nash equilibrium as well as the dynamics on the boundary of codimension 1 - that is when one of the strategies is not played by any agent. For the analysis of the asymptotic behaviour close to the boundary of state space, we provide the description of a naturally appearing heteroclinic network. Due to the symmetry, the heteroclinic network induces a quotient network. This quotient network is investigated as well. In the literature this model was already studied, but unfortunately the results stated there are invalid, as we will show in this paper. It turns out that certain types of behaviour are never possible or appear in the system only for some parameter values, contradicting what was stated before in the literature. Moreover, the parameter space (-1,1)x(-1,1) is divided into four regions where we observe either irregular behaviour, or preference to follow an itinerary consisting of strategies for which one or the other agent does not lose, or they alternate in winning. These regions in parameter space are separated by two analytical curves and lines where the game is either symmetric (system is not $C^1$ linearisable at stationary saddle points) or is zero-sum and the system is Hamiltonian. On each of these curves we observe different bifurcation scenarios: e.g. transition from order to chaos, or from one kind of stability to another kind, or just loss of one dimension of the local stable manifold of the subcycle.

math.DS↗