Paradoxical games are nonconstant sum conflicts, where individual and collective rationalities are at variance and refer to a dyadic antagonism where the contestants blackmail each other. The state space dynamics of a class of such games has been previously studied in the planar case of two variables x 1, x 2 (representing the propensities of the two parties to cooperate), for which phase space portraits have been obtained for a wide range of control parameters. In this paper, we extend the analysis to 3 dimensions, by allowing two of these parameters (the so-called «tempting factors») to oscillate in time. We observe on a Poincaré surface of section that the invariant manifolds of two unstable fixed points U 1 and U 1 intersect, and form heteroclinic and homoclinic orbits. Thus, sufficiently close to U 1 and U 2, one finds «horseshoe» chaos and extremely sensitive dependence to initial conditions. Moreover, since the equations of motion can be written in Hamiltonian form, all the known phenomena of periodic, quasi-periodic and chaotic orbits can be observed around two stable fixed points, where the two parties become «deadlocked» in an inconclusive exchange that never ends.
- General, theoretical and mathematical biophysics (including logic of biosystems, quantum biology and relevant aspects of thermodynamics, information theory, cybernetics and bionics)
ASJC Scopus subject areas
- Physics and Astronomy(all)