Conservative generalized bifurcation diagrams and phase space properties for oval-like billiards

Abstract

In this paper, we study some dynamic properties for oval-like billiards. These billiards have two control parameters, named ϵ, which controls the deformation of the boundary, and p, which changes the number of inflection points. The particle’s position (X,Y) uses Cartesian coordinates, and the angle μ gives us the particle’s direction. Here we consider a Poincare section, where we calculate the position X (in the horizontal axis) and angle μ every time a particle crosses Y=0 (in the vertical axis). We compute the phase space and the conservative generalized bifurcation diagrams (CGBD). These diagrams are obtained when changing the initial position X and the control parameter ϵ. We plot the respective maximum Lyapunov exponent for each combination of the control parameter and initial condition, which uses a customized color palette. These diagrams show how complex billiards dynamics are, where one can find the direct and inverse parabolic bifurcations. Moreover, one can highlight periodic, quasi-periodic, and chaotic regions. We found a fractal behavior (self-similar structure), where we verified the existence of period-adding structures logical sequences (periodic orbits) in the CGBD. These sequences accumulate in different regions depending on the control parameters, following the main body’s period and accumulating in different regions. When we set the control parameter p to 1, we observe that chaos dominates for a high enough value of the control parameter ϵ (which controls our billiard’s deformation). We also studied some orbits embedded in stochastic layers that appear near saddle points, which obey another period-adding logical sequence. These stochastic layers play a crucial role in the dynamics of billiard systems because that chaos grows in such regions, near saddle points, after increasing the control parameter’s value.