Basin entropy as an indicator of a bifurcation in a time-delayed system
arxiv(2024)
摘要
The basin entropy is a measure that quantifies, in a system that has two or
more attractors, the predictability of a final state, as a function of the
initial conditions. While the basin entropy has been demonstrated on a variety
of multistable dynamical systems, to the best of our knowledge, it has not yet
been tested in systems with a time delay, whose phase space is infinite
dimensional because the initial conditions are functions defined in a time
interval [-τ,0], where τ is the delay time. Here we consider a simple
time delayed system consisting of a bistable system with a linear delayed
feedback term. We show that the basin entropy captures relevant properties of
the basins of attraction of the two coexisting attractors. Moreover, we show
that the basin entropy can give an indication of the proximity of a Hopf
bifurcation, but fails to capture the proximity of a pitchfork bifurcation. Our
results suggest that the basin entropy can yield useful insights into the
long-term predictability of time delayed systems, which often have coexisting
attractors.
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