Chalmers Conferences, 9th European Conference on Mathematical and Theoretical Biology

Period-length estimate of Lyapunov exponent for probing transient chaos in model of population dynamics
Katarzyna Buszko, Krzysztof Stefański

Last modified: 2014-03-28


We analyze transient chaos in the  family of logistic maps: f(xn,r)=xn+1=rxn(1-xn),  which is the most “popular” model of population dynamics [1].  We test the period-length estimate of Lyapunov exponent as a tool for quantitative description of transient chaos generated by  1-dimensional maps from periodic windows.  We also define the  transient chaos duration for a single trajectory as a rambling time. To characterize the transient chaos generated by a particular map we average rambling time over a representative set of trajectories.  We  introduce the notion of period-length estimate of Lyapunov exponent, that for a 1-dimensional map f generating transient chaos and period-p attractor  is  given by the formula: λn(p)(x0) = 1/p log |(f p)(xn)|. Where xn= fn(x0) . It is natural to expect that for the rambling section of a trajectory λn(p) should remained positive and oscillate irregularly with n, but after the trajectory is stabilized, it should start to decrease gradually (and regularly), converging to the negative value. The beginning of this decrease can be used for determination of the rambling time for the trajectory springing from x0 . Numerical tests confirm such an image with a little caveat. The decrease is not smoothly monotonic but exhibits jagged structure with jags of the length equal p. Within each jag, beginning with a considerable drop of λn(p) , a mild and nearly linear increase  is observed.  This structure can be described in a simpler way with n-depended Pearson's coefficient rn computed for the set of pairs {(k, λk(p))} for k varying from n to n+p-1. We observed from numerical experiments  that  as long as the trajectory rambles, rn varies irregularly in general keeping away from the limiting values -1 and 1. With trapping, the pattern becomes  very regular, for n at which a new jag  begins rn is close to 1, and its next p-1 values are close to -1. The beginning of such a regular pattern can be used as a criterion for the end of rambling regime for a given trajectory.  We statistically analyze the obtained values of rambling time for  various samples of initials states x0. Finally we compare the results obtained  with the method of period-length estimate of Lyapunov exponent with results obtained with other methods used for measuring transient chaos [2-4].


population dynamics, transient chaos