On the mode-locking universality for critical circle maps

Abstract

The conjectured universality of the Hausdorff dimension of the fractal set formed by the set of the irrational winding parameter values for critical circle maps is shown to follow from the universal scalings for quadratic irrational winding numbers.

The qualitative behavior of coupled predator-prey oscillations as deduced from simple circle maps

“…Given that the elementary consumer/resource system is inherently oscillatory, it makes sense to analyze a coupled pair of such systems as a coupled oscillator (Vandermeer and Kaufmann, 1994a,b). Assuming consumer/ resource couplings are similar to others of nature's coupled oscillators, it is possible to borrow a powerful technique from physics, circle maps (Bohr et al, 1984;Bak, 1986;Cvitanovic et al, 1990), to gain qualitative insight into the apparently complicated behavior that may result from coupling oscillators. The purpose of this note is to explore some of the emergent qualitative patterns of non-equilibrium behavior of coupled consumer/resource systems, using the tool of the circle map.…”

Some Scaling Behaviors in a Circle Map With Two Inflection Points

“…[1][2][3] Jensen, Bak and Bohr, 4 in their study of the transition to chaos via quasiperiodicity, discovered the universality of the fractal dimension of the set of quasiperiodic windings at the onset of chaos, which was later thoroughly elucidated by Cvitanović et al using the renomalization group approach and cycle expansions. 5,6 These theoretical findings were further confirmed by the remarkable experiments on both Josephson junction simulators 7,8 and charge density waves. 9 On the other hand, Cumming and Linsay, 10 by analyzing the time series taken from a simple operational-amplifier relaxation oscillator driven by a sine wave, reported that the dimension of the quasiperiodic set at the transition was 0.795 ± 0.05, not 0.87 as had been predicted by the standard model based on the circle map in Ref.…”

“…5,[7][8][9] The universality of this fractal dimension, D 0.87, for the circle map with a cubic inflection point was proved. [4][5][6] Here we are interested in evaluating the corresponding fractal dimension of map (1), which has two cubic inflection points. The width of the mode locking with winding number W = P/Q, denoted by ∆Ω(P/Q), is determined by the following two equations:…”

Wada basins and qualitative unpredictability in ecological models: a graphical interpretation

“…By analyzing the transformed parameter Θ, it is possible to study the qualitative behavior of the underlying system (Vandermeer, 1994;Vandermeer et al, 2001) much as is done in the study of physical oscillators (Bak, 1986;Bohr et al, 1984;Cvitanovic et al, 1990;Jensen et al, 1984). It is a simple matter to describe the general behavior of the overall system simply by knowing the value of Θ.…”