A Farey triangle in the Belousov-Zhabotinskii reaction (original) (raw)
Related papers
Period doubling in a periodically forced Belousov-Zhabotinsky reaction
Physical Review E, 2007
Using an open-flow reactor periodically perturbed with light, we observe subharmonic frequency locking of the oscillatory Belousov-Zhabotinsky chemical reaction at one-sixth the forcing frequency ͑6:1͒ over a region of the parameter space of forcing intensity and forcing frequency where the Farey sequence dictates we should observe one-third the forcing frequency ͑3:1͒. In this parameter region, the spatial pattern also changes from slowly moving traveling waves to standing waves with a smaller wavelength. Numerical simulations of the FitzHugh-Nagumo equations show qualitative agreement with the experimental observations and indicate that the oscillations in the experiment are a result of period doubling.
Quint points lattice in a driven Belousov–Zhabotinsky reaction model
Chaos: An Interdisciplinary Journal of Nonlinear Science
We report the discovery of a regular lattice of exceptional quint points in a periodically driven oscillator, namely, in the frequency-amplitude control parameter space of a photochemically periodically perturbed ruthenium-catalyzed Belousov-Zhabotinsky reaction model. Quint points are singular boundary points where five distinct stable oscillatory phases coalesce. While spikes of the activator show a smooth and continuous variation, the spikes of the inhibitor show an intricate but regular branching into a myriad of stable phases that have fivefold contact points. Such boundary points form a wide parameter lattice as a function of the frequency and amplitude of light absorption. These findings revise current knowledge about the topology of the control parameter space of a celebrated prototypical example of an oscillating chemical reaction.
The Journal of Chemical Physics, 2009
We report a detailed numerical investigation of the relative abundance of periodic and chaotic oscillations in phase diagrams for the Belousov-Zhabotinsky ͑BZ͒ reaction as described by a nonpolynomial, autonomous, three-variable model suggested by Györgyi and Field ͓Nature ͑London͒ 355, 808 ͑1992͔͒. The model contains 14 parameters that may be tuned to produce rich dynamical scenarios. By computing the Lyapunov spectra, we find the structuring of periodic and chaotic phases of the BZ reaction to display unusual global patterns, very distinct from those recently found for gas and semiconductor lasers, for electric circuits, and for a few other familiar nonlinear oscillators. The unusual patterns found for the BZ reaction are surprisingly robust and independent of the parameter explored.
Non-Birkhoff Periodic Orbits of Farey Type and Dynamical Ordering in the Standard Mapping
Progress of Theoretical Physics, 2007
Using dynamical ordering for symmetric non-Birkhoff periodic orbits of Farey type (NBF), the properties of the limit at which the rotation numbers of NBFs accumulate to a rational number or irrational number are studied in the case of the standard mapping. Next, the relation between the critical value at which the KAM curve disappears and the critical value at which NBF appears is derived. The slow orbit connecting the two resonance chains is found.
Quantum intermittency in almost-periodic lattice systems derived from their spectral properties
Physica D: Nonlinear Phenomena, 1997
Hamiltonian tridiagonal matrices characterized by multi-fractal spectral measures in the family of Iterated Function Systems can be constructed by a recursive technique here described. We prove that these Hamiltonians are almost-periodic. They are suited to describe quantum lattice systems with nearest neighbours coupling, as well as chains of linear classical oscillators, and electrical transmission lines. We investigate numerically and theoretically the time dynamics of the systems so constructed. We derive a relation linking the long-time, power-law behaviour of the moments of the position operator, expressed by a scaling function β of the moment order α, and spectral multi-fractal dimensions, D q , via β(α) = D 1−α. We show cases in which this relation is exact, and cases where it is only approximate, unveiling the reasons for the discrepancies.
Fractional-period excitations in continuum periodic systems
Physical Review A, 2006
In the past few years, there has been considerable interest in both genuinely discrete and continuum but periodic systems 1. These arise in diverse physical contexts 2, including coupled waveguide arrays and photorefractive crystals in nonlinear optics 3, Bose-Einstein condensates BECs in optical lattices OLs in atomic physics 4, and DNA double-strand dynamics in biophysics 5. One of the most interesting themes that emerges in this context is the concept of “effective discreteness” induced by continuum periodic dynamics. There ...
Long periodic orbits of the triangle map
Proceedings of the American Mathematical Society, 1986
Let τ : [ 0 , 1 ] → [ 0 , 1 ] \tau :[0,1] \to [0,1] be defined by τ ( x ) = 2 x \tau (x) = 2x on [ 0 , 1 / 2 ] [0,1/2] and τ ( x ) = 2 ( 1 − x ) \tau (x) = 2(1 - x) on [ 1 / 2 , 1 ] [1/2,1] . We consider τ \tau restricted to the domain D N = { 2 a / p N , N ⩾ 1 , 0 ⩽ 2 a ⩽ p N , ( a , p ) = 1 } {D_N} = \{ 2a/{p^N},N \geqslant 1,0 \leqslant 2a \leqslant {p^N},(a,p) = 1\} where p p is any odd prime. Let k ⩾ 1 k \geqslant 1 be the minimum integer such that p N | 2 k ± 1 {p^N}|{2^k} \pm 1 . Then there are ( ( p − 1 ) ⋅ p N − 1 ) / 2 k (({\text {p}} - 1) \cdot {p^{N - 1}})/2k periodic orbits of τ | D N \tau {|_{{D_N}}} , having equal length, and there are k k points in each orbit. Furthermore, the proportion of points in any of these periodic orbits which lie in an interval ( c , d ) (c,d) approaches d − c d - c as p N − 1 → ∞ {p^{N - 1}} \to \infty . An application to the irreducibility of certain nonnegative matrices is given.