Chaotic Interval Research Articles

Mechanism for identification of chaotic firing interval in excitable cells

The experimental data of interspike intervals were recorded in a neural pacemaker,and the presence of period-adding bifurcation cascade was demonstrated.By detecting unstable periodic orbits from the chaotic interspike intervals located in between period-n bursting and period-(n+1) bursting,it is found that the unstable period-n orbit can be detected from the chaotic interspike intervals near the period-n bursting,and the unstable period-(n+1) orbit exists in the chaotic interspike intervals near period-(n+1) bursting.Moreover,the unstable period-n orbit also can be detected.For such a phenomenon the theoretical analyses are presented from the point of view of the nonlinear dynamics by way of the pancreatic β-cell model suggested by Sherman.The direct causes of emergence of the phenomenon are shown to be the intermittency of type Ⅰ and that of type Ⅲ,and they are induced by the saddle-node bifurcation and period-doubling bifurcation,respectively.Consequently,the mechanism for identification of chaotic firing interval is revealed.At the same time,the universality of the phenomenon is elucidated to a certain extent.Furthermore,a novel method is put forward for identifying the chaotic firing interval.

A chaotic interval map with zero sequence entropy

We construct a chaotic interval map f such that hm, A(f) = 0 for any increasing sequence of positive integers A and any invariant measure 7. This provides a counterexample for the variational principle for topological sequence entropy.

Chaotic force in Brownian motion

Brownian motion, driven by a chaotic force, is dicussed: v(t) = -γv(t) + ƒ(t). The force ƒ(t) takes the value y + 1/√T for tn ⩽ t < tn + 1 (n = 0, 1, 2…,), where tn=T σn-1j=0 xj. The effect of the chaotic time interval {xn} and the chaotic magnitude {yn} on the stationary distribution of v is investigate in the following cases: (i) yn is chaotic and xn is constant, (ii) yn is regular and xn is chaotic, (iii) yn and xn are chaotic and (iv) xn is strongly correlated with yn: xn = yn = 0.5. the T-dependence of the stationary distribution of v is discussed. For γT ∠ 1 the stationary distribution in each case is shown to have a Gaussian form. The diffusion constant is calculated. The theoretical results are shown to be in a good agreement with the numerical ones.

Dynamics of a Discrete Quantum Nonlinear Schrödinger Equation

Numerical solution of the density matrix formulation of a discrete quantum non-linear Schrödinger equation, which has been used to describe transport in one-dimensional molecular chains, has been carried out in the case of a trimer. Results show that the occupation probabilities tend to their initial values in the limit of large nonlinearity. How the asymptotic values are approached depends strongly on the initial conditions. It has been seen that the occupation probabilities behave quasiperiodically or chaotically with varying nonlinearity, and periodically if there is no nonlinearity. Amplitudes of oscillation of the site occupation probabilities are larger in chaotic intervals than in quasiperiodic intervals. Transition to localisation of the excitation is also marked by a sharp change in the probability amplitudes, if this transiton is preceeded by a chaotic interval. The degree of localisation is never bigger than initially. If the excitation is well, but not necessarily completely, localised initially, transition to localisation is preceeded by one or more prelocalised intervals. In a prelocalised interval excitation oscillates mainly between two sites.

Every chaotic interval map has a scrambled set in the recurrent set

Let I denote a compact real interval and let f ∈ C0(I, I). In this note we show that if f chaotic in the sense of Li and Yorke, then there is an uncountable perfect δ-scrambled set S for f in the recurrent set of f. Furthermore, the ω-limit set of every x ∈ S under f contains S and contains infinitely many periodic points of f with arbitrarily large periods.

Entropy Production in a Chaotic Chemical System

The average rate of entropy production in a homogenous chemical system is investigated in oscillating periodic and chaotic modes as well as in coexisting stationary states. The simulations are based on an abstract model of a chemical reaction system with three freely varying concentrations. Five concentrations are assumed to be kept constant by suitable flows across the boundary. A fixed concentration is used as a control parameter. Second order mass action kinetics with reverse reaction is used. An unexpected result is that periodic modes in some windows in the chaotic interval have higher average rate of entropy production than the surrounding chaotic modes. A chaotic mode coexists with a stable stationary state with smaller entropy production. A unique (unstable) stationary state produces more entropy than the corresponding oscillating mode.