Narrow Results

The percentage of organized motion of the chaotic zone (which shall from now on be referred to as percentage of order) for the logistic, the sine-square and the 4-exponent map, is calculated. The calculations are reached via a sampling method that incorporates the Lyapunov exponent. Although these maps are specially selected examples of one-dimensional ones, the conclusions can also be applied to any other one-dimensional map. Since the metric characteristics of a bifurcation diagram of a unimodal map, such as the referred percentage of order, are dependent on the order of the maximum, this dependence is verified for several maps. Once the chaotic zone can be separated into regions between the sequential band mergings, the percentage of order corresponding to each region is calculated for the logistic map. In each region, the resultant area occupied by order, or the supplementary area occupied by chaos, participates in a sequence similar to Feigenbaum's one, which converges to the same respective Feigenbaum's constant.

The effects of periodic forcing and impulsive perturbations on the predator–prey model with Beddington–DeAngelis functional response are investigated. We assume periodic variation in the intrinsic growth rate of the prey as well as periodic constant impulsive immigration of the predator. The dynamical behavior of the system is simulated and bifurcation diagrams are obtained for different parameters. The results show that periodic forcing and impulsive perturbation can very easily give rise to complex dynamics, including quasi-periodic oscillating, a period-doubling cascade, chaos, a period-halving cascade, non-unique dynamics, and period windows.

We investigate the dynamics of a single-species model with birth pulses and pulse harvesting in a polluted environment. Using the discrete dynamical system determined by the stroboscopic map, we obtain an exact 1-period solution of the system whose birth function is a Ricker or Beverton–Holt function and obtain the threshold conditions for their stability. Further, we show the effects of the time of pulse harvesting on the maximum annual-sustainable yield. Our results show that the best time for harvesting is immediately after the birth pulses. Numerical simulation results also show that birth pulses and pulse harvesting make the single-species model in a polluted environment we consider more complex, and the system is dominated by periodic and chaotic solutions.

Biological networks are often modeled by systems of ordinary differential equations. In chemical reaction kinetics, for instance, sigmoid functions represent the regulation of gene expression via transcription factors. Solutions of these models tend to converge to a unique steady state, and feedback control mechanisms are required for a more complex dynamic behavior.

This paper focuses on periodic behavior in two-component regulatory networks. Here, a key issue is that oscillations in chemical reaction systems are usually not robust with respect to parameter variations. Small variations lead to bifurcations that change the system's overall qualitative dynamic behavior. This concerns the mechanisms stabilizing periodic behavior in living cells. Using a small sample network, we demonstrate that oscillations can efficiently be stabilized by large time scale differences that correspond to reactions with different velocities. Furthermore, the inclusion of a time delay, reflecting transport and diffusion processes, has a similar effect. This suggests that processes of this kind potentially play a crucial role in biological oscillators.

This paper investigates an susceptible, infectious, recovered (SIR) model with childhood vaccination demand modeled as a function of prior levels of disease prevalence. Stability conditions for endemic and disease-free equilibria are uncovered, and both Hopf bifurcations and chaos are found with certain constellations of parameter values. Policy interventions directed toward increasing vaccination are explored, and the implied magnitudes of the external benefit of vaccination are modeled.

We propose that a regulation mechanism based on Hebbian covariance plasticity may cause the brain to operate near criticality. We analyze the effect of such a regulation on the dynamics of a network with excitatory and inhibitory neurons and uniform connectivity within and across the two populations. We show that, under broad conditions, the system converges to a critcal state lying at the common boundary of three regions in parameter space; these correspond to three modes of behavior: high activity, low activity, oscillation.

The dynamics of multiple competing political parties under spatial voting is explored. Parties are allowed to modify their positions adaptively in order to gain more votes. The parties in this model are opportunistic, in the sense that they try to maximize their share of votes regardless of any ideological position. Each party makes small corrections to its current platform in order to increase its own utility by means of the steepest ascent in the variables under its own control, i.e., by locally optimizing its own platform.

We show that in models with more than two parties bifurcations at the trivial equilibrium occur if only the voters are critical enough, that is, if they respond strongly to small changes in relative utilities. A numerical survey in a three-party model yields multiple bifurcations, multi-stability, and stable periodic at tractors that arise through Hopf bifurcations. Models with more than two parties can thus differ substantially from the two-party case, where it has been shown that under the assumptions of quadratic voter utilities and complete voter participation there is always a globally stable equilibrium that coincides with the mean voter position.