Multiple Attractors in Juvenile-adult Single Species Models

The role of age-structure and the Allee effect in generating multiple attractors in juvenile-adult single species single patch discrete-time models without dispersal are studied. In the presence of the Allee effect juvenile-adult single patch models support multiple attractors. However, in the absence of the Allee effect single attractors are supported when the dynamics are compensatory while multiple attractors are supported under overcompensatory dynamics. When the governing dynamics are compensatory, the boundaries of the basins of attraction have simple structure while complicated fractal basin boundaries are supported under overcompensatory dynamics.     

Truncated fractal basin boundaries in the pendulum with nonperiodic forcing

Summary It is well known that oscillators such as the pendulum can have fractal basin boundaries when they are periodically forced with the consequence that the long term behavior of the system may be unpredictable. In engineering and physical applications, the forcing is often nonperiodic and eventually decays to zero, and simulation of the pendulum with decaying forcing (M. Varghese, J. S. Thorp, Physical Review Letters, vol. 60, no. 8, pp. 665–668, Feb. 1988) exhibits truncated fractal basin boundaries which also limit the system predictability. We develop a coordinate change for the pendulum with decaying forcing that allows us to apply standard qualitative methods to study the basin boundaries. We prove that the basin boundaries cannot be fractal and show by example how the extreme stretching and folding leading to a truncated fractal basin boundary may arise.     

Nonlinear dynamics and the two-slit delayed experiment

The two-slit delayed experiment is re-examined from the standpoint of nonlinear dynamics in the presence of multiple attractors and fractal basin boundaries. It is suggested that the results may be interpreted as the response of the underlying system to a temporary switch of one control parameter, rather than as a retroaction between this system and the observer.     

Discrete-time Metapopulation Dynamics and Unidirectional Dispersal

The effects of unidirectional dispersal on single pioneer species discrete-time metapopulations where the pre-dispersal local patch dynamics are of the same (compensatory or overcompensatory) or mixed (compensatory and overcompensatory) types are studied. Single-species unidirectional metapopulation models behave as single-species single-patch models whenever all pre-dispersal local patch dynamics are compensatory and the dispersal rate is low. The pioneer species goes extinct in at least one patch when the dispersal rate is high, while it persists when the rate is low. Unidirectional dispersal can generate multiple attractors with fractal basin boundaries whenever the pre-dispersal local patch dynamics are overcompensatory, and is capable of altering the local patch dynamics in mixed systems from compensatory to overcompensatory dynamics and vice versa.     

Efficient synchronization of structurally adaptive coupled Hindmarsh–Rose neurons

The use of spikes to carry information between brain areas implies complete or partial synchronization of the neurons involved. The degree of synchronization reached by two coupled systems and the energy cost of maintaining their synchronized behavior is highly dependent on the nature of the systems. For non-identical systems the maintenance of a synchronized regime is energetically a costly process. In this work, we study conditions under which two non-identical electrically coupled neurons can reach an efficient regime of synchronization at low energy cost. We show that the energy consumption required to keep the synchronized regime can be spontaneously reduced if the receiving neuron has adaptive mechanisms able to bring its biological parameters closer in value to the corresponding ones in the sending neuron.     

Bursting synchronization in scale-free networks

Neuronal networks in some areas of the brain cortex present the scale-free property, i.e., the neuron connectivity is distributed according to a power-law, such that neurons are more likely to couple with other already well-connected ones. Neuron activity presents two timescales, a fast one related to action-potential spiking, and a slow timescale in which bursting takes place. Some pathological conditions are related with the synchronization of the bursting activity in a weak sense, meaning the adjustment of the bursting phase due to coupling. Hence it has been proposed that an externally applied time-periodic signal be applied in order to control undesirable synchronized bursting rhythms. We investigated this kind of intervention using a two-dimensional map to describe neurons with spiking–bursting activity in a scale-free network.     

Modulation of synchronization dynamics in a network of self-sustained systems

This paper addresses the combined modulatory effects of non-nearest neighbor oscillators and local injection on synchronized states dynamics with their corresponding stability boundaries in a network of self-sustained systems. The Whittaker method and Floquet theory are used to predict analytically the stability of these states for identical and non-identical coupling parameters. Charts revealing the modulation of synchronized states and their stability boundaries at the second order of interaction in the cases of identical and non-identical coupling parameters are constructed with and without an external signal locally injected in the network. Numerical simulations validate and complement the results of analytical surveys. The limits of the stability regions are numerically explored when a small amount of Gaussian white noise is also injected in the network.     

Dynamic complexities in predator–prey ecosystem models with age-structure for predator

Natural populations, whose generations are non-overlapping, can be modelled by difference equations that describe how the populations evolve in discrete time-steps. In the 1970s ecological research detected chaos and other forms of complex dynamics in simple population dynamics models, initiating a new research tradition in ecology. However, in former studies most of the investigations of complex population dynamics were mainly concentrated on single populations instead of higher dimensional ecological systems. This paper reports a recent study on the complicated dynamics occurring in a class of discrete-time models of predator–prey interaction based on age-structure of predator. The complexities include (a) non-unique dynamics, meaning that several attractors coexist; (b) antimonotonicity; (c) basins of attraction (defined as the set of the initial conditions leading to a certain type of an attractor) with fractal properties, consisting of pattern of self-similarity and fractal basin boundaries; (d) intermittency; (e) supertransients; and (f) chaotic attractors.     

Fractal characterization of acupuncture-induced spike trains of rat WDR neurons

The experimental and the clinical studies have showed manual acupuncture (MA) could evoke multiple responses in various neural regions. Characterising the neuronal activities in these regions may provide more deep insights into acupuncture mechanisms. This paper used fractal analysis to investigate MA-induced spike trains of Wide Dynamic Range (WDR) neurons in rat spinal dorsal horn, an important relay station and integral component in processing acupuncture information. Allan factor and Fano factor were utilized to test whether the spike trains were fractal, and Allan factor were used to evaluate the scaling exponents and Hurst exponents. It was found that these two fractal exponents before and during MA were different significantly. During MA, the scaling exponents of WDR neurons were regulated in a small range, indicating a special fractal pattern. The neuronal activities were long-range correlated over multiple time scales. The scaling exponents during and after MA were similar, suggesting that the long-range correlations not only displayed during MA, but also extended to after withdrawing the needle. Our results showed that fractal analysis is a useful tool for measuring acupuncture effects. MA could modulate neuronal activities of which the fractal properties change as time proceeding. This evolution of fractal dynamics in course of MA experiments may explain at the level of neuron why the effect of MA observed in experiment and in clinic are complex, time-evolutionary, long-range even lasting for some time after stimulation.     

Burst-duration mechanism of in-phase bursting in inhibitory networks

We study the emergence of in-phase and anti-phase synchronized rhythms in bursting networks of Hodgkin-Huxley-type neurons connected by inhibitory synapses. We show that when the state of the individual neuron composing the network is close to the transition from bursting into tonic spiking, the appearance of the network’s synchronous rhythms becomes sensitive to small changes in parameters and synaptic coupling strengths. This bursting-spiking transition is associated with codimension-one bifurcations of a saddle-node limit cycle with homoclinic orbits, first described and studied by Leonid Pavlovich Shilnikov. By this paper, we pay tribute to his pioneering results and emphasize their importance for understanding the cooperative behavior of bursting neurons. We describe the burst-duration mechanism of inphase synchronized bursting in a network with strong repulsive connections, induced by weak inhibition. Through the stability analysis, we also reveal the dual property of fast reciprocal inhibition to establish in- and anti-phase synchronized bursting.     

Multiple attractors and crisis route to chaos in a model food-chain

An attempt has been made to identify the mechanism, which is responsible for the existence of chaos in narrow parameter range in a realistic ecological model food-chain. Analytical and numerical studies of a three species food-chain model similar to a situation likely to be seen in terrestrial ecosystems has been carried out. The study of the model food chain suggests that the existence of chaos in narrow parameter ranges is caused by the crisis-induced sudden death of chaotic attractors. Varying one of the critical parameters in its range while keeping all the others constant, one can monitor the changes in the dynamical behaviour of the system, thereby fixing the regimes in which the system exhibits chaotic dynamics. The computed bifurcation diagrams and basin boundary calculations indicate that crisis is the underlying factor which generates chaotic dynamics in this model food-chain. We investigate sudden qualitative changes in chaotic dynamical behaviour, which occur at a parameter value a₁=1.7804 at which the chaotic attractor destroyed by boundary crisis with an unstable periodic orbit created by the saddle-node bifurcation. Multiple attractors with riddled basins and fractal boundaries are also observed. If ecological systems of interacting species do indeed exhibit multiple attractors etc., the long term dynamics of such systems may undergo vast qualitative changes following epidemics or environmental catastrophes due to the system being pushed into the basin of a new attractor by the perturbation. Coupled with stochasticity, such complex behaviours may render such systems practically unpredictable.     

Contour Integrals and Vector Calculus on Fractal Curves and Interfaces

This article develops the definition of contour integrals over fractal curves in the plane by introducing the notion of oriented Iterated Function Systems and directional pseudo-measures. An expression for the contour integral of continuous functions over fractal interfaces is obtained through renormalization. As a result, a vector calculus on fractal interfaces which are boundaries of regular two-dimensional domains is developed by extending Greens theorem in the plane, also to fractal curves.The use of moment analysis makes it possible to obtain recursive relations and closed-form expressions for contour integrals of algebraic functions. Several physical applications are analyzed, including the properties of double-layer potentials and connections with the solution of the Dirichlet problem on bounded two-dimensional domains possessing fractal boundaries.     

Bifurcations of basins of attraction from the view point of prime ends

In dynamical systems examples are common in which two or more attractors coexist, and in such cases the basin boundary is nonempty and the basins often have fractal basin boundaries. The purpose of this paper is to describe the structure and properties of unbounded basins and their boundaries for two-dimensional diffeomorphisms. Frequently, if not always, there is a periodic saddle on the boundary that is accessible from the basin. Carathéodory and many others developed an approach in which an open set (in our case a basin) is compactified using so-called prime end theory. Under the prime end compactification of the basin, boundary points of the basin (prime ends) can be characterized as either type 1, 2, 3, or 4. In all well-known examples, most points are of type 1. Many two-dimensional basins have a basin cell, that is, a trapping region whose boundary consists of pieces of the stable and unstable manifolds of a well chosen periodic orbit. Then the basin consists of a central body (the basin cell) and a finite number of channels attached to it, and the basin boundary is fractal. We present a result that says {a basin has a basin cell} if and only if {every prime end that is defined by a chain of unbounded regions (in the basin) is a prime end of type 3 and furthermore all other prime ends are of type 1}. We also prove as a parameter is varied, the basin cell for a basin B is created (or destroyed) if and only if either there is a saddle node bifurcation or the basin B has a prime end that is defined by a chain of unbounded regions and is a prime end of either type 2 or type 4.     

Appearance of horseshoes chaos on a buckled beam controlled by disseminated couple forces

This paper shows the appearance of horseshoes chaos on a buckled beam controlled by concentrated moments applied at suitable points. The distance between the stable and unstable manifolds of the control system are derived and analyzed taking into account the asymmetric characteristics of the potential brought by the control design. The structure of basin boundaries are obtained and confirmed by the analytical investigation.     

Endogenous chaotic dynamics in transitional economies

This paper examines a model of labor market dynamics in an economy undergoing transition from command socialism to market capitalism. State sector layoffs are modeled as a function of forecasts made by state planners of private sector wages where the laidoff workers are to be re-employed. The state switches between using a high information cost perfect forecast and a free naive forecast in a system that resembles a cobweb supply-demand model. Under certain specifications and parameter values chaotic dynamics are shown to endogenously emerge along with several other varieties of complex dynamics including strange attractors, coexistence of infinitely many stable cycles, cascades of infinitely many period doubling bifurcations and fractal basin boundaries between coexisting non-chaotic attractors.     

A scheme of de-synchronization in globally coupled neural networks and its possible implications for vagus nerve stimulation

A scheme of de-synchronization via pulse stimulation is numerically investigated in the Hindmarsh Rose globally coupled neural networks. The simulations show that synchronization evolves into de-synchronization in the globally coupled HR neural network when a part (about 10%) of neurons are stimulated with a pulse current signal. The network de-synchronization appears to be sensitive to the stimulation parameters. For the case of the same stimulation intensity, those weakly coupled networks reach de-synchronization more easily than strongly coupled networks. There exists a homologous asymptotic behavior in the region of higher frequency, and exist the optimal stimulation interval and period of continuous stimulation time when other stimulation parameters remain invariable.     

Switching diffusion logistic models involving singularly perturbed Markov chains: Weak convergence and stochastic permanence

Focusing on stochastic dynamics involve continuous states as well as discrete events, this article investigates stochastic logistic model with regime switching modulated by a singular Markov chain involving a small parameter. This Markov chain undergoes weak and strong interactions, where the small parameter is used to reflect rapid rate of regime switching among each state class. Two-time-scale formulation is used to reduce the complexity. We obtain weak convergence of the underlying system so that the limit has much simpler structure. Then we utilize the structure of limit system as a bridge, to invest stochastic permanence of original system driving by a singular Markov chain with a large number of states. Sufficient conditions for stochastic permanence are obtained. A couple of examples and numerical simulations are given to illustrate our results.     

Optimal control of sea level in a tidal basin by means of the Pontryagin maximum principle

The maximum principle has been applied in optimizing the sea level, induced by a tidal component, in a small basin. The water level control has been simulated by means of gate operations acting at the open mouth of the tidal basin. The main feature of the model is that the control operations do not require complete closure of the basin but only a variable reduction of its mouth. Although the equations describing the dynamics of the basin have been simplified, the results obtained are expected to be of practical use.     

Controlling unpredictability in the randomly driven Hénon–Heiles system

Noisy scattering dynamics in the randomly driven Hénon–Heiles system is investigated in the range of initial energies where the motion is unbounded. In this paper we study, with the help of the exit basins and the escape time distributions, how an external perturbation, be it dissipation or periodic forcing with a random phase, can enhance or mitigate the unpredictability of a system that exhibit chaotic scattering. In fact, if basin boundaries have the Wada property, predictability becomes very complicated, since the basin boundaries start to intermingle, what means that there are points of different basins close to each other. The main responsible of this unpredictability is the external forcing with random phase, while the dissipation can recompose the basin boundaries and turn the system more predictable. Therefore, we do the necessary simulations to find out the values of dissipation and external forcing for which the exit basins present the Wada property. Through these numerical simulations, we show that the presence of the Wada basins have a specific relation with the damping, the forcing amplitude and the energy value. Our approach consists on investigating the dynamics of the system in order to gain knowledge able to control the unpredictability due to the Wada basins.     

Autonomous strange nonchaotic oscillations in a system of mechanical rotators

We investigate strange nonchaotic self-oscillations in a dissipative system consisting of three mechanical rotators driven by a constant torque applied to one of them. The external driving is nonoscillatory; the incommensurable frequency ratio in vibrational-rotational dynamics arises due to an irrational ratio of diameters of the rotating elements involved. It is shown that, when losing stable equilibrium, the system can demonstrate two- or three-frequency quasi-periodic, chaotic and strange nonchaotic self-oscillations. The conclusions of the work are confirmed by numerical calculations of Lyapunov exponents, fractal dimensions, spectral analysis, and by special methods of detection of a strange nonchaotic attractor (SNA): phase sensitivity and analysis using rational approximation for the frequency ratio. In particular, SNA possesses a zero value of the largest Lyapunov exponent (and negative values of the other exponents), a capacitive dimension close to 2 and a singular continuous power spectrum. In general, the results of this work shed a new light on the occurrence of strange nonchaotic dynamics.