Comparison of the energetic properties and the dynamical stability in a mathematical model of the stretch reflex

The previous studies on finite-time thermodynamic engines have shown that some of the parameters affecting their thermodynamic performance also affect their stability. Moreover, such parameters have to be tuned to reach an optimal trade-off between these two generic properties. In the present work we carry out a similar analysis on a mathematical model of the stretch reflex regulatory pathway, which is a simplified version of a previously published model. We show that the model has a unique stable fixed point in the absence of time delays. However, when the system inherent time delays are considered, they can destabilize the fixed point and engender a stable limit cycle. We further explore the parameter space to analyse the sensitivity of the system stability to variations in the parameter values. Particular attention is paid to the parameter here denoted as α, which has been shown to determine the muscle thermodynamic properties during steady-state contractions: larger values of α mean more powerful and less efficient muscles. Our results indicate that the stretch reflex pathway is less stable in the more powerful and less efficient muscles. We finally compare these observations with those obtained on thermal engines.     

Trivariate analysis of two qubit symmetric separable state

We consider the simplest model in the family of discrete predator–prey system and introduce for the first time an environmental factor in the evolution of the system by periodically modulating the natural death rate of the predator. We show that with the introduction of environmental modulation, the bifurcation structure becomes much more complex with bubble structure and inverse period doubling bifurcation. The model also displays the peculiar phenomenon of coexistence of multiple limit cycles in the domain of attraction for a given parameter value that combine and finally gets transformed into a single strange attractor as the control parameter is increased. To identify the chaotic regime in the parameter plane of the model, we apply the recently proposed scheme based on the correlation dimension analysis. We show that the environmental modulation is more favourable for the stable coexistence of the predator and the prey as the regions of fixed point and limit cycle in the parameter plane increase at the expense of chaotic domain.     

On a simple recursive control algorithm automated and applied to an electrochemical experiment

We review a simple recursive proportional feedback (RPF) control strategy for stabilizing unstable periodic orbits found in chaotic attractors. The method is generally applicable to high-dimensional systems and stabilizes periodic orbits even if they are completely unstable, i.e., have no stable manifolds. The goal of the control scheme is the fixed point itself rather than a stable manifold and the controlled system reaches the fixed point in d+1 steps, where d is the dimension of the state space of the Poincare map. We provide a geometrical interpretation of the control method based on an extended phase space. Controllability conditions or special symmetries that limit the possibility of using a single control parameter to control multiply unstable periodic orbits are discussed. An automated adaptive learning algorithm is described for the application of the control method to an experimental system with no previous knowledge about its dynamics. The automated control system is used to stabilize a period-one orbit in an experimental system involving electrodissolution of copper. (c) 1997 American Institute of Physics.     

Impacts of strong Allee effect and hunting cooperation for a Leslie-Gower predator-prey system

In this paper, we propose a Leslie-Gower predator-prey system with strong Allee effect in prey and hunting cooperation among predators. To discuss the impacts of Allee effect and hunting cooperation, we choose the severity of Allee effect and the cooperative hunting coefficient as the main control parameters. First, using the topology equivalent method, type of singular point (0,0) is obtained. Moreover, the basin of attraction of (0,0) is studied. Then, the stability of all nonnegative equilibrium points, including type of degenerate equilibrium point, is discussed. Based on Sotomayor’s theorem, the existence of saddle-node bifurcation is derived. To determine the stability of limit cycles arising from the Hopf bifurcation, the first Lyapunov number is calculated. Meanwhile, by the rigorous mathematical proofs, we obtain that the bifurcating limit cycle is stable if the hunting cooperation coefficient and the severity of Allee effect are sufficiently small. Finally, numerical simulations are presented to validate the theoretical results and further explore the influences of Allee effect and hunting cooperation. These results show that strong Allee effect and hunting cooperation have significant impacts on dynamical behaviors of the system.     

A simple two-component reaction-diffusion system showing rich dynamic behavior: spatially homogeneous aspects and selected bifurcation scenarios

We present a simple reaction-diffusion model for two variables. The model was originally designed to have a stable localized solution for a range of parameters as a consequence of the coexistence of a stable limit cycle and a stable fixed point. We classify the spatially homogeneous solutions of the model. In addition we describe several bifurcation scenarios for particle-like solutions as a function of two of the parameters.     

Orbital dynamics in the photogravitational restricted four-body problem: Lagrange configuration

We study the effect of the radiation parameter in the location, stability and orbital dynamics in the Lagrange configuration of the restricted four-body problem when one of the primaries is a radiating body. The equations of motion for the test particle are derived by assuming that the primaries revolve in the same plane with uniform angular velocity, lying at the vertices of an equilateral triangle. The insertion of the radiation factor in the restricted four-body problem, let us model the dynamics of a test particle orbiting an astrophysical system with an active star. The dynamical mechanisms responsible for the smoothening on the basin structures of the configuration space is related to the decrease in the total number of fixed points with increasing values of the radiation parameter. In our model of the Sun-Jupiter-Trojan Asteroid system, it is found that despite the solar radiation pressure, there exist two stable libration points.     

Hopf bifurcation analysis of an aeroelastic model using stochastic normal form

We investigate the effects of parameter uncertainties on the dynamical response of an aeroelastic model representing an oscillating airfoil in pitch and plunge with linear aerodynamics and cubic structural nonlinearities. An approach based on the stochastic normal form is proposed to determine the effects due to the variations in the flow speed and the structural stiffness terms on the stability of the aeroelastic system near the Hopf bifurcation point. This approach allows us to study analytically the bifurcation scenario and to predict the amplitude and frequency of the limit cycle oscillation (LCO). The results show that the amplitude of LCO corresponding to the supercritical Hopf bifurcation increases with the intensity of the noise perturbing the pitch and plunge cubic terms, but there is almost no effect on the LCO frequency. Uncertainties in the flow speed produce a shift in the bifurcation point, and unstable subcritical behavior may occur for values of parameters for which the corresponding deterministic model is stable. The stochastic normal form confirms and extends previously known numerical results regarding the effect of parameter variations, and offers an effective way to perform sensitivity analysis of the system's response.     

Invariant integral and the transition to steady states in separable dynamical systems

We show that the transition between fixed points in a separable dynamical system is fully described by an invariant integral. We discuss in detail the case of a system with two temporal variables with bilinear coupling, where the new stable state is attained asymptotically through spiraling into the fixed point. Through the invariance, it is possible to establish conditions for the control parameter that permit a (targeted) transition in finite time and without relaxation oscillations.     

Limit Cycles near Stationary Points in the Lorenz System

The limit cycles in the Lorenz system near the stationary points are analysed numerically. A plane in phase space of the linear Lorenz system is used to locate suitable initial points of trajectories near the limit cycles. The numerical results show a stable and an unstable limit cycle near the stationary point. The stable limit cycle is smaller than the unstable one and has not been previously reported in the literature. In addition, all the limit cycles in the Lorenz system are theoreticallv Proven not to be planar.     

Local bifurcation analysis of a four-dimensional hyperchaotic system

Local bifurcation phenomena in a four-dlmensional continuous hyperchaotic system, which has rich and complex dynamical behaviours, are analysed. The local bifurcations of the system are investigated by utilizing the bifurcation theory and the centre manifold theorem, and thus the conditions of the existence of pitchfork bifurcation and Hopf bifurcation are derived in detail. Numerical simulations are presented to verify the theoretical analysis, and they show some interesting dynamics, including stable periodic orbits emerging from the new fixed points generated by pitchfork bifurcation, coexistence of a stable limit cycle and a chaotic attractor, as well as chaos within quite a wide parameter region.     

Noise-induced mixed-mode oscillations in a relaxation oscillator near the onset of a limit cycle

A detailed asymptotic study of the effect of small Gaussian white noise on a relaxation oscillator undergoing a supercritical Hopf bifurcation is presented. The analysis reveals an intricate stochastic bifurcation leading to several kinds of noise-driven mixed-mode oscillations at different levels of amplitude of the noise. In the limit of strong time-scale separation, five different scaling regimes for the noise amplitude are identified. As the noise amplitude is decreased, the dynamics of the system goes from the limit cycle due to self-induced stochastic resonance to the coherence resonance limit cycle, then to bursting relaxation oscillations, followed by rare clusters of several relaxation cycles (spikes), and finally to small-amplitude oscillations (or stable fixed point) with sporadic single spikes. These scenarios are corroborated by numerical simulations.     

Field theory of long time behaviour in driven diffusive systems

We present a renormalisation group study for the long time behaviour of a diffusive system with a single conserved density which is subjected to an external driving force. In the asymptotic long wavelength limit the system approaches an infrared stable fixed point where detailed balance is satisfied. We obtain the exact scaling form of the density correlation function. In one dimension, the corresponding universal amplitude agrees excellently with a recent Monte Carlo simulation.     

Control of chaotic systems using an on-line trained linear neural controller

We propose a control system including an on-line trained linear neural controller to control chaotic systems. The control system stabilizes a chaotic orbit onto an unstable fixed point without using the knowledge of the location of the point and the local linearized dynamics at the point. Furthermore, the control system can track the stabilized orbit to the unstable fixed point whose location and local dynamics vary slowly with a variation of the system parameter. This paper extends a previous paper (Konishi and Kokame, 1995) for more general situations and improves the neural controller proposed in the previous paper both to simplify the training algorithm and to guarantee the convergence of the neural controller. The stability analysis of the control system reveals that some unstable fixed points cannot be stabilized in the control system. Numerical experiments show that the control system works well for controlling high-dimensional chaotic systems.     

Effect of noise and perturbations on limit cycle systems

This paper demonstrates that the influence of noise and of external perturbations on a nonlinear oscillator can vary strongly along the limit cycle and upon transition from limit cycle to stationary point behaviour. For this purpose we consider the role of noise on the Bonhoeffer-van der Pol model in a range of control parameters where the model exhibits a limit cycle, but the parameters are close to values corresponding to a stable stationary point. Our analysis is based on the van Kampen approximation for solutions of the Fokker-Planck equation in the limit of weak noise. We investigate first separately the effect of noise on motion tangential and normal to the limit cycle. The key result is that noise induces diffusion-like spread along the limit cycle, but quasistationary behaviour normal to the limit cycle. We then describe the coupled motion and show that noise acting in the normal direction can strongly enhance diffusion along the limit cycle. We finally argue that the variability of the system's response to noise can be exploited in populations of nonlinear oscillators in that weak coupling can induce synchronization as long as the single oscillators operate in a regime close to the transition between oscillatory and excitatory modes.     

Local enrichment and its nonlocal consequences for victim-exploiter metapopulations

The stabilizing effects of local enrichment are revisited. Diffusively coupled host-parasitoid and predator-prey metapopulations are shown to admit a stable fixed point, limit cycle or stable torus with a rich bifurcation structure. A linear toy model that yields many of the basic qualitative features of this system is presented. The further nonlinear complications are analyzed in the framework of the marginally stable Lotka-Volterra model, and the continuous time analog of the unstable, host-parasitoid Nicholson-Bailey model. The dependence of the results on the migration rate and level of spatial variations is examined, and the possibility of “nonlocal” effect of enrichment, where local enrichment induces stable oscillations at a distance, is studied. A simple method for basic estimation of the relative importance of this effect in experimental systems is presented and exemplified.     

On the Dynamics of a Model with Coexistence of Three Attractors: A Point,a Periodic Orbit and a Strange Attractor

For a dynamical system described by a set of autonomous differential equations, an attractor can be either a point, or a periodic orbit, or even a strange attractor. Recently a new chaotic system with only one parameter has been presented where besides a point attractor and a chaotic attractor, it also has a coexisting attractor limit cycle which makes evident the complexity of such a system. We study using analytic tools the dynamics of such system. We describe its global dynamics near the infinity, and prove that it has no Darboux first integrals.     

单模激光Haken-Lorenz系统的振荡解析解

研究了单模激光Haken-Lorenz系统在Hopf 分歧点处的动力学行为.求出了Haken-Lorenz系统的定态解,采用线性稳定性原理对定态解进行了稳定性分析,获得了本征值方程,进而确定了系统的Hopf 分歧点μc.利用级数法求出了系统在分歧点处的时间周期振荡解的解析表达式.通过计算机对系统分歧点处的动力学行为进行了数值模拟,结果表明,系统在分歧点处存在一个极限环,即时间周期振荡解.与理论分析的解析结果相一致.     

Strategic Interaction in Trend-Driven Dynamics

We propose a discrete-time stochastic dynamics for a system of many interacting agents. At each time step agents aim at maximizing their individual payoff, depending on their action, on the global trend of the system and on a random noise; frictions are also taken into account. The equilibrium of the resulting sequence of games gives rise to a stochastic evolution. In the limit of infinitely many agents, a law of large numbers is obtained; the limit dynamics consist in an implicit dynamical system, possibly multiple valued. For a special model, we determine the phase diagram for the long time behavior of these limit dynamics and we show the existence of a phase, where a locally stable fixed point coexists with a locally stable periodic orbit.     

Quantum electrodynamics in 2 + 1 dimensions, confinement, and the stability of U(1) spin liquids

Compact quantum electrodynamics in 2 + 1 dimensions often arises as an effective theory for a Mott insulator, with the Dirac fermions representing the low-energy spinons. An important and controversial issue in this context is whether a deconfinement transition takes place. We perform a renormalization group analysis to show that deconfinement occurs when N > Nc = 36/pi3 approximately to 1.161, where N is the number of fermion replica. For N < Nc, however, there are two stable fixed points separated by a line containing a unstable nontrivial fixed point: a fixed point corresponding to the scaling limit of the noncompact theory, and another one governing the scaling behavior of the compact theory. The string tension associated with the confining interspinon potential is shown to exhibit a universal jump as N --> Nc-. Our results imply the stability of a spin liquid at the physical value N = 2 for Mott insulators.