Stability and attractive basins of multiple equilibria in delayed two-neuron networks

Multiple stability for two-dimensional delayed recurrent neural networks with piecewise linear activation flmctions of 2r （r 〉 1） corner points is studied. Sufficient conditions are established for checking the existence of （2r ＋ 1）2 equilibria in delayed recurrent neural networks. Under these conditions, （r ＋ 1）2 equilibria are locally exponentially stable, and （2r＋ 1）2 -（r ＋ 1）2 -r2 equilibria are unstable. Attractive basins of stable equilibria are estimated, which are larger than invariant sets derived by decomposing state space. One example is provided to illustrate the effectiveness of our results.

A THEORETICAL STUDY ON THE MULTIPLE EQUILIBRIA OF TROPICAL ATMOSPHERE AND THEIR RELATION TO THE ONSET OF THE SUMMER MONSOON

Multiple equilibria and their stability in tropical atmosphere are investigated through β-plane barotropic models with consideration of heating and dissipation. We have derived the solutions of the model equations corresponding to the multiple equilibria or the steady flows first, and then establish the criteria for the stability of steady flow by use of the Liapunov direct Method. When these criteria are applied to the solutions of equilibria obtained, stable flows, which are closely related to the different patterns of quasi-stationary circulation in the tropical region, are lbund. The configurations of these stable flows and the shift between two of them as season changes provide quite reasonable explanations to many fundarnental problems of tropical circulation features such as the catastrophe mechanism of the onset and the break-active cycle of the Asian summer monsoon. It fbllows that the onset or the abrupt transition of the Asian summer monsoon could be attributed to the multiple equilibrium property of the tropical circulation resulted from the advcetive nonlinearity, which provide another explanation among others.

Exploring multiple equilibria for symmetric 2×2 CES/LES pure exchange economies

This paper explores the existence of 3 equilibria for symmetric 2-individual 2-good CES/LES pure exchange economies. For certain parameterizations in the economies, we show analytically that there are no more than 3 equilibria. We generalize our analytical results of existence of 3 equilibria for a wide range of parameterizations. Then we provide examples of 3 equilibria and parameter zones of 3 equilibria for CES and CES/LES economies.

Sensitivity of the Multiple Equilibria to Gorverning System, Mode Chosen and Parameter

Multiple equilibria of a forced, dissipative atmosphere system studied by Charney andothers have provided a new insight into the dynamics of the atmospheric circulation. But thetheoretical results remain some uncertainty for different approaches. Charney and Devore (1979) obtained two stable equilibria for certain range of externalforcing in the barotropic model, one of which is a high--index circulation, and the other alow--index for a blocking state. However Charney and Straus (1980) found that only thelow--index (blocking) state is stable and there do not exist multiple equilibria in the baroclinicmodel.

μ-stability of multiple equilibria in Cohen-Grossberg neural networks and its application to associative memory

In this paper, the μ-stability of multiple equilibrium points(EPs) in the Cohen-Grossberg neural networks(CGNNs) is addressed by designing a kind of discontinuous activation function(AF). Under some criteria, CGNNs with this AF are shown to possess at least 5^(n)EPs, of which 3^(n)EPs are locally μ-stable. Compared with the saturated AF or the sigmoidal AF, CGNNs with the designed AF can produce many more total/stable EPs. Therefore, when CGNNs with the designed discontinuous AF are applied to associative memory, they can store more prototype patterns. Moreover, the AF is expanded to a more general version to further increase the number of total/stable equilibria. The CGNNs with the expanded AF are found to produce(2k+3)^(n)EPs, of which (k+2)^(n)EPs are locally μ-stable. By adjusting two parameters in the AF, the number of sufficient conditions ensuring the μ-stability of multiple equilibria can be decreased. This finding implies that the computational complexity can be greatly reduced.Two numerical examples and an application to associative memory are illustrated to verify the correctness of the obtained results.

Multiple Equilibria in a Land–Atmosphere Coupled System

Many low-order modeling studies indicate that there may be multiple equilibria in the atmosphere induced by thermal and topographic forcings.However,most work uses uncoupled atmospheric model and just focuses on the multiple equilibria with distinct wave amplitude,i.e.,the high-and low-index equilibria.Here,a low-order coupled land–atmosphere model is used to study the multiple equilibria with both distinct wave phase and wave amplitude.The model combines a two-layer quasi-geostrophic channel model and an energy balance model.Highly truncated spectral expansions are used and the results show that there may be two stable equilibria with distinct wave phase relative to the topography:one(the other)has a lower layer streamfunction that is nearly in(out of)phase with the topography,i.e.,the lower layer ridges(troughs)are over the mountains,called ridge-type(trough-type)equilibria.The wave phase of equilibrium state depends on the direction of lower layer zonal wind and horizontal scale of the topography.The multiple wave phase equilibria associated with ridge-and trough-types originate from the orographic instability of the Hadley circulation,which is a pitch-fork bifurcation.Compared with the uncoupled model,the land–atmosphere coupled system produces more stable atmospheric flow and more ridge-type equilibrium states,particularly,these effects are primarily attributed to the longwave radiation fluxes.The upper layer streamfunctions of both ridge-and trough-type equilibria are also characterized by either a high-or low-index flow pattern.However,the multiple wave phase equilibria associated with ridge-and trough-types are more prominent than multiple wave amplitude equilibria associated with high-and low-index types in this study.

MULTIPLE EQUILIBRIA IN BAROCLINIC ATMOSPHERE BY THERMAL FORCING RELATED TO SUBTROPICAL HIGH

By using Lorenz’s moist general circulation model, a nonlinear and dissipative system describing atmospheric motion has been obtained in approximation of Low Order.The multiple equilibria and the transformation between the flow patterns of winter and summer, the latitudinal oscillation of subtropical high by thermal forcing and nonlinear interaction of general circulation are studied in this paper.The results show that the transformation of flow patterns is a discontinuous leaping, and is a process of resetting new flow pattern by rapid exchange inflow field.In the corresponding dry model, we cannot find the latitudinal oscillation of the center of subtropical high.In the moist model, after the thermal effect of water vapor is drawn into the model, nonlinear interaction appears between flow and heating fields.This effect helps to bring about the latitudinal oscillation of subtropical high.

CONSTRUCTING EXAMPLES WITH 5 EQUILIBRIA FOR SYMMETRIC 3×2 CES /LES PURE EXCHANGE ECONOMIES

This paper explores the existence of multiple equilibria for symmetric 3 indi- vidual, 2 good CES / LES pure exchange economies. Analytically, we show that there are no more than 5 equilibria in such economies. The number of equilibria varies from 5 to 3 then to 1. We generalize our analytical results of existence of 1, 3, 5 equilibria for a wide range of parametrizations. We also provide concrete examples of 1, 3, 5 equilibria with parameter zones specified.

Nonlinear dynamics analysis of cluster-shaped conservative flows generated from a generalized thermostatted system

The thermostatted system is a conservative system different from Hamiltonian systems,and has attracted much attention because of its rich and different nonlinear dynamics.We report and analyze the multiple equilibria and curve axes of the cluster-shaped conservative flows generated from a generalized thermostatted system.It is found that the cluster-shaped structure is reflected in the geometry of the Hamiltonian,such as isosurfaces and local centers,and the shapes of cluster-shaped chaotic flows and invariant tori rely on the isosurfaces determined by initial conditions,while the numbers of clusters are subject to the local centers solved by the Hessian matrix of the Hamiltonian.Moreover,the study shows that the cluster-shaped chaotic flows and invariant tori are chained together by curve axes,which are the segments of equilibrium curves of the generalized thermostatted system.Furthermore,the interesting results are vividly demonstrated by the numerical simulations.

Wind-Driven,Double-Gyre,Ocean Circulation in a Reduced-Gravity,2.5-Layer,Lattice Boltzmann Model

A coupled lattice Boltzmann （LB） model with second-order accuracy is applied to the reduced-gravity, shallow water, 2.5-layer model for wind-driven double-gyre ocean circulation. By introducing the secondorder integral approximation for the collision operator, the model becomes fully explicit. The Coriolis force and other external forces are included in the model with second-order accuracy, which is consistent with the discretization accuracy of the LB equation. The feature of the multiple equilibria solutions is found in the numerical experiments under different Reynolds numbers based on this LB scheme. With the Reynolds number increasing from 3000 to 4000, the solution of this model is destabilized from the anti-symmetric double-gyre solution to the subtropic gyre solution and then to the subpolar gyre solution. The transitions between these equilibria states are also found in some parameter ranges. The time-dependent variability of the circulation based on this LB simulation is also discussed for varying viscosity regimes. The flow of this model exhibits oscillations with different timescales varying from subannual to interannual. The corresponding statistical oscillation modes are obtained by spectral analysis. By analyzing the spatiotemporal structures of these modes, it is found that the subannual oscillation with a 9-month period originates from the barotropic Rossby basin mode, and the interarmual oscillations with periods ranging from 1.5 years to 4.6 years originate from the recirculation gyre modes, which include the barotropic and the baroclinic recirculation gyre modes.

Multi-scroll hidden attractors and multi-wing hidden attractors in a 5-dimensional memristive system

A novel 5-dimensional（5D） memristive chaotic system is proposed, in which multi-scroll hidden attractors and multiwing hidden attractors can be observed on different phase planes. The dynamical system has multiple lines of equilibria or no equilibrium when the system parameters are appropriately selected, and the multi-scroll hidden attractors and multi-wing hidden attractors have nothing to do with the system equilibria. Particularly, the numbers of multi-scroll hidden attractors and multi-wing hidden attractors are sensitive to the transient simulation time and the initial values. Dynamical properties of the system, such as phase plane, time series, frequency spectra, Lyapunov exponent, and Poincar′e map, are studied in detail. In addition, a state feedback controller is designed to select multiple hidden attractors within a long enough simulation time. Finally, an electronic circuit is realized in Pspice, and the experimental results are in agreement with the numerical ones.

Information Acquisition: Fundamental and Non-Fundamental

We present a parsimonious information acquisition model in which two types of traders can produce either fundamental or non-fundamental information.Fundamental information is related to asset liquidation value,whereas non-fundamental information is related to the noise caused by traders'sentiment.Opening access to non-fundamental information increases the coordination possibilities among sentiment-informed traders and can yield two equilibrium-displaying properties:substitutability and complementarity.We find that the dominated mass of one type of informed trader can attenuate their information advantage,resulting in low ex ante expected utility associated with such traders.We further find that there is a crowding-out effect in information acquisition between the two types of informed traders,which offers some significant insights in explaining why bubbles burst when market sentiment is dominant.

Two-Dimensional Energy Balance Model and Its Application to Some Climatic Issues

Based on a two-dimensional energy balance model, the studies on some climatic issues such as the re- lationship between ice cap latitude and solar constant, desertifieation, and the warming effect of carbon dioxide, have been reviewed and discussed. The phenomenon that a fixed solar constant might correspond to different equilibrium ice cap latitudes is determined by the continuity of albedo distribution. The disconti- nuity in albedo distribution increases the number of equilibrium ice cap latitudes. Desert would expand both northward and southward when desert surface albedo is increasing. This would deteriorate the ecological environment in border regions, and then threaten the existence of local inhabitants. Melting of the polar ice would not be accelerated, with increasing carbon dioxide concentration. The ice cap latitude would move northward slowly, with some “hiatus” periods, under the slowly increasing global average surface tempera- ture. According to the current research, future development of the two-dimensional energy balance model and possible progress are also forecasted.

Fold Bifurcation Caused by Pollution Emission in an OLG Economy

The purpose of this paper is to investigate the macroeconomic effect of pollution emission in an overlapping generation system. This is done through constructing an endogenous economy which generates pollution and environment deterioration. It shows how different level of pollution emission could cause complexity such as fold bifurcation in this system. In addition, numerical simulations are provided to show the emergency of complexity.

Competitive Equilibrium in an Overlapping Generations Model with Production Loans

The paper shows that there do exist two kinds of steady states equilibria in the overlapping generations models with consumption and production loans, similar to the pure exchange economies examined by Gale （1973）. Furthermore, the local stability properties of these two （kinds of） steady states are also investigated： In the classical case, the golden-rule steady state is stable and the balanced steady state is saddle-point stable; however, in the Samuelson case, the golden-rule steady state is saddle-point stable and the balanced steady state is stable.