Unique Equilibrium State Research Articles

The equilibrium states for semigroups of rational maps

We consider the dynamics of skew product maps associated with finitely generated semigroups of rational maps on the Riemann sphere. We show that under some conditions on the dynamics and the potential function ψ, there exists a unique equilibrium state for ψ and a unique exp(P(ψ) − ψ)-conformal measure, where P(ψ) denotes the topological pressure of ψ.

Extendability of Equilibria of Nematic Polymers

The purpose of this paper is to study the extendability of equilibrium states of rodlike nematic polymers with the Maier-Saupe intermolecular potential. We formulate equilibrium states as solutions of a nonlinear system and calculate the determinant of the Jacobian matrix of the nonlinear system. It is found that the Jacobian matrix is nonsingular everywhere except at two equilibrium states. These two special equilibrium states correspond to two points in the phase diagram. One point is the folding point where the stable prolate branch folds into the unstable prolate branch; the other point is the intersection point of the nematic branch and the isotropic branch where the unstable prolate state becomes the unstable oblate state. Our result establishes the existence and uniqueness of equilibrium states in the presence of small perturbations away from these two special equilibrium states.

A unified treatment for stability preservation in computer simulations of impulsive BAM networks

This paper is concerned with the stability preservation in computer simulations of an impulsive bidirectional associative memory (BAM) network. The simulations are provided by difference equations formulated from a semi-discretisation technique and impulsive maps as discrete-time representations of the nonlinear impulses which attempt to destabilise the BAM network at fixed moments of time. Prior to producing the computer simulations, the analogue is analysed for its exponential convergence towards a unique equilibrium state. The analysis exploits the method of Lyapunov sequences to derive several sufficient conditions that govern the network parameters and the impulse magnitude and frequency. As special cases, one can obtain from our results, those corresponding to the non-impulsive discrete-time BAM networks and also those corresponding to continuous-time (impulsive and non-impulsive) systems. The treatment of the analysis leads us to a relation between the Lyapunov exponent of the non-impulsive system and that of the impulsive system involving the size of the impulses and the inter-impulse intervals.

Computer simulations of exponentially convergent networks with large impulses

This paper demonstrates the use of a semi-discretization technique for obtaining a discrete-time analogue of an exponentially convergent network that is subject to impulses with large magnitude. Prior to implementing the analogue for computer simulations, we investigate its exponential convergence towards a unique equilibrium state and thereby obtain a family of sufficiency conditions governing the network parameters and the impulse magnitude and frequency. Although the time-step does not appear in the conditions that govern the network parameters, its value needs to be sufficiently small in order for the analogue displays correct convergence behaviour of the network when subjected particularly to large impulses.

Exponential stability of artificial neural networks with distributed delays and large impulses

This paper illustrates that there is a globally exponentially stable unique equilibrium state in an artificial neural network that is subject to delays distributed over unbounded intervals, and also to large impulses that are not too frequent. The activation functions, which may be unbounded, nondifferentiable and/or nonmonotonic, are assumed to be globally Lipschitz continuous. The stability analysis exploits the method of Lyapunov functions and the technique of Halanay inequalities to derive a family of easily verifiable sufficient conditions for convergence to the unique equilibrium state. The sufficiency conditions, in the norm either ∥ · ∥ p where p ⩾ 1 or ∥ · ∥ ∞ , include those that govern the network parameters and the impulse magnitude and frequency.

Existence, uniqueness and stability of the equilibrium points of a SHARON bioreactor model

This paper addresses the dynamics of a SHARON reactor, a promising technology for ammonium removal from concentrated wastewater streams. The contraction mapping theorem is used to determine which operating conditions of a SHARON reactor with pH-control result in a unique equilibrium state. However, this approach only identifies the case of very large dilution rates, in practice corresponding with complete biomass wash-out, i.e. with complete loss of biological activity. Practical operation of a SHARON reactor aims at reaching ammonium conversion to nitrite. To identify such interesting operating points, the equilibrium points are subsequently calculated directly in terms of input variables for a simplified SHARON reactor model. The stability of the obtained equilibrium points is assessed and the corresponding phase portraits are analyzed. The influence of slightly varying parameter and input values is investigated as well.

Exponential stability in Hopfield-type neural networks with impulses

Abstract This paper demonstrates that there is an exponentially stable unique equilibrium state in a Hopfield-type neural network that is subject to quite large impulses that are not too frequent. The activation functions are assumed to be globally Lipschitz continuous and unbounded. The analysis exploits an homeomorphic mapping and an appropriate Lyapunov function, and also either a geometric–arithmetic mean inequality or a Young inequality, to derive a family of easily verifiable sufficient conditions for convergence to the unique globally stable equilibrium state. These sufficiency conditions, in the norm ∥·∥ p where p ⩾ 1, include those governing the network parameters and the impulse magnitude and frequency.

Convergence to Equilibrium for the Linearized Cometary Flow Equation

We study convergence to equilibrium for certain spatially inhomogenous kinetic equations, such as discrete velocity models or a linearization of a kinetic model for cometary flow. For such equations, the convergence to a unique equilibrium state is the result of, first, the dissipative effects of the collision operator, which morphs the solution towards an entropy‐minimizing local equilibrium state, and second, the transport operator as well as the imposed periodic boundary conditions, which repulse the solution from the set of local equilibria as long as the approached local equilibrium is not the global one. This behavior is quantified in a system of differential inequalities of relative entropies with respect to different (sub)classes of local equilibria, respectively, the global equilibrium. We introduce projection operators leading to a convenient notation.

A dynamical proof for the convergence of Gibbs measures at temperature zero

We give a dynamical proof of a result due to Brémont (2003 Nonlinearity 16 419–26). It concerns the problem of maximizing measures for some given observable ϕ: for a subshift of finite type, and when ϕ only depends on a finite number of coordinates, it was proved in Brémont (2003) that the unique equilibrium state associated with βϕ converges to some measure when β goes to +∞. This measure has maximal entropy among the maximizing measures for ϕ. We give here a dynamical proof of this result and we improve it. We prove that for any Hölder continuous function (not necessarily locally constant), f, the unique equilibrium state associated with f + βϕ converges to some measure with maximal f-pressure among the maximizing measures. Moreover we also identify the limit measure.

The generalized Markov measure as an equilibrium state

In this paper, we continue development of the theory of contractive Markov systems (CMSs) initiated Werner (2005 Contractive Markov systems J. Lond. Math. Soc. 71 236–58). Also, this work can be seen as a small contribution to the theory of equilibrium states.We construct an energy function on the code space, using the coding map from Werner (2006 Math. Proc. Camb. Phil. Soc. 140 at press), and show that the generalized Markov measure associated with an irreducible CMS is a unique equilibrium state for this energy function if the vertex sets form an open partition of the state space of the CMS and the restrictions of the probability functions on their vertex sets are Dini-continuous and bounded away from zero.

Non-linear incidence and stability of infectious disease models

In this paper we consider the impact of the form of the non-linearity of the infectious disease incidence rate on the dynamics of epidemiological models. We consider a very general form of the non-linear incidence rate (in fact, we assumed that the incidence rate is given by an arbitrary function f (S, I, N) constrained by a few biologically feasible conditions) and a variety of epidemiological models. We show that under the constant population size assumption, these models exhibit asymptotically stable steady states. Precisely, we demonstrate that the concavity of the incidence rate with respect to the number of infective individuals is a sufficient condition for stability. If the incidence rate is concave in the number of the infectives, the models we consider have either a unique and stable endemic equilibrium state or no endemic equilibrium state at all; in the latter case the infection-free equilibrium state is stable. For the incidence rate of the form g(I)h(S), we prove global stability, constructing a Lyapunov function and using the direct Lyapunov method. It is remarkable that the system dynamics is independent of how the incidence rate depends on the number of susceptible individuals. We demonstrate this result using a SIRS model and a SEIRS model as case studies. For other compartment epidemic models, the analysis is quite similar, and the same conclusion, namely stability of the equilibrium states, holds.

Comparative studies on magnetic flux relaxations of varied spherical toroids produced by co‐helicity merging

AbstractThe co‐helicity merging operations of compact toroid (CT) and spherical tokamak (ST) have been performed with external toroidal fields in the CT/ST merging device TS‐4. The low‐q (safety factor) CT merging as the compact RFP merging and the spheromak merging show the flux conversion from toroidal to poloidal in the course of the reconstruction of the Taylor force free state. The relaxation to the Taylor state proceeds through the following three states: (1) axisymmetric merging with increasing toroidal flux; (2) increase in the poloidal flux Ψ; and (3) relaxation to the Taylor state. The high‐q ST merging shows different relaxation process from those of the compact RFP and the spheromak mergings. Increases in Ψ were not clearly observed in the ST merging. The measured eigenvalues λ show that ST's, especially high‐q ST's, approach a unique intrinsic equilibrium state that has a λ proportional to Ψ with a longer lifetime than that of CT's. When external toroidal field is set in a certain range between the low‐q operation and the high‐q operation for ST's, an abnormal phenomenon was found in the ST formation, namely, a drastic decrease in the plasma lifetime. This phenomenon is characterized by very weak poloidal flux generations during the initial plasma production phase and the subsequent plasma separation phase when the plasma starts detaching from the flux core. © 2005 Wiley Periodicals, Inc. Electr Eng Jpn, 151(4): 7–15, 2005; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/eej.20069

REGULARITY CONDITIONS AND BERNOULLI PROPERTIES OF EQUILIBRIUM STATES AND $g$-MEASURES

When T : X → X is a one-sided topologically mixing subshift of finite type and φ : X → R is a continuous function, one can define the Ruelle operator Lφ : C(X) → C(X) on the space C(X) of real-valued continuous functions on X. The dual operator L φ * always has a probability measure ν as an eigenvector corresponding to a positive eigenvalue ( L φ * ν = λν with λ > 0). Necessary and sufficient conditions on such an eigenmeasure ν are obtained for φ to belong to two important spaces of functions, W(X, T) and Bow (X, T). For example, φ ∈ Bow(X, T) if and only if ν is a measure with a certain approximate product structure. This is used to apply results of Bradley to show that the natural extension of the unique equilibrium state μφ of φ ∈ Bow(X, T) has the weak Bernoulli property and hence is measure-theoretically isomorphic to a Bernoulli shift. It is also shown that the unique equilibrium state of a two-sided Bowen function has the weak Bernoulli property. The characterizations mentioned above are used in the case of g-measures to obtain results on the ‘reverse’ of a g-measure.

On circulant populations. I. The algebra of semelparity

We consider a class of nonlinear Leslie matrix models, describing the population dynamics of an age-structured semelparous species. Semelparous species are those whose individuals reproduce only once and die afterwards. Competitive interaction between individuals is modelled via a one-dimensional environmental quantity. Age classes are characterised by their impact on, and their sensitivity to, the environment. We do not restrict ourselves to some particular form of functional dependence and keep the model otherwise as general as possible. The system possesses a cyclic symmetry. Due to the symmetry it exhibits so-called vertical bifurcations, where a manifold filled with periodic orbits appears in the phase space for specific parameter combinations. This bifurcation serves as a switch between the main types of behaviour: coexistence of all year classes or a periodic regime with some year classes missing. In particular, the vertical bifurcation takes place when a certain circulant matrix is singular. We also analyse the local stability of the unique coexistence equilibrium state and derive a characteristic equation for it. The dynamics of populations with two and, especially, with three age classes are analysed in detail.

Initial formation and long-term evolution of channel–shoal patterns

A complex process-based model (Delft3D) is used to simulate the formation of channels and shoals in a schematised estuary. The model set-up enables a comparison with results obtained from other studies using idealised models. Initial (one tidal cycle) as well as long-term (up to 300 years) simulations are made. Dominant wavelengths of channel–shoal patterns are investigated together with their dependency on width and depth of the basin and the local maximum velocity. Prevalent wavelengths range between 5 and 10km and increase with increasing width and flow velocity. Initial model results are compared with those of idealised models. The dominant wavelengths are of the same order of magnitude as those found in idealised models and their dependency on width and velocity agrees qualitatively. (Exponential) growth of the perturbations during long-term simulations provides information on the limits of validity of the idealised models. Subsequent pattern formation and changes of dominant wavelengths are attributed to non-linear interactions. Morphological developments during long-term simulations suggest that channel–shoal patterns evolve into a unique morphodynamic equilibrium state, independent of the initial perturbation.

同極性合体を用いて生成された各種球状トーラスの磁束緩和に関する比較研究

The co-helicity merging operations of compact toroid (CT) and spherical tokamak (ST) have been performed with external toroidal fields in the CT/ST merging device TS-4. The low-q (safety factor) CT merging as the compact RFP merging and the spheromak merging show the flux conversion from toroidal to poloidal in the course of the reconstruction of the Taylor force free state. The relaxation to the Taylor state proceeds through the following three states: (1) axisymmetric merging with increasing toroidal flux; (2) increase in the poloidal flux Ψ; and (3) relaxation to the Taylor state. The high-q ST merging shows different relaxation process from those of the compact RFP and the spheromak mergings. Increases in Ψ were not clearly observed in the ST merging. The measured eigenvalues λ show that ST's, especially high-q ST's, approach a unique intrinsic equilibrium state that has a λ proportional to Ψ with a longer lifetime than that of CT's. When external toroidal field is set in a certain range between the low-q operation and the high-q operation for ST's, an abnormal phenomenon was found in the ST formation, namely, a drastic decrease in the plasma lifetime. This phenomenon is characterized by very weak poloidal flux generations during the initial plasma production phase and the subsequent plasma separation phase when the plasma starts detaching from the flux core. © 2005 Wiley Periodicals, Inc. Electr Eng Jpn, 151(4): 7–15, 2005; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/eej.20069

Sinai-Ruelle-Bowen Measures for Lattice Dynamical Systems

For weakly coupled expanding maps on the unit circle, Bricmont and Kupiainen showed that the Sinai-Ruelle-Bowen (SRB) measure exists as a Gibbs state. Via thermodynamic formalism, we prove that this SRB measure is indeed the unique equilibrium state for a Holder continuous potential function on the infinite dimensional phase space. For a more general class of lattice systems that are small perturbations of the uncoupled map lattice, we present the variational principle, the entropy formula, and the formula for the potential function for the SRB measures. For coupled map lattices with nearest neighbor interactions, we give an explicit formula of the potential function for the SRB measure and consequently, obtain the entropy in terms of coupling parameters.

A biophysical model of nonlinear dynamics underlying plateau potentials and calcium spikes in purkinje cell dendrites.

Computational capabilities of Purkinje cells (PCs) are central to the cerebellum function. Information originating from the whole nervous system converges on their dendrites, and their axon is the sole output of the cerebellar cortex. PC dendrites respond to weak synaptic activation with long-lasting, low-amplitude plateau potentials, but stronger synaptic activation can generate fast, large amplitude calcium spikes. Pharmacological data have suggested the involvement of only the P-type of Ca channels in both of these electric responses. However, the mechanism allowing this Ca current to underlie responses with such different dynamics is still unclear. This mechanism was explored by constraining a biophysical model with electrophysiological, Ca-imaging, and single ion channel data. A model is presented here incorporating a simplified description of [Ca](i) regulation and three ionic currents: 1) the P-type Ca current, 2) a delayed-rectifier K current, and 3) a generic class of K channels activating sharply in the sub-threshold voltage range. This model sustains fast spikes and long-lasting plateaus terminating spontaneously with recovery of the resting potential. Small depolarizing, tonic inputs turn plateaus into a stable membrane state and endow the dendrite with bistability properties. With larger tonic inputs, the plateau remains the unique equilibrium state, showing long traces of transient inhibitory inputs that are called "valley potentials" because their dynamics mirrors that of inverted, finite-duration plateaus. Analyzing the slow subsystem obtained by assuming instantaneous activation of the delayed-rectifier reveals that the time course of plateaus and valleys is controlled by the slow [Ca](i) dynamics, which arises from the high Ca-buffering capacity of PCs. A bifurcation analysis shows that tonic currents modulate sub-threshold dynamics by displacing the resting state along a hysteresis region edged by two saddle-node bifurcations; these bifurcations mark transitions from finite-duration plateaus to bistability and from bistability to valley potentials, respectively. This low-dimensionality model may be introduced into large-scale models to explore the role of PC dendrite computations in the functional capabilities of the cerebellum.

Can a moment-curvature relationship describe the flexion of softening beams?

Abstract The flexure of a non-linear elastic beam is described through a relationship between the bending moment M and the corresponding curvature χ of the centroid line and neglecting shear deformations. We consider an odd softening M−χ law with horizontal asymptotes M=±M0 for χ→±∞, so that the corresponding strain-energy density U(χ) is an even, non-nonconvex, function with oblique asymptotes. Considering the limit behavior of the total energy functional on sequences converging to singular functions, it is shown that the linear growth at infinity of U(·) is not sufficient to bind the curvature field, i.e., while maintaining the energy a finite value, infinite curvatures may occur at points of the beam. In the example considered, such singularities cannot be of the Cantor type but are represented by Dirac's masses concentrated at isolated points, which are interpreted as localized rotations, similar-in-type to those corresponding to the formation of plastic hinges in the engineering theory of elastic-plastic beams. The equilibrium configurations of the beam are discussed by considering the stationary points of a properly extended energy functional, introduced to take into account the possibility of concentrated rotations. Solutions, in general, are not unique, but unstable, locally-stable and globally-stable equilibrium states are recognized through their correspondence to saddle points, local minima or global minima of the energy functional. In particular, it is demonstrated that the globally stable equilibrium states can be equivalently investigated by introducing an auxiliary minimum problem involving the relaxed energy functional, obtained by replacing U(·) with its lower convex envelope U★★(·) (the corresponding M−χ relationship is similar in type to that of an elastic-perfectly-plastic beam, presenting a plateaux coinciding with the horizontal asymptotes). The possible difficulties derived from having assumed a softening moment-curvature relationship as the only constitutive descriptor of the beam behavior are finally summarized and, as an additional result, a new explanation of Wood and Roberts' paradox is obtained.

A Generic C 1 Expanding Map¶has a Singular S-R-B Measure

We show that for a generic C1 expanding map T of the unit circle, there is a unique equilibrium state for − log T′ that is an S–R–B measure for T, and whose statistical basin of attraction has Lebesgue measure 1. We also present some results related to the question of whether a generic C1 expanding map preserves a σ-finite measure, absolutely continuous with respect to Lebesgue measure.