Kolmogorov Systems Research Articles

Torsion and attractors in the Kolmogorov hydrodynamical system

Abstract The geometrical structure of the Kolmogorov system is studied. Considering a divergence-free geodesic motion on a Riemann-Cartan manifold, it is shown that the torsion tensor is related via group theory to the quadratic part of this system. Kolmogorov equations can be considered as the dissipative Euler-Poincare equations on the Lie algebra of the associated group manifold. The relationship with Navier-Stokes equations and their truncated models is discussed.

Coexistence States for Periodic Competitive Kolmogorov Systems

We prove the existence of positive periodic solutions for a class of nonautonomous competitive periodic Kolmogorov systems which generalize the May–Leonard model. A necessary and sufficient condition is also obtained.

Limit cycles of a cubic Kolmogorov system

We show that for certain cubic Kolmogorov systems, four, and no more than four, limit cycles can bifurcate from a critical point in the first quadrant; moreover, three fine foci of order one can coexist and a limit cycle can bifurcate from each.

Nonlocality of the Misra-Prigogine-Courbage semigroup

We show that the Markov semigroups constructed by Misra, Prigogine, and Courbage through nonunitary similarity transformations of Kolmogorov systems are not implementable by local point transformations, i.e., they are not the Frobenius-Perron semigroups associated with noninvertible point transformations, in contrast with the semigroups obtained by coarse-graining projections. Our result is a straightforward generalization of the proof of the nonlocality of the similarity transformation given by Goldstein, Misra, and Courbage and also of the previous illustration by Misra and Prigogine for the baker transformation and completes the characterization of the Misra-Prigogine-Courbage semigroups.

Relaxation and localization in interacting quantum maps

Quantum relaxation is studied in coupled quantum baker's maps. The classical systems are exactly solvable Kolmogorov systems, for which the exponential decay to equilibrium is known. They model the fundamental processes of transport in classically chaotic phase space. The quantum systems, in the absence of global symmetry, show a marked saturation in the level of transport, as the suppression of diffusion in the quantum kicked rotor, and eigenfunction localization in the position basis. In the presence of a global symmetry we study another model that has classically an identical decay to equilibrium, but-quantally shows resonant transport, no saturation, and large fluctuations around equilibrium. We generalize the quantization to finite multibaker maps. As a byproduct we introduce some simple models of quantal tunneling between classically chaotic regions of phase space.

Intrinsic irreversibility and integrability of dynamics

Irreversibility as the emergence of a priviledged direction of time arises in an intrinsic way at the fundamental level for highly unstable dynamical systems, such as Kolmogorov systems or large Poincaré systems. The presence of resonances in large Poincaré systems causes a breakdown of the conventional perturbation methods analytic in the coupling parameter. These difficulties are manifestations of general limitations to computability for unstable dynamical systems. However, a natural ordering of the dynamical states leads to a well-defined prescription for the regularization of the propagators which lifts the divergence and gives rise to an extension of the eigenvalue problem to the complex plane. The extension acquires meaning in suitable rigged Hilbert spaces which are constructed explicitly for the Friedrichs model. We show that the unitary evolution group, when extended, splits into two semigroups, one decaying in the future and the other in the past. Irreversibility emerges as the selection of the semigroup compatible with our future observations. In this way the problems of integration and irreversibility both enjoy a common solution in the extended space.

Permanence and Positive Periodic Solutions for Kolmogorov Competing Species Systems

We prove the existence of periodic solutions in a compact attractor of (R+)n for the Kolmogorov system x′i= xifi(t, x1, …, xn), i = l, …, n in the competitive case. Extension to differential delay equations are con- sidered too. Applications are given to Lotka-Volterra systems with periodic coefficients.

Deterministic quantum noise and Kolmogorov systems

Two irreversible quantum evolutions with zero dynamical entropy, when dilated to reversible automorphisms, provide quantum Kolmogorov systems of both the algebraic and entropic type with infinite-dynamical entropy.

A necessary and sufficient condition for convergence to equilibrium in Kolmogorov systems

A necessary and sufficient condition is established for a state to converge to equilibrium in a dynamical system described as a Kolmogorov system. Because the time reversal of a state converging to equilibrium does not necessarily share this property, this opens the possibility of spontaneous breaking of time reversal symmetry.

Quasi-ergodicity for non-homogeneous continuous-time Markov chains

We consider a non-homogeneous continuous-time Markov chain X(t) with countable state space. Definitions of uniform and strong quasi-ergodicity are introduced. The forward Kolmogorov system for X(t) is considered as a differential equation in the space of sequences l1. Sufficient conditions for uniform quasi-ergodicity are deduced from this equation. We consider conditions of uniform and strong ergodicity in the case of proportional intensities.

Quasi-ergodicity for non-homogeneous continuous-time Markov chains

We consider a non-homogeneous continuous-time Markov chain X(t) with countable state space. Definitions of uniform and strong quasi-ergodicity are introduced. The forward Kolmogorov system for X(t) is considered as a differential equation in the space of sequences l 1 . Sufficient conditions for uniform quasi-ergodicity are deduced from this equation. We consider conditions of uniform and strong ergodicity in the case of proportional intensities.

A method for approximating the probability functions of a Markov chain

This paper presents a method of approximating the state probabilities for a continuous-time Markov chain. This is done by constructing a right-shift process and then solving the Kolmogorov system of differential equations recursively. By solving a finite number of the differential equations, it is possible to obtain the state probabilities to any degree of accuracy over any finite time interval.

A method for approximating the probability functions of a Markov chain

This paper presents a method of approximating the state probabilities for a continuous-time Markov chain. This is done by constructing a right-shift process and then solving the Kolmogorov system of differential equations recursively. By solving a finite number of the differential equations, it is possible to obtain the state probabilities to any degree of accuracy over any finite time interval.

Discussion: On Harold Jeffreys' Axioms

It is argued that models of H. Jeffreys' axioms of probability (Jeffreys [1939] 1967) are not monotone even with I. J. Good's proposed modification (Good 1950). Hence the additivity axiom seems essential to a theory of probability as it is with Kolmogorov's system (Kolmogorov 1950).

Existence and uniqueness of limit cycles on a cubic kolmogorov differential system in the Predator-Prey relation

In this paper, we analyse qualitatively a cubic Kolmogorov system:\(\left\{ \begin{gathered} \frac{{dx}}{{dt}} = x[a_0 + a_1 x - a_3 x^2 - a_4 y + a_5 xy], \hfill \\ \frac{{dy}}{{dt}} = y(x - 1)(1 + by), \hfill \\ \end{gathered} \right.\) which is one of the mathematical models in ecology describing the interaction between Predator-Prey of two populations; and give the conditions of nonexistence, existence and uniqueness of limit cycles for three different cases.

Infinite Channel Queueing System with Controlled Input

We study a queueing model of type G⃗/M/∞ with a nonrecurrent input. The intensity of the input stream is regulated at epochs of successive starts of customers moving from the source to the system, and it is dependent on the number of busy channels. We consider three groups of channels. Such a division is connected with different expenditures of servicing within each of the groups. It is worthwhile to control the input rate in order to minimize the long-run average cost. The basic process is the number of busy channels. We derive its limiting distribution explicitly by introducing an auxiliary two-dimensional Markov process and treating the corresponding Kolmogorov system of differential equations. We also use some properties of semiregenerative processes. The limiting distributions are used to solve an optimization problem.

On the multiserver queue with finite waiting room and controlled input

In this paper we study a multi-channel queueing model of type with N waiting places and a non-recurrent input flow dependent on queue length at the time of each arrival. The queue length is treated as a basic process. We first determine explicitly the limit distribution of the embedded Markov chain. Then, by introducing an auxiliary Markov process, we find a simple relationship between the limiting distribution of the Markov chain and the limiting distribution of the original process with continuous time parameter. Here we simultaneously combine two methods: solving the corresponding Kolmogorov system of the differential equations, and using an approach based on the theory of semi-regenerative processes. Among various applications of multi-channel queues with state-dependent input stream, we consider a closed single-server system with reserve replacement and state-dependent service, which turns out to be dual (in a certain sense) in relation to our model; an optimization problem is also solved, and an interpretation by means of tandem systems is discussed.

On the multiserver queue with finite waiting room and controlled input

In this paper we study a multi-channel queueing model of type with N waiting places and a non-recurrent input flow dependent on queue length at the time of each arrival. The queue length is treated as a basic process. We first determine explicitly the limit distribution of the embedded Markov chain. Then, by introducing an auxiliary Markov process, we find a simple relationship between the limiting distribution of the Markov chain and the limiting distribution of the original process with continuous time parameter. Here we simultaneously combine two methods: solving the corresponding Kolmogorov system of the differential equations, and using an approach based on the theory of semi-regenerative processes. Among various applications of multi-channel queues with state-dependent input stream, we consider a closed single-server system with reserve replacement and state-dependent service, which turns out to be dual (in a certain sense) in relation to our model; an optimization problem is also solved, and an interpretation by means of tandem systems is discussed.

Reversible dynamics and the macroscopic rate law for a solvable Kolmogorov system: The three bakers' reaction

We investigate a piecewise linear (area-preserving) mapT describing two coupled baker transformations on two squares, with coupling parameter 0⩽c⩽1. The resulting dynamical system is Kolmogorov for anyc≠0. For rational values ofc, we construct a generating partition on whichT induces a Markov chain. This Markov structure is used to discuss the decay of correlation functions: exponential decay is found for a class of functions related to the partition. Explicit results are given forc=2−n. The macroscopic analog of our model is a leaking process between two (badly) stirred containers: according to the Markov analysis, the corresponding progress variable decays exponentially, but the rate coefficients characterizing this decay are not those determined from the one-way flux across the cell boundary. The validity of the macroscopic rate law is discussed.

Periodic Kolmogorov Systems

The existence of nontrivial periodic solutions of periodic Kolmogorov systems of ordinary differential equations is considered. Under very general conditions, a global continuum of solutions is shown to bifurcate from a noncritical periodic solution of a reduced system using the average inherent per unit growth rate of one component of the system as a bifurcation parameter. The positivity and stability of the bifurcating branch solutions are studied. The stability is shown to depend on the stability of the solution of the reduced system as well as the direction of bifurcation. These results extend and generalize earlier work on periodic Volterra–Lotka systems. Applications to mathematical ecology are given.