The principal methods for the definition of thermodynamic entropy are discussed with special reference to those developed by Carathéodory, the Keenan School, Lieb and Yngvason, and the present authors. An improvement of the latter method is then presented. Seven basic axioms are employed: three Postulates, which are considered as having a quite general validity, and four Assumptions, which iden...

For finite-dimensional maps and periodic systems, Palmer rigorously proved Smale horseshoe theorem using shadowing lemma in 1988 [20]. For infinitedimensional maps and periodic systems, such a proof was completed by Steinlein and Walther in 1990 [30], and Henry in 1994 [9]. For finite-dimensional autonomous systems, such a proof was accomplished by Palmer in 1996 [17]. For infinite-dimensional ...

An SEI autonomous model with logistic growth rate and its corresponding nonautonomous model are investigated. For the autonomous case, we give the attractive regions of equilibria and perform some numerical simulations. Basic demographic reproduction number R d is obtained. Moreover, only the basic reproduction number R 0 cannot ensure the existence of the positive equilibrium, which needs addi...

<p style='text-indent:20px;'>A non-autonomous random dynamical system is called to be controllable if there a pullback attractor (PRA) such that each fibre of the PRA converges upper semi-continuously nonempty compact set (called controller) as time-parameter goes minus infinity, while asymptotically autonomous for another (autonomous) controller. We establish criteria ensuring existence ...

Autonomous machines have fascinated people for millennia. The Ancient Greeks told of Talos, the Man of Bronze, whose supposedly mythical existence has been given unexpected credence by the recent discovery of the Antikythera Mechanism. Could the Greeks really have created a giant mechanical man to protect Europa in Crete against pirates and invaders? Long before that, the Ancient Egyptians crea...

We study the existence of uniform correlated equilibrium payoffs in stochastic games. The correlation devices that we use are either autonomous (they base their choice of signal on previous signals, but not on previous states or actions) or stationary (their choice is independent of any data and is drawn according to the same probability distribution at every stage). We prove that any n-player ...

For a class of hhgher dimensional autonomous systems that have an invariant aane manifold, conditions are derived to preclude the existence of periodic solutions on the invariant manifold. It is established that these conditions are robust under certain type of local perturbations of the vector eld. As a consequence, each bounded semitrajectory on the invariant manifold is shown to converge to ...

Abstract Based on our studies done on two-dimensional autonomous systems, forced non-autonomous systems and time-delayed systems, we propose a unified methodology — that uses renormalization group theory — for finding out existence of periodic solutions in a plethora of nonlinear dynamical systems appearing across disciplines. The technique will be shown to have a non-trivial ability of classif...

In this paper we extend the well-known bifurcation theory for autonomous logistic equations to the non-autonomous equation ut −∆u = λu− b(t)u with b(t) ∈ [b0, B0], 0 < b0 < B0 < 2b0. In particular, we prove the existence of a unique uniformly bounded trajectory that bifurcates from zero as λ passes through the first eigenvalue of the Laplacian, which attracts all other trajectories. Although it...

Annals of Probability 31(2003), 2109-2135. Invariant man-ifolds provide the geometric structures for describing and understanding dynamics of nonlinear systems. The theory of invariant manifolds for both finite and infinite dimensional autonomous deterministic systems, and for stochastic ordinary differential equations is relatively mature. In this paper, we present a unified theory of invarian...