Monotone Systems

Let 7r(Ar, t) denote the dynamical system generated by the autonomous system of differential equations 

Sufficient conditions for (C.l) to generate a dynamical system which is monotone (strongly monotone) are given in Appendix B. Corollary B.2, Theorem B.3, Corollary B.5, and Theorem B.6 imply the following result. 

Theorem C.l. If (C.l) is cooperative in D, then ir is a monotone dynamical system with respect to in D. If (C.l) is cooperative and irreducible in D, then tt is a strongly monotone system with respect to in D. If 

Hereafter, results will be stated only for the partial order k, it will be understood that they hold as well for the order . 

Theorem C.2. Let 7 (a ) be an orbit of the monotone dynamical system (C.l) which has compact closure in D. Then either of the following conditions is sufficient for u(x) to be a rest point  

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Such systems are said to be competitive (as noted in Chapter 1). When (6.1) represents a population growth equation, (6.2) indicates that an increase in the size of one component inhibits the growth of the others. Such a system is not necessarily order-preserving, so the theory of monotone systems does not apply. However, if solutions exist for all time and if one runs time backwards (more correctly, if one makes the change of variables / = —t and regards t as time ), then the corresponding dynamical system is monotone. More formally, the system... 

System (2.4) is the one that will receive most of the analysis. Several of the results in the appendices will be used the theory of monotone systems and the persistence results will be particularly useful. It is generally not possible to analyze a four-dimensional system such as (2.4) because the dynamics can be very complicated indeed, they can be chaotic. One must work very hard, using the theory developed, to show rigorously that the dynamics are, in fact, very simple. From the standpoint of dynamical systems, this is extraordinary luck from the standpoint of the biology, it is expected. What is new, biologically, is that coexistence is possible and the competition uncomplicated. 

A Matrices and Their Eigenvalues B Differential Inequalities C Monotone Systems D Persistence... 

Theorem C.5 places strong restrictions on how a limit set is imbedded in K". Note in particular that a periodic orbit may always be considered as a limit set of one of its points and hence Theorem C.5 applies to a periodic orbit. This should convince the reader that periodic orbits are ruled out for two-dimensional monotone systems. 

In order to illustrate the main ideas we use discrete events in order to analyze a multicomponent multistate monotone system of repairable components. The present paper extends (Huseby et al., 2008) and (Huseby et al., 2009) covering the binary case. 

Korezak, E., 2007. New formula for the failure/repair frequency of multi-state monotone systems and its applications. Control and Cybernetics, 36(1) 219-239. 

This paper has introduced the concept of persistence in systems subjected to IFC. The ICM has been proposed to incorporate the coverage of the non-persistent components whenever they become irrelevant, which opens up a cost-effective approach to improve the system reliability without increasing redundance. As for the future directions, we are going to extend the ICM to non-monotonic systems and multi-fault models. 

Snch a generalization is consistent with the Second Law of Thennodynamics, since the //theorem and the generalized definition of entropy together lead to the conchision that the entropy of an isolated non-eqnilibrium system increases monotonically, as it approaches equilibrium. 

Since Tis positive for systems in thennodynamic equilibrium,. S and hence log S should both be monotonically increasing fiinctions of E. This is the case as discussed above. 

From stochastic molecnlar dynamics calcnlations on the same system, in the viscosity regime covered by the experiment, it appears that intra- and intennolecnlar energy flow occur on comparable time scales, which leads to the conclnsion that cyclohexane isomerization in liquid CS2 is an activated process [99]. Classical molecnlar dynamics calcnlations [104] also reprodnce the observed non-monotonic viscosity dependence of ic. Furthennore, they also yield a solvent contribntion to the free energy of activation for tlie isomerization reaction which in liquid CS, increases by abont 0.4 kJ moC when the solvent density is increased from 1.3 to 1.5 g cm T Tims the molecnlar dynamics calcnlations support the conclnsion that the high-pressure limit of this unimolecular reaction is not attained in liquid solntion at ambient pressure. It has to be remembered, though, that the analysis of the measnred isomerization rates depends critically on the estimated valne of... 

Figure A3.14.3. Example bifurcation diagrams, showing dependence of steady-state concentration in an open system on some experimental parameter such as residence time (inverse flow rate) (a) monotonic dependence (b) bistability (c) tristability (d) isola and (e) musliroom.

Knowledge-based systems typically use quaHtative methods rather than quantitative ones. For example, consider a simple tank system. The equation describing the flow rate of Hquid out of the tank is given below, where C is the orifice coefficient, d is the diameter of the orifice, and h is the height of Hquid in the tank. Based solely on the form of the equation, a human reasoner can infer that the flow rate F increases monotonically with the height b of Hquid in the tank. 

Order 0 minimization methods do not take the slope or the curvature properties of the energy surface into account. As a result, such methods are crude and can be used only with very simple energy surfaces, i.e., surfaces with a small number of local minima and monotonic behavior away from the minima. These methods are rarely used for macro-molecular systems. 

Addition of a plasticizer decreases the Tg of the polymer and, in partially crystalline polymers, also influences both crystallization and melting. The amount of plasticizer affects its effectiveness. Thus, while the Tg of the polymer is strongly depressed by small plasticizer additions, the increase in the plasticizer content leads to lower decrease in To and in several systems two T values can be found [36J. Therefore, the increase in the plasticizer content in polymers does not show a monotonic decrease in Tg. 

The function /[0(r)] has three minima by construction and guarantees three-phase coexistence of the oil-rich phase, water-rich phase, and microemulsion. The minima for oil-rich and water-rich phases are of equal depth, which makes the system symmetric, therefore fi is zero. Varying the parameter /o makes the microemulsion more or less stable with respect to the other two bulk uniform phases. Thus /o is related to the chemical potential of the surfactant. The constant g2 depends on go /o and is chosen in such a way that the correlation function G r) = (0(r)0(O)) decays monotonically in the oil-rich and water-rich phases [12,13]. This is the case when gi > 4y/l +/o - go- Here we take, arbitrarily, gj = 4y l +/o - go + 0.01. 

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