Convergence in strongly monotone systems with an increasing first integral

arXiv:0906.0272v1 [math.DS] 1 Jun 2009 CONVERGENCE IN STRONGLY MONOTONE SYSTEMS WITH AN INCREASING FIRST INTEGRAL MURAD BANAJI∗§ AND DAVID ANGELI† Abstract. In this paper we generalise a useful result due to J. Mierczy´nski which states that for a strictly cooperative system on the positive orthant, with increasing first integral, all bounded orbits are convergent. Moreover any equilibrium attracts its entire level set, and there can be no more than one equilibrium on any level set. Here, more general state spaces and more general orderings are considered. Let Y ⊂K ⊂Rn be any two proper cones. Given a local semiflow φ on Y which is strongly monotone with respect to K, and which preserves a K-increasing first integral, we show that every bounded orbit converges. Again, each equilibrium attracts its entire level set, and there can be no more than one equilibrium on any level set. An application from chemical dynamics is provided. Key words. First integral; strongly monotone system; global convergence AMS subject classifications. 34A26; 34C12; 34D23; 06A06 1. Introduction. The study of the qualitative behaviour of dynamical systems is a vast subject with applications in many fields. In particular monotone systems, i.e. systems which preserve some partial order on the state space, have been intensively studied, with a range of qualitative results on asymptotic behaviour in these systems. See [8] for a recent survey or [12] for an earlier monograph on the subject. When the state space is some subset of Euclidean space, and the preserved partial order is the “natural” order generated by the positive orthant, we get so-called cooperative sys- tems. The fundamental notions connected with cooperativity extend to more general orderings ([14] for example). Monotonicity constrains the behaviour of dynamical systems, for example ruling out attracting nontrivial periodic orbits, provided at least one point of any periodic orbit it accessible from above or below [8]. When a dynamical system is strongly monotone (to be defined below) behaviour is constrained further: for almost all initial conditions bounded solutions converge to the set of equilibria, a result initially proved for strongly cooperative systems by M. Hirsch in [7]. Sometimes generic convergence claims can be strengthened, provided additional structure is available. For instance, global convergence (i.e. convergence of every bounded orbit) can be obtained in a variety of special cases: for tridiagonal strongly cooperative systems [11]; when a system enjoys so-called “positive translation invariance” [2]; and when a strongly cooperative system is endowed with a strictly increasing first integral (the result of Mierczy´nski [9] to be generalised here). In this latter case, the conclusions are stronger still: there can be no more than one equilibrium on each level set of the first integral, and when it exists, such an equilibrium attracts the whole level set. In the same spirit is Theorem 5 of [6], which shows how for lattice state spaces, and provided a unique equilibrium exists, all bounded solutions converge to this equilibrium. The importance of Mierczy´nski’s result stems from the fact that in a variety of applications natural constraints lead to order preservation, while at the same time ∗Dept. of Mathematics, University College London, UK, and Dept. of Biological Sciences, University of Essex, Colchester, UK. †Dept. of Electrical and Electronic Engineering, Imperial College London, UK, and Dip. Sistemi e Informatica, University of Florence, Italy. §m.banaji@ucl.ac.uk. Research funded by EPSRC grant EP/D060982/1. 1 conservation laws define preserved functions. However, as shown for chemical reac- tion networks in [3, 1], the preserved partial orders may not be induced by orthants, and indeed, may not be induced by simplicial cones. Thus appropriate generalisations of Mierczy´nski’s result potentially have useful application in these areas. A small ex- ample of such an application will be presented later. 2. The main result. We state the main result and outline the proof. Definition 2.1. A proper cone in Rn will be defined as a closed, convex, pointed cone with nonempty interior [5]. Let Y, K be proper cones in Rn with K ⊃Y . From now on, all inequalities are with respect to the ordering defined by K, i.e. x ≤y will mean y −x ∈K, x < y will mean x ≤y and x ̸= y, x ≪y will mean y −x ∈int(K), etc. Define K∗to be dual cone to K, i.e. K∗= {y ∈Rn | ⟨y, k⟩≥0 for all k ∈K}. Consider a system ˙x = F(x) (2.1) on Y , where F(x) is locally Lipschitz and so defines a local semiflow φ on Y . Assume that: 1. φ is strongly monotone with respect to K, i.e. x > y ⇒φt(x) ≫φt(y) for all t > 0 such that φt(x) and φt(y) are defined). 2. The system has a C1 first integral H : Y →R, such that for each y ∈Y , i) ⟨∇H(y), F(y)⟩= 0 and ii) ∇H(y) ∈int(K∗). Remarks. Since K is proper, K∗is automatically a proper cone [5], and hence has nonempty interior. From here on if we refer to φt(y), the assum

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