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# An Introduction to Monotone Dynamical Systems The Time Discrete Case Sina Straub Prof

Dr Rupert Lasser Center for Mathematical Sciences Technische Universit57512at M unchen sina straubwebde Chair of Biomathematics Abstract Monotone dynamical systems which are dynamical systems on an or dered metric space having the property that orde

## An Introduction to Monotone Dynamical Systems The Time Discrete Case Sina Straub Prof

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An Introduction to Monotone Dynamical Systems: The Time Discrete Case Sina Straub, Prof. Dr. Rupert Lasser Center for Mathematical Sciences, Technische Universitat M unchen, sina straub@web.de Chair of Biomathematics Abstract Monotone dynamical systems, which are dynamical systems on an or dered metric space having the property that ordered initial s tates lead to ordered subsequent states, occur in many biological, chemical, physical and economic models. Co nsidering , a limit set trichotomy for monotone nonlinear dynamical syst ems can be shown by using some sort of

weak concavity condition, working piecewise on the so called parts of a convex cone and employing the part metric, i.e. there is one center of attraction for all orbits which is inﬁnity, or some positive position in between. Under more general cond itions, every attractor in a strongly ordered metric space co ntains a cycle which attracts a nonempty open set. Provided the dynamical system is strongly monotone, every a ttractor contains a stable periodic orbit. Thus the dynamics of such systems can�t be arbitrarily chaotic. Deﬁnitions Cones and ordering Let be a Banach space. is

an order cone if it is a nonempty closed subset of with: ) = Then if and only if x < y if and only if and if and only if IntY Ordered spaces An ordered metric space is a metric space with an order cone. Strongly ordered spaces A metric space is strongly ordered if for every open subset in the following holds: (SO1) There exist a, b with and (SO2) There exists with a, b with Monotonicity and strong monotonicity A continuous map Φ : is (strongly) monotone on an ordered metric space if whenever x < y and Limit Set Trichotomy Theorem 1 Let Φ : be a monotone and continuous (for the

Euclidean topology) map satisfying some sort of concavity condition, called (k,P)-prop erty. Then exactly one of the following holds if contains an orbit at all: 1. each nonzero orbit in is unbounded 2. each orbit in is bounded with at least one limit point that is not contained in 3. each nonzero orbit in converges to the unique ﬁxed point of in Example Theorem 1 is applicable to Φ ( ) = for = ( , x , and it can be seen in ﬁgure 1 that the third case holds. Figure 1: Iteration plot (2., p. 391) Stable Cycles for Strongly Monotone Maps Theorem 2 Let be a strongly ordered

metric space, Φ : continuous and monotone map, an attractor and the nonwandering set. If is a maximal or a minimal nonwandering point in , then is a trap Theorem 3 Let be a strongly ordered metric space, an attractor for the continuous and strongly monotone map Φ : and such that contains a T-cycle. Then the following holds: 1. contains a T-cycle which is order-stable 2. contains a T-cycle which is asymptotically order-stable if the number of T- cycles in is ﬁnite . Corollary 1 Let be a strongly ordered metric space, Φ : a continuous and strongly monotone map. If is an

attractor which has no isolated points, then no ﬁnite collection of orbits is dense in Corollary 2 Let be a strongly ordered metric space, Φ : a continuous and strongly monotone map. If is an attractor having no isolated points and for some the union of the T-cycles in is ﬁnite and nonempty, then there exists a wandering point in Applications Figure 2: Signaling circuitry of the mammalian cell (3., p.10) Figure 2 shows signaling pathways for growth, diﬀerentiation, and ap optosis commands, which instruct the cell to die. Large classes of such signaling systems may be

proﬁtably stud ied by ﬁrst decomposing them into several subsystems, each of which is endowed with cer tain qualitative mathematical properties which are input/output (I/O) monotonicity and existence of I/O characteristics meaning that there is a monostable stea dy-state response to any constant external input. For example, mitogen-activated protein kinase (MAPK) cascade s satisfy these condi- tions. References 1. Hirsch, M. W., Attractors for discrete-time monotone dynami cal systems in strongly ordered spaces; Geometry and Topology, Lecture Notes in Mathematics vol. 1167, J.

Alexander and J. Harder eds., 14 1 - 153. New York: Springer Verlag, 1985. 2. Krause, U. and Ranft, P., A limit set trichotomy for monotone n onlinear dynamical systems; Nonlinear Analysis, Theory, Metho ds and Application, Vol.19, No. 4, pp. 375 - 392, 1992. 3. Sontag, E. D., Some new directions in control theory inspir ed by systems in Biology; The IEE Systems Biology, Vol. 1, No. 1, June 2004, ISSN 1741-2471.