Smith, H.L. (1995): "Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems", Mathematical Surveys and Monographs 41, American Mathematical Society, Providence, Rhode Island.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....with respect to x, then any fence is nonporous. 3 3 Moving Forward 3.1 Multi Dimensional Fences The theory of fences discussed above is given only for one dimensional statespaces, i.e. x 2 R in Equation (3) above. A similar theory exists for a class of ODEs in R n known as monotone systems [18]. We outline a more general approach, motivated by the well known concept of a Lyapunov function, below. The approach is distinct from Lyapunov functions in the usual sense, e.g. the function need not be positive definite. The method starts by identifying with the specification set S ae R n a ....

H.L. Smith. Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems. Mathematical Surveys and Monographs, Vol. 41. American Mathematical Society, Providence, RI, 1995.

....Later it was discovered that the same result holds in a much more general framework, namely that of strongly order preserving systems . This is a class of dynamical systems for which the comparison principle holds in a certain strong sense, whose concept was introduced in [13] 20] see also [32], which gives a comprehensive survey on early developments of this theory) Mierczy nski Pol acik [24] for the time continuous case) and Tak ac [33] for the time discrete case) considered strongly order preserving dynamical systems with a symmetry property associated with a compact connected ....

....Section 3 holds. The standard parabolic estimate shows that (82) 83) are fulfilled. Clearly (G1) G3) are also fulfilled, and condition (f 1) implies (G2 0 ) Further the strong maximum principle shows that equation (5. 1) forms a strongly order preserving dynamical system (see [13] 19] [32]) Hence, as we noted in Remark 3.1, every solution u of (5.2) satisfies condition (E 0 ) See also Lemma 5.2 below for a more direct verification of condition (E 0 ) Applying Corollary C 0 , we obtain the following : Theorem 5.1. Let Omega be bounded. Then any stable solution u of ....

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H. L. Smith, Monotone dynamical systems: an introduction to the theory of competitive and cooperative systems, Math. Surveys and Monographs, 41, Amer. Math. Soc., Providence, 1995.

....# while # # # will mean that #(#) # #(#) for all # # [ #, 0] The ordering assigned to C as above is called the standard ordering for C. Now, when we come to the delayed system (1.5) the negativity of diagonal entries and # ii 0 would make (1. 5) fail to be monotone in the sense of Smith [23] under the standard ordering in C. Motivated by Smith and Thieme [24] we equip C with another nonstandard ordering in C and find conditions under which the semiflows generated by the solutions of (1.5) are strongly order preserving in terms of this nonstandard ordering in C. Let D be an n n ....

<F3.762e+05> H. Smith,<F3.413e+05> Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems,<F3.796e+05> Math. Surveys Monographs 41, AMS, Providence, RI, 1995.

....principle. Remark 2.1. In the case where the mapping F is strongly order preserving, E ) and hence (E) are automatically fulfilled. Here a mapping F is called strongly orderpreserving if u OE v implies F ( u) OE F ( v) for any u, v that are sufficiently close to u, v, respectively ( 14] [26]) To derive (E ) note that the strongly order preserving property and (u) OE hu imply F (F k (u) OE F (ghu) ghu for sufficiently large k and any g 2 G sufficiently close to e. It follows that F k 1 (u) OE ghu for all large k, hence (u) ghu. Considering that (u) is compact and ....

....in Section 2 holds. The standard parabolic estimate shows that (82) 83) are fulfilled. Clearly (G1) G3) are also fulfilled, and condition (f 1) implies (G2 0 ) Further the strong maximum principle shows that equation (4. 1) forms a strongly order preserving dynamical system (see [9] 14] [26]) Hence, as we have noted in Remark 2.8, every solution u of (4.2) satisfies condition (E 0 ) Since G is compact, so is the group orbit Gu. Therefore the alternative (b) in the monotonicity theorem (Theorem 2.9) does not hold. Thus we obtain : Theorem 4.1. 19, Theorem 5.1] Let Omega be ....

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H. L. Smith, "Monotone dynamical systems: an introduction to the theory of competitive and cooperative systems", Math. Surveys and Monographs 41, Amer. Math. Soc., Providence, 1995.

....should be consulted for more details. For min max functions, there are periodic orbits of size n C [ n 2 ] in dimension n and this is conjectured to be best possible, 24] The monotonicity property is also known, at least in the context of flows, to have a constraining influence on dynamics, [25, 43]. The relationship between these various constraints on dynamics has yet to be properly explored. The homogeneity property suggests a modification of the conventional notion of fixed, or periodic, point. Definition 1.3. Suppose that F : R n R n satisfies property H. We say that x 2 R n ....

H. L. Smith. Monotone Dynamical Systems: an Introduction to the Theory of Competitive and Cooperative Systems, volume 41 of Mathematical Surveys and Monographs. AMS, 1995.

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Smith, H.L. (1995): "Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems", Mathematical Surveys and Monographs 41, American Mathematical Society, Providence, Rhode Island.

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H. L. Smith. `Monotone dynamical systems: An introduction to the theory of competitive and cooperative systems', AMS Mathematical Surveys and Monographs, 41, 31-53, (1995).

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Smith, H.L.: Monotone dynamical systems: An introduction to the theory of competitive and cooperative systems, Mathematical Surveys and Monographs, vol. 41, (AMS, Providence, RI, 1995).

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H.L. Smith, Monotone Dynamical Systems: an Introduction to the Theory of Competitive and Cooperative systems, Mathematical Surveys and Monographs, Vol. 41, American Mathematical Society, Ann Arbor, 1995. Fig. 5. Oscillations seen in simulations (vs = 0.5, delay of 100, initial conditions all at 0.2), using MATLAB's dde23 package

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H. L. Smith, "Monotone dynamical systems: an introduction to the theory of competitive and cooperative systems," in Mathematical Surveys and Monographs. Providence, RI: Amer. Math. Soc., 1995, vol. 41.

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Smith H (1995) Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative systems. Mathematical Surveys and Monographs, Vol. 41, American Mathematical Society Ann Arbor

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H. L. Smith. Monotone dynamical systems: an introduction to the theory of competitive and cooperative systems., volume 41 of Mathematical surveys. American Mathematical Society, 1995.

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H.L. Smith, Monotone Dynamical Systems: an Introduction to the Theory of Competitive and Cooperative Systems, Math. Surv. Monographs, Vol. 41, AMS, Providence, RI, 1995.

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H. Smith. Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems. AMS, Providence, 1995.

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H.L. Smith, Monotone dynamical systems: An introduction to the theory of competitive and cooperative systems, Mathematical Surveys and Monographs, vol. 41, AMS, Providence, RI, 1995

No context found.

H.L. Smith, "Monotone Dynamical Systems, An Introduction to the Theory of Competitive and Cooperative Systems", Mathematical Surveys and Monographs 41, Amer. Math. Soc., Providence, RI, 1995.

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