Transversality for time-periodic competitive-cooperative tridiagonal systems

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August 2015, 20(6): 1821-1830. doi: 10.3934/dcdsb.2015.20.1821

Transversality for time-periodic competitive-cooperative tridiagonal systems

Yi Wang ^1, and Dun Zhou ^1,

1.	Wu Wen-Tsun Key Laboratory, School of Mathematical Science, University of Science and Technology of China, Hefei, Anhui, 230026, China, China

Received July 2014 Revised January 2015 Published June 2015

Abstract
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Transversality of the stable and unstable manifolds of hyperbolic periodic solutions is proved for tridiagonal competitive-cooperative time-periodic systems. We further show that such systems admit the Morse-Smale property provided that all the fixed points (of the corresponding Poincaré map) are hyperbolic. The main tools used here are the integer-valued Lyapunov function, as well as the Floquet theory developed in [1] for general time-dependent tridiagonal linear systems.

Keywords: Morse-Smale property., Floquet theory, competitive-cooperative systems, Transversality, hyperbolicity.

Mathematics Subject Classification: Primary: 37C20, 34D30, 92D25; Secondary: 37C25, 37C6.

Citation: Yi Wang, Dun Zhou. Transversality for time-periodic competitive-cooperative tridiagonal systems. Discrete and Continuous Dynamical Systems - B, 2015, 20 (6) : 1821-1830. doi: 10.3934/dcdsb.2015.20.1821

References:

[1]	C. Fang, M. Gyllenberg and Y. Wang, Floquet bundles for tridiagonal competitive-cooperative systems and the dynamics of time-recurrent systems, SIAM J. Math. Anal., 45 (2013), 2477-2498. doi: 10.1137/120878021.
[2]	G. Fusco and W. Oliva, Transversality between invariant manifolds of periodic orbits for a class of monotone dynamical systems, J. Dynam. Differential Equations, 2 (1990), 1-17. doi: 10.1007/BF01047768.
[3]	G. Fusco and W. Oliva, Jacobi matrices and transversality, Proc. Roy. Soc. Edinburgh Sect. A, 109 (1988), 231-243. doi: 10.1017/S0308210500027748.
[4]	J. Hale and A. Somolinos, Competition for fluctuating nutrient, J. Math. Biol., 18 (1983), 255-280. doi: 10.1007/BF00276091.
[5]	M. Hirsch, Systems of differential equations that are competitive or cooperative. V. Convergence in 3-dimensional systems, J. Differential Equations, 80 (1989), 94-106. doi: 10.1016/0022-0396(89)90097-1.
[6]	J. Mallet-Paret and G. Sell, Systems of differential delay equations: Floquet multipliers and discrete lyapunov functions, J. Dynam. Differential Equations, 125 (1996), 385-440. doi: 10.1006/jdeq.1996.0036.
[7]	J. Mallet-Paret and H. Smith, The poincare-bendixson theorem for monotone cyclic feedback systems, J. Dynam. Differential Equations, 2 (1990), 367-421. doi: 10.1007/BF01054041.
[8]	P. Mottoni and A. Schiaffino, Competition systems with periodic coefficients: A geometric approach, J. Math. Biol., 11 (1981), 319-335. doi: 10.1007/BF00276900.
[9]	J. Selgrade, Isolated invariant sets for flows on vector bundles, Trans. Amer. Math. Soc., 203 (1975), 359-390. doi: 10.1090/S0002-9947-1975-0368080-X.
[10]	J. Smillie, Competitive and cooperative tridiagonal systems of differential equations, SIAM J. Math. Anal., 15 (1984), 530-534. doi: 10.1137/0515040.
[11]	H. Smith, Periodic tridiagonal competitive and cooperative systems of differential equations, SIAM J. Math. Anal., 22 (1991), 1102-1109. doi: 10.1137/0522071.
[12]	Y. Wang, Dynamics of nonautonomous tridiagonal competitive-cooperative systems of differential equations, Nonlinearity, 20 (2007), 831-843. doi: 10.1088/0951-7715/20/4/002.

show all references

References:

[1]	C. Fang, M. Gyllenberg and Y. Wang, Floquet bundles for tridiagonal competitive-cooperative systems and the dynamics of time-recurrent systems, SIAM J. Math. Anal., 45 (2013), 2477-2498. doi: 10.1137/120878021.
[2]	G. Fusco and W. Oliva, Transversality between invariant manifolds of periodic orbits for a class of monotone dynamical systems, J. Dynam. Differential Equations, 2 (1990), 1-17. doi: 10.1007/BF01047768.
[3]	G. Fusco and W. Oliva, Jacobi matrices and transversality, Proc. Roy. Soc. Edinburgh Sect. A, 109 (1988), 231-243. doi: 10.1017/S0308210500027748.
[4]	J. Hale and A. Somolinos, Competition for fluctuating nutrient, J. Math. Biol., 18 (1983), 255-280. doi: 10.1007/BF00276091.
[5]	M. Hirsch, Systems of differential equations that are competitive or cooperative. V. Convergence in 3-dimensional systems, J. Differential Equations, 80 (1989), 94-106. doi: 10.1016/0022-0396(89)90097-1.
[6]	J. Mallet-Paret and G. Sell, Systems of differential delay equations: Floquet multipliers and discrete lyapunov functions, J. Dynam. Differential Equations, 125 (1996), 385-440. doi: 10.1006/jdeq.1996.0036.
[7]	J. Mallet-Paret and H. Smith, The poincare-bendixson theorem for monotone cyclic feedback systems, J. Dynam. Differential Equations, 2 (1990), 367-421. doi: 10.1007/BF01054041.
[8]	P. Mottoni and A. Schiaffino, Competition systems with periodic coefficients: A geometric approach, J. Math. Biol., 11 (1981), 319-335. doi: 10.1007/BF00276900.
[9]	J. Selgrade, Isolated invariant sets for flows on vector bundles, Trans. Amer. Math. Soc., 203 (1975), 359-390. doi: 10.1090/S0002-9947-1975-0368080-X.
[10]	J. Smillie, Competitive and cooperative tridiagonal systems of differential equations, SIAM J. Math. Anal., 15 (1984), 530-534. doi: 10.1137/0515040.
[11]	H. Smith, Periodic tridiagonal competitive and cooperative systems of differential equations, SIAM J. Math. Anal., 22 (1991), 1102-1109. doi: 10.1137/0522071.
[12]	Y. Wang, Dynamics of nonautonomous tridiagonal competitive-cooperative systems of differential equations, Nonlinearity, 20 (2007), 831-843. doi: 10.1088/0951-7715/20/4/002.

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