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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

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 & 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.  Google Scholar

[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.  Google Scholar

[3]

G. Fusco and W. Oliva, Jacobi matrices and transversality, Proc. Roy. Soc. Edinburgh Sect. A, 109 (1988), 231-243. doi: 10.1017/S0308210500027748.  Google Scholar

[4]

J. Hale and A. Somolinos, Competition for fluctuating nutrient, J. Math. Biol., 18 (1983), 255-280. doi: 10.1007/BF00276091.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

J. Smillie, Competitive and cooperative tridiagonal systems of differential equations, SIAM J. Math. Anal., 15 (1984), 530-534. doi: 10.1137/0515040.  Google Scholar

[11]

H. Smith, Periodic tridiagonal competitive and cooperative systems of differential equations, SIAM J. Math. Anal., 22 (1991), 1102-1109. doi: 10.1137/0522071.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

G. Fusco and W. Oliva, Jacobi matrices and transversality, Proc. Roy. Soc. Edinburgh Sect. A, 109 (1988), 231-243. doi: 10.1017/S0308210500027748.  Google Scholar

[4]

J. Hale and A. Somolinos, Competition for fluctuating nutrient, J. Math. Biol., 18 (1983), 255-280. doi: 10.1007/BF00276091.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

J. Smillie, Competitive and cooperative tridiagonal systems of differential equations, SIAM J. Math. Anal., 15 (1984), 530-534. doi: 10.1137/0515040.  Google Scholar

[11]

H. Smith, Periodic tridiagonal competitive and cooperative systems of differential equations, SIAM J. Math. Anal., 22 (1991), 1102-1109. doi: 10.1137/0522071.  Google Scholar

[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.  Google Scholar

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