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

Transversality for time-periodic competitive-cooperative tridiagonal systems

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.
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. 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. doi: 10.1007/BF01047768.

[3]

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

[4]

J. Hale and A. Somolinos, Competition for fluctuating nutrient,, J. Math. Biol., 18 (1983), 255. 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. 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. 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. doi: 10.1007/BF01054041.

[8]

P. Mottoni and A. Schiaffino, Competition systems with periodic coefficients: A geometric approach,, J. Math. Biol., 11 (1981), 319. doi: 10.1007/BF00276900.

[9]

J. Selgrade, Isolated invariant sets for flows on vector bundles,, Trans. Amer. Math. Soc., 203 (1975), 359. 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. doi: 10.1137/0515040.

[11]

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

[12]

Y. Wang, Dynamics of nonautonomous tridiagonal competitive-cooperative systems of differential equations,, Nonlinearity, 20 (2007), 831. 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. 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. doi: 10.1007/BF01047768.

[3]

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

[4]

J. Hale and A. Somolinos, Competition for fluctuating nutrient,, J. Math. Biol., 18 (1983), 255. 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. 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. 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. doi: 10.1007/BF01054041.

[8]

P. Mottoni and A. Schiaffino, Competition systems with periodic coefficients: A geometric approach,, J. Math. Biol., 11 (1981), 319. doi: 10.1007/BF00276900.

[9]

J. Selgrade, Isolated invariant sets for flows on vector bundles,, Trans. Amer. Math. Soc., 203 (1975), 359. 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. doi: 10.1137/0515040.

[11]

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

[12]

Y. Wang, Dynamics of nonautonomous tridiagonal competitive-cooperative systems of differential equations,, Nonlinearity, 20 (2007), 831. doi: 10.1088/0951-7715/20/4/002.

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