On the dynamics of strongly tridiagonal competitive-cooperative system

Transcription

1 On the dynamics of strongly tridiagonal competitive-cooperative system Chun Fang University of Helsinki International Conference on Advances on Fractals and Related Topics, Hong Kong December 14, 2012 Chun Fang ( University of Helsinki International Conference Competitive-Cooperative on Advances on Fractals System and Related Topics, December Hong Kong 14, 2012 ) 1 / 15

2 Setting Consider the nonautonomous tridiagonal system ẋ 1 = f 1 (t, x 1, x 2 ), ẋ i = f i (t, x i 1, x i, x i+1 ), 2 i n 1 (1) ẋ n = f n (t, x n 1, x n ) where f = (f 1, f 2,, f n ) : R R n R n satisfies following conditions. hun Fang ( University of Helsinki International Conference Competitive-Cooperative on Advances on Fractals System and Related Topics, December Hong Kong 14, 2012 ) 2 / 15

3 Setting Consider the nonautonomous tridiagonal system ẋ 1 = f 1 (t, x 1, x 2 ), ẋ i = f i (t, x i 1, x i, x i+1 ), 2 i n 1 (1) ẋ n = f n (t, x n 1, x n ) where f = (f 1, f 2,, f n ) : R R n R n satisfies following conditions. (A1) f is C 1 -admissible, i.e. f together with f x are bounded and uniformly continuous on R K for any compact set K R n. hun Fang ( University of Helsinki International Conference Competitive-Cooperative on Advances on Fractals System and Related Topics, December Hong Kong 14, 2012 ) 2 / 15

4 Setting Consider the nonautonomous tridiagonal system ẋ 1 = f 1 (t, x 1, x 2 ), ẋ i = f i (t, x i 1, x i, x i+1 ), 2 i n 1 (1) ẋ n = f n (t, x n 1, x n ) where f = (f 1, f 2,, f n ) : R R n R n satisfies following conditions. (A1) f is C 1 -admissible, i.e. f together with f x are bounded and uniformly continuous on R K for any compact set K R n. (A2) There are ε 0 > 0 such that for all (t, x) R R n, f i x i+1 (t, x) ε 0, f i+1 x i (t, x) ε 0 1 i n 1. for all (t, x) R R n. hun Fang ( University of Helsinki International Conference Competitive-Cooperative on Advances on Fractals System and Related Topics, December Hong Kong 14, 2012 ) 2 / 15

5 Question and Idea Question Let x(t, x 0, f ) be a bounded solution of (1). What does its asymptotic behavior looks like? Can we characterize the structure of its ω-limit set ω(x 0, f )? Chun Fang ( University of Helsinki International Conference Competitive-Cooperative on Advances on Fractals System and Related Topics, December Hong Kong 14, 2012 ) 3 / 15

6 Question and Idea Question Let x(t, x 0, f ) be a bounded solution of (1). What does its asymptotic behavior looks like? Can we characterize the structure of its ω-limit set ω(x 0, f )? ẋ = f (t, x) Chun Fang ( University of Helsinki International Conference Competitive-Cooperative on Advances on Fractals System and Related Topics, December Hong Kong 14, 2012 ) 3 / 15

7 Question and Idea Question Let x(t, x 0, f ) be a bounded solution of (1). What does its asymptotic behavior looks like? Can we characterize the structure of its ω-limit set ω(x 0, f )? ẋ = f (t, x) H(f ) = {g : R R n R n τ R, s.t. g(t, x) = f (τ + t, x)} Chun Fang ( University of Helsinki International Conference Competitive-Cooperative on Advances on Fractals System and Related Topics, December Hong Kong 14, 2012 ) 3 / 15

8 Question and Idea Question Let x(t, x 0, f ) be a bounded solution of (1). What does its asymptotic behavior looks like? Can we characterize the structure of its ω-limit set ω(x 0, f )? ẋ = f (t, x) H(f ) = {g : R R n R n τ R, s.t. g(t, x) = f (τ + t, x)} For convenience, we assume the hull H(f ) is minimal under the shift action defined by f τ f (τ +, ). It is reasonable to characterize the structure of ω(x 0, f ) in terms of H(f ). Chun Fang ( University of Helsinki International Conference Competitive-Cooperative on Advances on Fractals System and Related Topics, December Hong Kong 14, 2012 ) 3 / 15

9 Question and Idea Question Let x(t, x 0, f ) be a bounded solution of (1). What does its asymptotic behavior looks like? Can we characterize the structure of its ω-limit set ω(x 0, f )? ẋ = f (t, x) H(f ) = {g : R R n R n τ R, s.t. g(t, x) = f (τ + t, x)} For convenience, we assume the hull H(f ) is minimal under the shift action defined by f τ f (τ +, ). It is reasonable to characterize the structure of ω(x 0, f ) in terms of H(f ). 1. Biologically, it describes the relationship between the environment and the variety of number of species; 2. Mathematically, it covert the complexity of ω(x 0, f ) to the one of H(f ). Chun Fang ( University of Helsinki International Conference Competitive-Cooperative on Advances on Fractals System and Related Topics, December Hong Kong 14, 2012 ) 3 / 15

10 (1) can generates a skew-product semiflow π on R n H(f ), by π(t, x 0, g) (x 0, g) t (x(t, x 0, g), g t), (2) where x(t, x 0, g) is the solution of initial value problem ẋ 1 = g(t, x), x(0) = x 0 Chun Fang ( University of Helsinki International Conference Competitive-Cooperative on Advances on Fractals System and Related Topics, December Hong Kong 14, 2012 ) 4 / 15

11 (1) can generates a skew-product semiflow π on R n H(f ), by π(t, x 0, g) (x 0, g) t (x(t, x 0, g), g t), (2) where x(t, x 0, g) is the solution of initial value problem ẋ 1 = g(t, x), x(0) = x 0 An omega limit set of system (1) corresponds to an omega limit set of system (2) (see George R. Sell [1]). Chun Fang ( University of Helsinki International Conference Competitive-Cooperative on Advances on Fractals System and Related Topics, December Hong Kong 14, 2012 ) 4 / 15

12 Hyperbolic Case Theorem (F.-Gyllenberg-Wang) Let π(t, x 0, g 0 ) be a positively bounded motion of system (2). If ω(x 0, g 0 ) is hyperbolic, then ω(x 0, g 0 ) is 1-cover of H(f ). Chun Fang ( University of Helsinki International Conference Competitive-Cooperative on Advances on Fractals System and Related Topics, December Hong Kong 14, 2012 ) 5 / 15

13 Hyperbolic Case Theorem (F.-Gyllenberg-Wang) Let π(t, x 0, g 0 ) be a positively bounded motion of system (2). If ω(x 0, g 0 ) is hyperbolic, then ω(x 0, g 0 ) is 1-cover of H(f ). Definition: A set Y R n H(f ) is said to be an 1-cover of H(f ) if #(Y P 1 (g)) = 1 for all g H(f ), where P : R n H(f ) H(f ), P(x, g) = g is the natural projection. Chun Fang ( University of Helsinki International Conference Competitive-Cooperative on Advances on Fractals System and Related Topics, December Hong Kong 14, 2012 ) 5 / 15

14 Hyperbolic Case Definition: Let Y R n H(f ) be a compact invariant set of (2). For each y = (x, g) Y, the linearized equation of (2) along y t = (x, g) t reads: ẋ = A(y t)x. (3) Y is hyperbolic if the system (3) admits an exponential dichotomy over Y, i.e. there is a projector Q : R n H(f ) R n H(f ) and positive constants K and α such that (i) Φ(t, y)q(y) = Q(y t)φ(t, y), t R, (ii) Φ(t, y)(1 Q(y)) Ke αt, t R +, Φ(t, y)q(y) Ke αt, t R, for all y H(f ). Chun Fang ( University of Helsinki International Conference Competitive-Cooperative on Advances on Fractals System and Related Topics, December Hong Kong 14, 2012 ) 6 / 15

15 Hyperbolic Case - Perturbation Theory Consider a perturbed system of (1) ẋ = f (t, x) + h(t, x). Assume the perturbation item h and its Jacobi matrix with respect to x are uniformly continuous and there exist 0 < δ < 1 such that for all t R and x R n. h(t, x) < δ, h(t, x)/ x < δ (4) hun Fang ( University of Helsinki International Conference Competitive-Cooperative on Advances on Fractals System and Related Topics, December Hong Kong 14, 2012 ) 7 / 15

16 Hyperbolic Case - Perturbation Theory Consider a perturbed system of (1) ẋ = f (t, x) + h(t, x). Assume the perturbation item h and its Jacobi matrix with respect to x are uniformly continuous and there exist 0 < δ < 1 such that for all t R and x R n. Theorem (F.-Gyllenberg-Wang) h(t, x) < δ, h(t, x)/ x < δ (4) Suppose the skew-product flow (2) generated by ẋ = f (t, x) admits a hyperbolic ω-limit set ω(x 0, f ). Then there exist a C 1 neighborhood F of f in the sense of (4) and a neighborhood U of ω(x 0, f ) such that for any g F, there exist an ω-limit set ω(x 0, g) Rn H(g) U, moreover, it is 1-cover of H(g). hun Fang ( University of Helsinki International Conference Competitive-Cooperative on Advances on Fractals System and Related Topics, December Hong Kong 14, 2012 ) 7 / 15

17 General Case Question How about the structural of a general ω-limit set of (1) or (2)? Chun Fang ( University of Helsinki International Conference Competitive-Cooperative on Advances on Fractals System and Related Topics, December Hong Kong 14, 2012 ) 8 / 15

18 General Case Question How about the structural of a general ω-limit set of (1) or (2)? Lemma (F.-Gyllenberg-Wang) For any π-invariant set Y R n H(f ) of (2), the linearized system (3): ẋ = A(y t)x admits an (1,, 1)-dominated splitting. Chun Fang ( University of Helsinki International Conference Competitive-Cooperative on Advances on Fractals System and Related Topics, December Hong Kong 14, 2012 ) 8 / 15

19 General Case Question How about the structural of a general ω-limit set of (1) or (2)? Lemma (F.-Gyllenberg-Wang) For any π-invariant set Y R n H(f ) of (2), the linearized system (3): ẋ = A(y t)x admits an (1,, 1)-dominated splitting. Definition: System (3) is said to admit a (n 1, n 2 )-dominated splitting if there exists π-invariant splitting X 1 (Y ) X 2 (Y ) of R n Y such that there exist positive numbers K and ν satisfies Φ(t, y)x 2 Φ(t, y)x 1 Ke νt, t 0, for all y Y and x 1 X 1 (y), x 2 X 2 (y) with x 1 = x 2 = 1. Chun Fang ( University of Helsinki International Conference Competitive-Cooperative on Advances on Fractals System and Related Topics, December Hong Kong 14, 2012 ) 8 / 15

20 General Case Next theorem shows that after a suitable functional distortion, dominated splitting becomes to hyperbolicity. Theorem (F.-Gyllenberg-Liu) Let f : M M be a diffeomorphism on a closed manifold M and Λ M be any compact f -invariant set. A splitting T Λ M = E 1 E k of tangent bundle over Λ is (n 1,, n k )-dominated if and only if there exist continuous real functions p i : Λ R +, i = 1,, k, with log p 1,, log p k are summably separated with respect to f, such that for i = 1,, k the linear cocycle (f, p i Df ) admits a hyperbolicity over Λ with stable subspace of dimension n n i. Chun Fang ( University of Helsinki International Conference Competitive-Cooperative on Advances on Fractals System and Related Topics, December Hong Kong 14, 2012 ) 9 / 15

21 General Case - Central Dimension One For any given λ R, consider ẋ = (A(y t) λid)x, y Y, (5) Definition: (Y ) = {λ R 1 (5) has no exponential dichotomy on Y } is called the Sacker-Sell spectrum of (3). Chun Fang ( University of Helsinki International Conference Competitive-Cooperative on Advances on Fractals System and Related Topics, December Hong Kong 14, 2012 ) 10 / 15

22 General Case - Central Dimension One For any given λ R, consider ẋ = (A(y t) λid)x, y Y, (5) Definition: (Y ) = {λ R 1 (5) has no exponential dichotomy on Y } is called the Sacker-Sell spectrum of (3). Remark 1. (Y ) = k i=1 I i, where I i = [a i, b i ] and {I i } is ordered from right to left. Denote the invariant subbundle associated with I i is X i (Y ), then X 1 (Y ) X k (Y ) = R n Y. 2. If Y is hyperbolic, 0 / (Y ). Chun Fang ( University of Helsinki International Conference Competitive-Cooperative on Advances on Fractals System and Related Topics, December Hong Kong 14, 2012 ) 10 / 15

23 General Case - Central Dimension One Definition: We say Y is of central dimension one, if 0 I i for some i and dim X i (Y ) = 1. Chun Fang ( University of Helsinki International Conference Competitive-Cooperative on Advances on Fractals System and Related Topics, December Hong Kong 14, 2012 ) 11 / 15

24 General Case - Central Dimension One Definition: We say Y is of central dimension one, if 0 I i for some i and dim X i (Y ) = 1. Theorem (F.-Wang) Suppose Y R n H(f ) is a minimal set and is of central dimension one. Then the flow (Y, ) is topologically conjugated to a scalar skew-product subflow of (R 1 H(f ), ). Chun Fang ( University of Helsinki International Conference Competitive-Cooperative on Advances on Fractals System and Related Topics, December Hong Kong 14, 2012 ) 11 / 15

25 General Case - Central Dimension One Definition: We say Y is of central dimension one, if 0 I i for some i and dim X i (Y ) = 1. Theorem (F.-Wang) Suppose Y R n H(f ) is a minimal set and is of central dimension one. Then the flow (Y, ) is topologically conjugated to a scalar skew-product subflow of (R 1 H(f ), ). Corollary Suppose f is almost periodic in t and let Y R n H(f ) be a minimal set and unique ergodic. Then (Y, ) is topologically conjugated to a scalar skew-product subflow of (R 1 H(f ), ). Chun Fang ( University of Helsinki International Conference Competitive-Cooperative on Advances on Fractals System and Related Topics, December Hong Kong 14, 2012 ) 11 / 15

26 Future Research Question How about the case that the central dimension is bigger than one, for example two? Chun Fang ( University of Helsinki International Conference Competitive-Cooperative on Advances on Fractals System and Related Topics, December Hong Kong 14, 2012 ) 12 / 15

27 Thank you for your attention! Chun Fang ( University of Helsinki International Conference Competitive-Cooperative on Advances on Fractals System and Related Topics, December Hong Kong 14, 2012 ) 13 / 15

28 References G. R. Sell (1971) Topological dynamics and ordinary differential equations. Van Nostrand-Reinhold. H. L. Smith (1991) Periodic tridiagonal competitive and cooperative systems of differential equations. SIAM J. MATH. ANAL. 22(4), V. M. Millionshchikov (1968) A criterion for a small change in the direction of the solutions of a linear system of differential equations as the result of small disturbances in the coefficients of the system. Mat. Zametki 4(2), I. U. Bronshtein and V. F. Chernii (1978) Linear extensions satisfying Perron s condition. I. Differential Equations 14, J. Smillie (1984) Competitive and cooperative tridiagonal sytems of differential equations. SIAM J. MATH. ANAL. 15, Chun Fang ( University of Helsinki International Conference Competitive-Cooperative on Advances on Fractals System and Related Topics, December Hong Kong 14, 2012 ) 14 / 15

29 References Y. Wang (2007) Dynamics of nonautonomous tridiagonal competitive-cooperative systems of differential equations. Nonlinearity 20, S.-N Chow, X.-B. Lin and K. Lu (1991) Smooth invariant foliations in infinite dimensional spaces. J. Diff. Eq. 94, D. Henry (1981) Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Mathematics Vol.840. Y. Yi (1993) Stability of integral manifold and orbital attraction of quasi-periodic motions. J. Diff. Eq. 102(2), W. Shen and Y. Yi (1995) Dynamics of almost periodic scalar parabolic equations. J. Diff. Eq. 122, Chun Fang ( University of Helsinki International Conference Competitive-Cooperative on Advances on Fractals System and Related Topics, December Hong Kong 14, 2012 ) 15 / 15