Monotone Dynamical Systems: A Quick Tour Hal Smith A R I Z O N A S - PowerPoint PPT Presentation

Monotone Dynamical Systems: A Quick Tour Hal Smith A R I Z O N A S T A T E U N I V E R S I T Y H.L. Smith (ASU) Monotone Dynamical Systems Sontagfest, May 23, 2011 1 / 16

Monotone Dynamical System State space: metric space ( X , d ) with a closed* partial order 1 relation ≤ . *( x n ≤ y n ∧ x n → x ∧ y n → y ⇒ x ≤ y ) Dynamics: discrete-time ( T = Z + ) or continuous-time 2 ( T = R + ) semiflow Φ : T × X → X . Notation Φ t ( x ) = Φ( t , x ) : Φ continuous. Φ 0 = id X t , s ∈ T Φ t ◦ Φ s = Φ t + s , Order-Preserving: x ≤ y ⇒ Φ t ( x ) ≤ Φ t ( y ) , t ∈ T , x , y ∈ X . 3 Trivial Examples: X = R , usual order ≤ , x ′ = f ( x ) , Φ t ( x 0 ) = x ( t , x 0 ) . X = BC ( R , R ) , usual order ≤ , u t = u xx + f ( x , u ) , Φ t ( u 0 ) = u ( t , · ) . X = R , f ր , x ( n + 1 ) = f ( x ( n )) , n ≥ 0 , Φ n ( x ( 0 )) = f ( n ) ( x ( 0 )) . standing assumptions: T = R + . ∀ x ∈ X , { Φ t ( x ) : t ≥ 0 } has compact closure in X . H.L. Smith (ASU) Monotone Dynamical Systems Sontagfest, May 23, 2011 2 / 16

Monotone Dynamical System State space: metric space ( X , d ) with a closed* partial order 1 relation ≤ . *( x n ≤ y n ∧ x n → x ∧ y n → y ⇒ x ≤ y ) Dynamics: discrete-time ( T = Z + ) or continuous-time 2 ( T = R + ) semiflow Φ : T × X → X . Notation Φ t ( x ) = Φ( t , x ) : Φ continuous. Φ 0 = id X t , s ∈ T Φ t ◦ Φ s = Φ t + s , Order-Preserving: x ≤ y ⇒ Φ t ( x ) ≤ Φ t ( y ) , t ∈ T , x , y ∈ X . 3 Trivial Examples: X = R , usual order ≤ , x ′ = f ( x ) , Φ t ( x 0 ) = x ( t , x 0 ) . X = BC ( R , R ) , usual order ≤ , u t = u xx + f ( x , u ) , Φ t ( u 0 ) = u ( t , · ) . X = R , f ր , x ( n + 1 ) = f ( x ( n )) , n ≥ 0 , Φ n ( x ( 0 )) = f ( n ) ( x ( 0 )) . standing assumptions: T = R + . ∀ x ∈ X , { Φ t ( x ) : t ≥ 0 } has compact closure in X . H.L. Smith (ASU) Monotone Dynamical Systems Sontagfest, May 23, 2011 2 / 16

Monotone Dynamical System State space: metric space ( X , d ) with a closed* partial order 1 relation ≤ . *( x n ≤ y n ∧ x n → x ∧ y n → y ⇒ x ≤ y ) Dynamics: discrete-time ( T = Z + ) or continuous-time 2 ( T = R + ) semiflow Φ : T × X → X . Notation Φ t ( x ) = Φ( t , x ) : Φ continuous. Φ 0 = id X t , s ∈ T Φ t ◦ Φ s = Φ t + s , Order-Preserving: x ≤ y ⇒ Φ t ( x ) ≤ Φ t ( y ) , t ∈ T , x , y ∈ X . 3 Trivial Examples: X = R , usual order ≤ , x ′ = f ( x ) , Φ t ( x 0 ) = x ( t , x 0 ) . X = BC ( R , R ) , usual order ≤ , u t = u xx + f ( x , u ) , Φ t ( u 0 ) = u ( t , · ) . X = R , f ր , x ( n + 1 ) = f ( x ( n )) , n ≥ 0 , Φ n ( x ( 0 )) = f ( n ) ( x ( 0 )) . standing assumptions: T = R + . ∀ x ∈ X , { Φ t ( x ) : t ≥ 0 } has compact closure in X . H.L. Smith (ASU) Monotone Dynamical Systems Sontagfest, May 23, 2011 2 / 16

Ordered Banach Space Induces ≤ X ⊂ Y , Y an ordered Banach space with closed positive cone Y + : ( R + ) Y + ⊂ Y + , Y + + Y + ⊂ Y + , ( Y + ) ∩ ( − Y + ) = { 0 } Partial order: y ≤ x ⇔ x − y ∈ Y + Y is strongly ordered if Int Y + � = ∅ . Then y ≪ x ⇔ x − y ∈ Int Y + . Examples: + × ( − R n − k Y = R n , Y + = R k ) , 0 ≤ k ≤ n : + x ≤ y ⇔ ( x i ≤ y i , i ≤ k ) ∧ ( x j ≥ y j , j > k ) Y = L p (Ω , R n ) , C r (Ω , R n ) , f ≤ g ⇔ f ( s ) ≤ g ( s ) , s ∈ Ω H.L. Smith (ASU) Monotone Dynamical Systems Sontagfest, May 23, 2011 3 / 16

Ordered Banach Space Induces ≤ X ⊂ Y , Y an ordered Banach space with closed positive cone Y + : ( R + ) Y + ⊂ Y + , Y + + Y + ⊂ Y + , ( Y + ) ∩ ( − Y + ) = { 0 } Partial order: y ≤ x ⇔ x − y ∈ Y + Y is strongly ordered if Int Y + � = ∅ . Then y ≪ x ⇔ x − y ∈ Int Y + . Examples: + × ( − R n − k Y = R n , Y + = R k ) , 0 ≤ k ≤ n : + x ≤ y ⇔ ( x i ≤ y i , i ≤ k ) ∧ ( x j ≥ y j , j > k ) Y = L p (Ω , R n ) , C r (Ω , R n ) , f ≤ g ⇔ f ( s ) ≤ g ( s ) , s ∈ Ω H.L. Smith (ASU) Monotone Dynamical Systems Sontagfest, May 23, 2011 3 / 16

Ordered Banach Space Induces ≤ X ⊂ Y , Y an ordered Banach space with closed positive cone Y + : ( R + ) Y + ⊂ Y + , Y + + Y + ⊂ Y + , ( Y + ) ∩ ( − Y + ) = { 0 } Partial order: y ≤ x ⇔ x − y ∈ Y + Y is strongly ordered if Int Y + � = ∅ . Then y ≪ x ⇔ x − y ∈ Int Y + . Examples: + × ( − R n − k Y = R n , Y + = R k ) , 0 ≤ k ≤ n : + x ≤ y ⇔ ( x i ≤ y i , i ≤ k ) ∧ ( x j ≥ y j , j > k ) Y = L p (Ω , R n ) , C r (Ω , R n ) , f ≤ g ⇔ f ( s ) ≤ g ( s ) , s ∈ Ω H.L. Smith (ASU) Monotone Dynamical Systems Sontagfest, May 23, 2011 3 / 16

Equilibria, Sub & Super Equilibria: E = { e ∈ X : ∀ t ≥ 0 , Φ t ( e ) = e } Sub-equilibria: E − = { x ∈ X : ∀ t ≥ 0 , Φ t ( x ) ≥ x } x ∈ E − ⇒ x ≤ Φ s ( x ) ≤ Φ t + s ( x ) , t , s ≥ 0 ∴ Φ t ( x ) ր e ∈ E , t ր ∞ . Super-equilibria: { x ∈ X : ∀ t ≥ 0 , Φ t ( x ) ≤ x } in applications, these can be identified by the semiflow generator H.L. Smith (ASU) Monotone Dynamical Systems Sontagfest, May 23, 2011 4 / 16

Equilibria, Sub & Super Equilibria: E = { e ∈ X : ∀ t ≥ 0 , Φ t ( e ) = e } Sub-equilibria: E − = { x ∈ X : ∀ t ≥ 0 , Φ t ( x ) ≥ x } x ∈ E − ⇒ x ≤ Φ s ( x ) ≤ Φ t + s ( x ) , t , s ≥ 0 ∴ Φ t ( x ) ր e ∈ E , t ր ∞ . Super-equilibria: { x ∈ X : ∀ t ≥ 0 , Φ t ( x ) ≤ x } in applications, these can be identified by the semiflow generator H.L. Smith (ASU) Monotone Dynamical Systems Sontagfest, May 23, 2011 4 / 16

Equilibria, Sub & Super Equilibria: E = { e ∈ X : ∀ t ≥ 0 , Φ t ( e ) = e } Sub-equilibria: E − = { x ∈ X : ∀ t ≥ 0 , Φ t ( x ) ≥ x } x ∈ E − ⇒ x ≤ Φ s ( x ) ≤ Φ t + s ( x ) , t , s ≥ 0 ∴ Φ t ( x ) ր e ∈ E , t ր ∞ . Super-equilibria: { x ∈ X : ∀ t ≥ 0 , Φ t ( x ) ≤ x } in applications, these can be identified by the semiflow generator H.L. Smith (ASU) Monotone Dynamical Systems Sontagfest, May 23, 2011 4 / 16

Equilibria, Sub & Super Equilibria: E = { e ∈ X : ∀ t ≥ 0 , Φ t ( e ) = e } Sub-equilibria: E − = { x ∈ X : ∀ t ≥ 0 , Φ t ( x ) ≥ x } x ∈ E − ⇒ x ≤ Φ s ( x ) ≤ Φ t + s ( x ) , t , s ≥ 0 ∴ Φ t ( x ) ր e ∈ E , t ր ∞ . Super-equilibria: { x ∈ X : ∀ t ≥ 0 , Φ t ( x ) ≤ x } in applications, these can be identified by the semiflow generator H.L. Smith (ASU) Monotone Dynamical Systems Sontagfest, May 23, 2011 4 / 16

Sub & Super Equilibria Bracket Basin x 1 is a sub-equilibrium with Φ t ( x 1 ) ր e ∈ E . Monotonicity implies B = { x ∈ X : x 1 ≤ x ≤ e } ⊂ Basin of attraction of e because it is “sandwiched": Φ t ( x 1 ) ≤ Φ t ( x ) ≤ Φ t ( e ) = e H.L. Smith (ASU) Monotone Dynamical Systems Sontagfest, May 23, 2011 5 / 16

Strong Monotonicity & Limit Set Dichotomy Φ strongly monotone (Hirsch) if Y is strongly ordered and x < y ⇒ Φ t ( x ) ≪ Φ t ( y ) , t > 0 . Φ is strongly order preserving (Matano) (SOP) if it is monotone and x < y ⇒ ∃ nbhds U , V , x ∈ U , y ∈ V , ∃ t 0 ≥ 0 such that Φ t 0 ( U ) ≤ Φ t 0 ( V ) . If x < y then either Theorem[LSD, Hirsch(1982)]: Let Φ be SOP (a) ω ( x ) < ω ( y ) , or (b) ω ( x ) = ω ( y ) ⊂ E ω ( x ) = omega limit set of x H.L. Smith (ASU) Monotone Dynamical Systems Sontagfest, May 23, 2011 6 / 16

Strong Monotonicity & Limit Set Dichotomy Φ strongly monotone (Hirsch) if Y is strongly ordered and x < y ⇒ Φ t ( x ) ≪ Φ t ( y ) , t > 0 . Φ is strongly order preserving (Matano) (SOP) if it is monotone and x < y ⇒ ∃ nbhds U , V , x ∈ U , y ∈ V , ∃ t 0 ≥ 0 such that Φ t 0 ( U ) ≤ Φ t 0 ( V ) . If x < y then either Theorem[LSD, Hirsch(1982)]: Let Φ be SOP (a) ω ( x ) < ω ( y ) , or (b) ω ( x ) = ω ( y ) ⊂ E ω ( x ) = omega limit set of x H.L. Smith (ASU) Monotone Dynamical Systems Sontagfest, May 23, 2011 6 / 16

Generic Convergence Theorem*: Assume X ⊂ Y , Y an ordered Banach space, and X is either convex or the closure of an open set. Let C = { x ∈ X : Φ t ( x ) → e , e ∈ Equilibria } If Φ is SOP on X and some mild smoothness and compactness assumptions hold ( † ), then Int C is dense in X . *Inspired by: M. Hirsch. Systems of differential equations which are competitive or cooperative II: convergence almost everywhere, SIAM J. Math. Anal., 16, 1985. ( † ) ∃ τ > 0: x 1 < x 2 ⇒ Φ τ x 1 ≪ Φ τ x 2 Φ τ is locally C 1 at each e ∈ E , Φ ′ τ ( e ) is Krein-Rutman operator. H.L. Smith (ASU) Monotone Dynamical Systems Sontagfest, May 23, 2011 7 / 16

ODEs-A Canonical Form + × ( − R n − k x ′ = F ( x ) is a monotone system w.r.t. orthant cone R k ) in + domain X if, on permuting variables x = ( x 1 , x 2 ) , x 1 ∈ R k , x 2 ∈ R n − k x ′ f 1 ( x 1 , x 2 ) = 1 x ′ f 2 ( x 1 , x 2 ) = 2 diagonal blocks ∂ f i ∂ x i ( x ) have nonnegative off-diagonal entries. off-diagonal blocks ∂ f i ∂ x j ( x ) ≤ 0 , i � = j have nonpositive entries.   ∗ + − − + ∗ − −   Jacobian =  , + ≥ 0 , − ≤ 0   − − ∗ +  − − + ∗ Components cluster into two subgroups. positive within-group interactions, negative between-group interactions. Strong monotonicity holds if the Jacobian is irreducible at a.e. x ∈ X ! H.L. Smith (ASU) Monotone Dynamical Systems Sontagfest, May 23, 2011 8 / 16