Self-excited Wave Processes in Chains of Unidirectionally Coupled - PowerPoint PPT Presentation

Self-excited Wave Processes in Chains of Unidirectionally Coupled Relaxation Oscillators Sergey Glyzin P.G. Demidov Yaroslavl State University November 17-19, 2015 S.D. Glyzin YarSU Self-excited Wave Processes in Chains of Unidirectionally Coupled Relaxation Oscillators

Introduction u = λ [ f ( u ( t − h )) − g ( u ( t − 1))] u. ˙ (1) u ( t ) > 0 , λ ≫ 1 , h ∈ (0 , 1) , f ( u ) , g ( u ) ∈ C 1 ( R + ) , R + = { u ∈ R : u ≥ 0 } , f (0) = 1 , g (0) = 0; f ( u ) = − a 0 + O (1 /u ) , uf ′ ( u ) = O (1 /u ) , u 2 f ′′ ( u ) = O (1 /u ) , g ( u ) = b 0 + O (1 /u ) , ug ′ ( u ) = O (1 /u ) , (2) u 2 g ′′ ( u ) = O (1 /u ) as u → + ∞ , a 0 > 0 , b 0 > 0 . S.D. Glyzin YarSU Self-excited Wave Processes in Chains of Unidirectionally Coupled Relaxation Oscillators

u = λf ( u ( t − 1)) u, ˙ (3) h = 1 f ( u ) − g ( u ) → f ( u ) , a 0 + b 0 → a. a > 1 . (4) u j = d ( u j +1 − u j ) + λf ( u j ( t − 1)) u j , ˙ j = 1 , . . . , m, u m +1 = u 1 , (5) d = const > 0 . λ ≫ 1 . u 1 ≡ . . . ≡ u m = u ∗ ( t, λ ) , (6) u ∗ ( t, λ ) S.D. Glyzin YarSU Self-excited Wave Processes in Chains of Unidirectionally Coupled Relaxation Oscillators

Main theorem j − 1 � � � u 1 = exp( x/ε ) , u j = exp x/ε + y k , j = 2 , . . . , m, ε = 1 /λ. (7) k =1 � � x = εd (exp y 1 − 1) + F ˙ x ( t − 1) , ε , (8) � � � � y j = d ˙ exp y j +1 − exp y j + G j x ( t − 1) , y 1 ( t − 1) , . . . , y j ( t − 1) , ε , (9) j = 1 , . . . , m − 1 , � � y m = − y 1 − y 2 − . . . − y m − 1 , F ( x, ε ) = f exp( x/ε ) , j j − 1 G j ( x, y 1 , . . . , y j , ε ) = 1 � � � �� � � ��� � � f exp x/ε + y k − f exp x/ε + y k , ε k =1 k =1 j = 1 , . . . , m − 1 . S.D. Glyzin YarSU Self-excited Wave Processes in Chains of Unidirectionally Coupled Relaxation Oscillators

� � 0 < σ 0 < a − 1 , F – Banach space of functions ϕ ( t ) = ϕ 1 ( t ) , . . . , ϕ m ( t ) on − 1 − σ 0 ≤ t ≤ − σ 0 . � � || ϕ || F = max − 1 − σ 0 ≤ t ≤− σ 0 | ϕ j ( t ) | max . (10) 1 ≤ j ≤ m � � � � S = ϕ ( t ) = ϕ 1 ( t ) , . . . , ϕ m ( t ) : ϕ 1 ∈ S 1 , ϕ 2 ∈ S 2 , . . . , ϕ m ∈ S m ⊂ F . S 1 = { ϕ 1 ( t ) ∈ C [ − 1 − σ 0 , − σ 0 ] | − q 1 ≤ ϕ 1 ( t ) ≤ − q 2 , ϕ 1 ( − σ 0 ) = − σ 0 } , q 1 > σ 0 , q 2 ∈ (0 , σ 0 ) , S 2 , . . . , S m ⊂ C [ − 1 − σ 0 , − σ 0 ] . � � x ϕ ( t, ε ) , y 1 ,ϕ ( t, ε ) , . . . , y m − 1 ,ϕ ( t, ε ) , t ≥ − σ 0 (11) Π ε : S → F � � Π ε ( ϕ ) = x ϕ ( t + T ϕ , ε ) , y 1 ,ϕ ( t + T ϕ , ε ) , . . . , y m − 1 ,ϕ ( t + T ϕ , ε ) , (12) − 1 − σ 0 ≤ t ≤ − σ 0 . S.D. Glyzin YarSU Self-excited Wave Processes in Chains of Unidirectionally Coupled Relaxation Oscillators

�� x 0 ( t ) , y 0 1 ( t + T 0 , z ) , . . . , y 0 � Π 0 ( ϕ ) = m − 1 ( t + T 0 , z ) z =( ϕ 2 ( − σ 0 ) ,...,ϕ m ( − σ 0 )) , � � − 1 − σ 0 ≤ t ≤ − σ 0 . (13)  t if 0 ≤ t ≤ 1 ,   x 0 ( t ) = 1 − a ( t − 1) if 1 ≤ t ≤ t 0 + 1 , x 0 ( t + T 0 ) ≡ x 0 ( t ) . (14)  − a + t − t 0 − 1 if t 0 + 1 ≤ t ≤ T 0 ,  S.D. Glyzin YarSU Self-excited Wave Processes in Chains of Unidirectionally Coupled Relaxation Oscillators

� � y j = d ˙ exp y j +1 − exp y j y j (1 + 0) = y j (1 − 0) − (1 + a ) y j (0) , y j ( t 0 + 1 + 0) = y j ( t 0 + 1 − 0) − (1 + 1 /a ) y j ( t 0 ) , j = 1 , . . . , m − 1 , (15) y m = − y 1 − y 2 − . . . − y m − 1 , � ( y 1 , . . . , y m − 1 ) t = − σ 0 = ( z 1 , . . . , z m − 1 ) , (16) � t 0 = 1 + 1 /a . S.D. Glyzin YarSU Self-excited Wave Processes in Chains of Unidirectionally Coupled Relaxation Oscillators

Theorem (on C 1 -convergence) There exist small enough ε 0 = ε 0 ( S ) > 0 such that for all 0 < ε ≤ ε 0 the operator Π ε are defined on S and ε → 0 sup lim || Π ε ( ϕ ) − Π 0 ( ϕ ) || F = 0 , ϕ ∈ S (17) ε → 0 sup lim || ∂ ϕ Π ε ( ϕ ) − ∂ ϕ Π 0 ( ϕ ) || F 0 → F 0 = 0 . ϕ ∈ S S.D. Glyzin YarSU Self-excited Wave Processes in Chains of Unidirectionally Coupled Relaxation Oscillators

def = ( y 0 1 ( t, z ) , y 0 2 ( t, z ) , . . . , y 0 � z → Φ( z ) m − 1 ( t, z )) t = T 0 − σ 0 , (18) � z = ( ϕ 2 ( − σ 0 ) , . . . , ϕ m ( − σ 0 )) . z = z ∗ ϕ ∗ 1 ( t ) , . . . , ϕ ∗ : ϕ ∗ 1 ( t ) = x 0 ( t ) , ϕ ∗ j ( t ) = y 0 � � ϕ ∗ ( t ) = m ( t ) j − 1 ( t + T 0 , z ∗ ) , j = 2 , . . . , m, − 1 − σ 0 ≤ t ≤ − σ 0 S.D. Glyzin YarSU Self-excited Wave Processes in Chains of Unidirectionally Coupled Relaxation Oscillators

Theorem (Compliance Theorem) For any fixed point z = z ∗ of map Φ( z ) (18) , such that det ( I − Φ ′ ( z ∗ )) � = 0 , there exist relaxation cycle of system (8) , (9) . This cycle exists for all small enough ε > 0 and is exponentially orbitally stable (unstable) if r ∗ < 1 ( > 1) , where r ∗ – spectral radius of matrix Φ ′ ( z ∗ ) . S.D. Glyzin YarSU Self-excited Wave Processes in Chains of Unidirectionally Coupled Relaxation Oscillators

a > m − 1 . (19) z j = − 1 a ln 1 d + v j , j = 1 , . . . , m − 1 , (20) d → 0 y j ( t, v, d ) = − 1 a ln 1 d + v j + O ( d 1 − ( m − 1) /a ) if 0 ≤ t < 1 , (21) y j ( t, v, d ) = ln 1 d + ω 0 j ( t, v ) + O ( d 1 − ( m − 1) /a ) if 1 ≤ t < t 0 + 1 , (22) y j ( t, v, d ) = − 1 a ln 1 d + ψ j ( v ) + O ( d 1 − ( m − 1) /a ) if t 0 + 1 ≤ t ≤ T 0 , (23) S.D. Glyzin YarSU Self-excited Wave Processes in Chains of Unidirectionally Coupled Relaxation Oscillators

ω j = exp ω j +1 − exp ω j , ˙ j = 1 , . . . , m − 2 , ω m − 1 = − exp ω m − 1 , ˙ � ω j t =1 = − a v j , j = 1 , . . . , m − 1 � ω 0 m − 1 ( t, v ) + . . . + ω 0 m − s ( t, v ) = � s − 1 � s − ℓ �� ( t − 1) s ( t − 1) ℓ � � = − ln + exp a v m − j , (24) s ! ℓ ! ℓ = 0 j =1 s = 1 , . . . , m − 1 . S.D. Glyzin YarSU Self-excited Wave Processes in Chains of Unidirectionally Coupled Relaxation Oscillators

ψ j ( v ) = ω 0 t = t 0 +1 − (1 + 1 /a ) ω 0 � � j ( t, v ) j ( t, v ) t = t 0 , j = 1 , . . . , m − 1 . (25) � � v j → ψ j ( v ) , j = 1 , . . . , m − 1 . (26) S.D. Glyzin YarSU Self-excited Wave Processes in Chains of Unidirectionally Coupled Relaxation Oscillators

α s = − v m − 1 − . . . − v m − s , s = 1 , . . . , m − 1 � � � � α k → ln r 1 ,k + exp( − aα k ) − (1 + 1 /a ) ln r 2 ,k + exp( − aα k ) , (27) k = 1 , . . . , m − 1 , where r 1 , 1 = 1 + 1 /a, r 2 , 1 = 1 /a, (28) k − 1 r 1 ,k ( α 1 , . . . , α k − 1 ) = (1 + 1 /a ) k (1 + 1 /a ) ℓ � + exp( − aα k − ℓ ) , k ! ℓ ! ℓ =1 k − 1 1 1 � r 2 ,k ( α 1 , . . . , α k − 1 ) = a k k ! + a ℓ ℓ ! exp( − aα k − ℓ ) , k = 2 , . . . , m − 1 . ℓ =1 (29) ( α ∗ 1 , . . . , α ∗ m − 1 ) , S.D. Glyzin YarSU Self-excited Wave Processes in Chains of Unidirectionally Coupled Relaxation Oscillators

j = − 1 a ln 1 z ∗ = ( z ∗ 1 , . . . , z ∗ z ∗ d + v ∗ j + O ( d 1 − ( m − 1) /a ) , m − 1 ) , (30) j = 1 , . . . , m − 1 , d → 0 , where v ∗ m − 1 = − α ∗ 1 , v ∗ m − s = α ∗ s − 1 − α ∗ s , s = 2 , . . . , m − 1 . S.D. Glyzin YarSU Self-excited Wave Processes in Chains of Unidirectionally Coupled Relaxation Oscillators

Thank you for attention! S.D. Glyzin YarSU Self-excited Wave Processes in Chains of Unidirectionally Coupled Relaxation Oscillators