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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 Self-excited Wave Pro cesses in Chains of Unidirectionally Coupled Relaxation Oscillators

YarSU


Intro duction

u = [f (u(t - h)) - g (u(t - 1))]u. u (t ) > 0, 1, h (0, 1), f (u), g (u) C (R+ ), R+ = {u R : u 0},
1

(1)

f (0) = 1, g (0) = 0; f (u) = -a0 + O(1/u), uf (u) = O(1/u), u2 f (u) = O(1/u), g (u) = b0 + O(1/u), ug (u) = O(1/u), u g (u) = O(1/u) as u +,
2

(2)

a0 > 0, b0 > 0.

S.D. Glyzin Self-excited Wave Pro cesses in Chains of Unidirectionally Coupled Relaxation Oscillators

YarSU


u = f (u(t - 1))u, h=1 f (u) - g (u) f (u), a > 1. a0 + b0 a.

(3)

(4)

u j = d (u

j +1

- uj ) + f (uj (t - 1))uj , 1. u1 . . . u
m

j = 1, . . . , m,

u

m+1

= u1 ,

(5)

d = const > 0.

= u (t, ),

(6)

u (t, )

S.D. Glyzin Self-excited Wave Pro cesses in Chains of Unidirectionally Coupled Relaxation Oscillators

YarSU


Main theorem

j -1

u1 = exp(x/),

uj = exp x/ +
k=1

yk

,

j = 2, . . . , m,

= 1/. (7)

x = d (exp y1 - 1) + F x(t - 1), , yj = d exp y y
j +1

(8) (9)

- exp yj + Gj x(t - 1), y1 (t - 1), . . . , yj (t - 1), , j = 1 , . . . , m - 1, , F (x, ) = f exp(x/) ,
j j -1

m

= -y1 - y2 - . . . - ym

-1

Gj (x, y1 , . . . , yj , ) =

1 f exp x/ +

y
k=1

k

- f exp x/ +
k=1

y

k

,

j = 1, . . . , m - 1.
S.D. Glyzin Self-excited Wave Pro cesses in Chains of Unidirectionally Coupled Relaxation Oscillators YarSU


0 < 0 < a - 1, F ­ Banach space of functions (t) = 1 (t), . . . , m (t) on -1 - 0 t -0 . ||||F = max
1j m -1-0 t-0

max

|j (t)| .
m

(10) S
m

S = (t) = 1 (t), . . . , m (t) : 1 S1 , 2 S2 , . . . , S1 = {1 (t) C [-1 - 0 , -0 ]| - q1 1 (t) -q2 , q1 > 0 , q2 (0, 0 ), S2 , . . . , Sm C [-1 - 0 , -0 ]. x (t, ), y1, (t, ), . . . , ym : S F () = x (t + T , ), y1, (t + T , ), . . . , ym -1 - 0 t -0 .
-1, -1,

F.

1 (-0 ) = -0 },

(t, ) ,

t -0

(11)

(t + T , ) ,

(12)

S.D. Glyzin Self-excited Wave Pro cesses in Chains of Unidirectionally Coupled Relaxation Oscillators

YarSU


0 0 () = x0 (t), y1 (t + T0 , z ), . . . , y

0 m- 1

(t + T0 , z )

,
z =(2 (-0 ),...,m (-0 ))

-1 - 0 t -0 . (13) t x 0 (t ) = 1 - a(t - 1) -a + t - t0 - 1 if 0 t 1, if 1 t t0 + 1, if t0 + 1 t T0 , x0 (t + T0 ) x0 (t). (14)

S.D. Glyzin Self-excited Wave Pro cesses in Chains of Unidirectionally Coupled Relaxation Oscillators

YarSU


yj = d exp y

j +1

- exp y

j

yj (1 + 0) = yj (1 - 0) - (1 + a) yj (0), j = 1, . . . , m - 1, (15)
-1

yj (t0 + 1 + 0) = yj (t0 + 1 - 0) - (1 + 1/a) yj (t0 ), ym = -y1 - y2 - . . . - ym (y1 , . . . , y t0 = 1 + 1/a.
m-1

,
m- 1

)

t= -0

= ( z1 , . . . , z

),

(16)

S.D. Glyzin Self-excited Wave Pro cesses in Chains of Unidirectionally Coupled Relaxation Oscillators

YarSU


Theorem (on C1 -convergence) There exist small enough 0 = 0 (S ) > 0 such that for all 0 < 0 the operator are defined on S and
0 S 0 S

lim sup || () - 0 ()||

F

= 0,
F0

lim sup || () - 0 ()||F0

= 0.

(17)

S.D. Glyzin Self-excited Wave Pro cesses in Chains of Unidirectionally Coupled Relaxation Oscillators

YarSU


0 0 z (z ) = (y1 (t, z ), y2 (t, z ), . . . , y

def

0 m-1

(t, z ))

t=T0 -0

,

(18)

z = (2 (-0 ), . . . , m (-0 )). z = z
0 (t) = (t), . . . , (t) : (t) = x0 (t), (t) = yj -1 (t + T0 , z ), 1 m 1 j

j = 2, . . . , m,

-1 - 0 t -0

S.D. Glyzin Self-excited Wave Pro cesses in Chains of Unidirectionally Coupled Relaxation Oscillators

YarSU


Theorem (Compliance Theorem) For any fixed point z = z of map (z ) (18), such that det (I - (z )) = 0, there exist relaxation cycle of system (8), (9). This cycle exists for all small enough > 0 and is exponentially orbitally stable (unstable) if r < 1 (> 1), where r ­ spectral radius of matrix (z ).

S.D. Glyzin Self-excited Wave Pro cesses in Chains of Unidirectionally Coupled Relaxation Oscillators

YarSU


a > m - 1. zj = - d0 yj (t, v , d) = - yj (t, v , d) = ln 1 1 ln + vj + O(d a d
1-(m-1)/a

(19) (20)

1 1 ln + vj , a d

j = 1 , . . . , m - 1,

) if 0 t < 1,

(21) (22) (23)

1 0 + j (t, v ) + O(d1-(m-1)/a ) if 1 t < t0 + 1, d 1 1 yj (t, v , d) = - ln + j (v ) + O(d1-(m-1)/a ) if t0 + 1 t T0 , a d

S.D. Glyzin Self-excited Wave Pro cesses in Chains of Unidirectionally Coupled Relaxation Oscillators

YarSU


j = exp j

j +1

- exp j ,

j = 1 , . . . , m - 2,
-1


t=1

m-1

= - exp m

,

= -a vj ,

j = 1, . . . , m - 1
-s

0 m-1

0 (t, v ) + . . . + m s-1

(t, v ) =
s-

= - ln

(t - 1) + s!

s

=0

(t - 1) exp a !

vm
j =1

-j

,

(24)

s = 1, . . . , m - 1.

S.D. Glyzin Self-excited Wave Pro cesses in Chains of Unidirectionally Coupled Relaxation Oscillators

YarSU


0 j (v ) = j (t, v )

t= t0 +1

0 - (1 + 1/a)j (t, v )

t= t0

,

j = 1, . . . , m - 1. (25) (26)

vj j (v ),

j = 1, . . . , m - 1.

S.D. Glyzin Self-excited Wave Pro cesses in Chains of Unidirectionally Coupled Relaxation Oscillators

YarSU


s = -v

m-1

- . . . - vm

-s

,

s = 1, . . . , m - 1

k ln r

1,k

+ exp(-ak ) - (1 + 1/a) ln r k = 1, . . . , m - 1,

2,k

+ exp(-ak ) ,

(27)

where r r
1,k 1,1

= 1 + 1/a, (1 + 1/a)k + k!
k-1

r

2,1

= 1/a, (1 + 1/a) exp(-ak !
- -

(28) ),

k-1

(1 , . . . ,

k-1

)=

=1

r

2,k

(1 , . . . ,
m-1

k-1

)=

1 + ak k!

=1

1 exp(-ak a!

),

k = 2 , . . . , m - 1. (29)

(1 , . . . ,

),

S.D. Glyzin Self-excited Wave Pro cesses in Chains of Unidirectionally Coupled Relaxation Oscillators

YarSU


z = ( z1 , . . . , z

m-1

1 1 ln + vj + O(d a d j = 1, . . . , m - 1, d 0, ),
zj = -

1-(m-1)/a

),

(30)

where v

m-1

= -1 , vm

-s

= s-1 - s , s = 2, . . . , m - 1.

S.D. Glyzin Self-excited Wave Pro cesses in Chains of Unidirectionally Coupled Relaxation Oscillators

YarSU


Thank you for attention!

S.D. Glyzin Self-excited Wave Pro cesses in Chains of Unidirectionally Coupled Relaxation Oscillators

YarSU