Theory of Ordinary Differential Equations. Stability and Bifurcation II. John A. Burns

Transcription

1 Theory of Ordiary Differetial Equatios Stability ad Bifurcatio II Joh A. Burs Ceter for Optimal Desig Ad Cotrol Iterdiscipliary Ceter for Applied Mathematics Virgiia Polytechic Istitute ad State Uiversity Blacksburg, Virgiia MATH FALL 1

2 Iitial Value Problem (IVP) { (Σ) (IC) x( t) f x( t), q x( t) x R m f ( xq, ) : D R R R d dt x1( t) f1( x1( t), x( t),... x( t), q1, q,... qm) x ( t) f ( x ( t), x ( t),... x ( t), q, q,... q ) R x ( t) f ( x ( t), x ( t),... x ( t), q, q,... q ) 1 1 m 1 1 m

3 Autoomous Systems (IVP) { (Σ) (IC) x( t) f x( t), q x( t) x R m f ( xq, ) : D R R R Let x e = x e (q) be a equilibrium for some parameter q, i.e. f ( x, q) f( x ( q), q) e e We will assume x e = x e (q) is a isolated equilibrium

4 Isolated Equilibrium f ( x, q) f ( x ( q), q), j 1,,3,... j j there exists a such that x B( x, ) if i j j i j j x ( ) q x ( ) 1 q R x ( ) 4 q x ( ) 3 q NON-ISOLATED EQUILIBRIUM CAN NOT BE ASYMPTOTICALLY STABLE

5 First Order Liear x( t) qx( t) x() t e qt x x( t) qx( t) f ( x( t), q) q = q = q = q = -.5 q =

6 Equilibrium x e =, q < : Stable qx f ( x, q) q < x e xe

7 Equilibrium x e =, q > : Ustable qx f ( x, q) q > x e xe

8 Equilibrium x e =, q = : Stable qx f ( x, q) q = xe x 1 ANY xe xe.5 x.1 xe.1 x.5 e e x e is NOT isolated

9 Example 4. x ( t) x ( t) 1 x ( t) qx ( t) x ( t) [ x ( t)] f x x 1 [ ] 3 x qx x x 1 1 x 1 1 x ( q [ x ] ) x or ( q [ x ] ) 1 1

10 Example 4. ( [ ] x q x ) 1 1 x q x 1 x q > x q 1 x

11 Example 4. q = 1 f1( x, y) y ( xy, ) : 3 f( x, y) ax y x ( x, y) : f ( x, y) y 1 3 ( x, y) : f( x, y) ax y x ( x, y) : y ax x 3

12 Epidemic Models Susceptible Ifected Removed

13 Epidemic Models SIR Models (Kermak McKedrick, 197) Susceptible Ifected Recovered/Removed d S ( t ) S ( t ) I ( t ) dt d I ( t ) S ( t ) I ( t ) I ( t ) dt d R ( t ) I ( t ) dt S( t) I( t) R( t) N costat

14 SIR Models d R ( t ) I ( t ) ad S ( t ) I ( t ) R ( t ) N costat dt d S ( t ) S ( t ) I ( t ) dt d I ( t ) S ( t ) I ( t ) I ( t ) I ( t ) S ( t ) dt x ( t) S( t), x ( t) I( t) ad q 1 T d dt x1 ( t) q1x1 ( t) x( t) x ( t) q x ( t) x ( t) q x ( t) 1 1

15 SIR Model: Equilibrium d dt x1 ( t) q1x1 ( t) x( t) x ( t) q x ( t) x ( t) q x ( t) 1 1 q1x 1x q1x 1x q x x q x ( q x q ) x q x x x1 or x 1 1 x1 q x hece x 1 x x 1 ca be ay value

16 SIR Model: Equilibrium x R x 1 EQUILIBRIUM x e x x 1 : x x NONE ARE ISOLATED

17 SIR Model x (t) x 1 (t) + x (t) N = 1 x 1 (t)

18 Stability of Equilibrium f x x( t) f x( t) e () t xe x x e x(t) (t)= x e t? HOW DO WE KNOW IF x e IS ASYMPTOTICALLY STABLE?

19 Fudametal Stability Theorem Let 1,, 3, be the eigevalues of J x f(x e ), i.e det( I J f( )) k x xe i (Re( ), Im( ) ) k k k k k k k Theorem S1: If Re( k ) < for all k=1,,., the x e is a asymptotically stable equilibrium for the o-liear system x( t) f x( t). I particular, there exist > such that if x() x, the lim ( t). e t x x e

20 No-Stability Theorem Theorem S: If there is oe eigevalue p such that Re( p ) >, the x e is a ustable equilibrium for the o-liear system x t f x t ( ) ( ). The two theorems above may be foud i: Richard K. Miller ad Athoy N. Michel, Ordiary Differetial Equatios, Academic Press, 198. (see pages 58 53) ad Earl A. Coddigto ad Norma Leviso, Theory of Ordiary Differetial Equatios, McGraw-Hill, (see pages )

21 Critical Case If there is oe eigevalue p of [ J f x ( x, q e )] such that Re( p ) =, the x e the liearizatio theorems do ot apply ad other methods must be used to determie the stability properties of the equilibrium for the oliear system x( t) f x( t), q

22 Example 5.1 f 3 d x1( t) x1( t) [ x1( t)] x( t) f 3 dt x( t) x( t) x1( t) [ x( t)] 3 x1 [ x 1] x 3 x x 1[ x] [ x ] x 3 1 ad [ x ] 1 x 3 [ x ] [ x ] x or 8 [ x1 ] 1 x e x1 x x?? IS STABLE?? e

23 Example 5.1 Try the liearizatio theorems J f ( ) x 1 1 3[ x1 ] 1 1 3[ x] x, x det I J( ) det( ) det i Re( i ) for i 1, Theorem S1 ad Theorem S do ot apply

24 Example 5.1 LOOKS ASYMPTOTICALLY STABLE

25 Example 5. d dt x1( t) x1( t) x( t) f 3 x( t) x( t) q[ x1( t)] 3 x( t) f x1 x 3 x q[ x1 ] 3x x e x1 x x?? IS ASYMPTOTICALLY STABLE?? e Try the liearizatio Theorem J f ( ) x [ ] 3 3 qx 1 x, x 1

26 Example det I J( ) det( ) det ( 3) ( 3) ad 3 1 Re( ) Liearizatio Theorems do ot apply BUT

27 Example 5. ZOOM IN

28 Example 5. LOOKS ASYMPTOTICALLY STABLE

29 Isolated Equilibrium H R x 3 x e R a ope set x e x 1 xˆe H R ad x H, for x V( x) : H R R e e

30 Lyapuov Fuctios H R e a ope set x e x x V( x) : H R R V( x) V( x, x,... x ) 1 If V ad whe ( ) V ( x), x, the V ( x) is said to be positive defiite

31 Lyapuov Fuctios d dt x() t f f f 1 ( x) ( x) ( x) R x x x x R 1 T x 1 V( ) : H R R V( x) V( x, x,... x ) We defie the fuctio. V( x) : H R R by. V ( x) V ( x) V ( x) V ( x) f1( x) f( x)... f( x) x x x 1

32 Lyapuov Fuctios V( x) : H R R is called a Lyapuov fuctio for the equilibrium (Σ) x e of the system x( t) f x( t) if ad ( i) V ( x) is positive defiite i H. ( ii) V ( x) for all x H

33 Lyapuov Theorems Theorem L1. If there exists a Lyapuov fuctio for the equilibrium x of the system the the equilibrium e (Σ) x( t) f x( t), x e is stable. Theorem L. If there exists a Lyapuov fuctio for the equilibrium x of the system ad. e (Σ). x( t) f x( t), V( ) ad V( x) for all x H, x, the the equilibrium x e is asymptotically stable.

34 Example 5.1 AGAIN 3 d x1( t) x1( t) [ x1( t)] x( t) f 3 dt x( t) x( t) x1( t) [ x( t)] x?? IS STABLE?? e H R is a ope set V( x) V( x, x ) [ x ] [ x ] 1 1 V( ) ad V( x) V( x, x ) [ x ] [ x ] if x 1 1 Hece, V( x) is positive defiite

35 Example 5.1 AGAIN. V( x) V( x, x ) [ x ] [ x ] 1 1 V ( x, x ) V ( x, x ) V x x f x x f x x 1 1 ( 1, ) 1( 1, ) ( 1, ) x x 1 V ( x1, x) x 1 x 1 V ( x1, x) x x f 3 x1 f1( x1, x) [ x1] x 3 x f( x1, x) x1[ x]. V ( x, x ) V ( x, x ) V x x f x x x ( fx x [ xx] ) ( 1, ) x1 ( [ 1x( 1] 1, x ) 1( 1, ) x x 1

36 Example 5.1 AGAIN.. V ( x, x ) V ( x, x ) V x x f x x x ( fx x [ xx] ) ( 1, ) x1 ( [ 1x( 1] 1, x ) 1( 1, ) x x 1. V( x, x ) [ x ] x x x x [ x ] V (,) ([] [] ) x x V( x, x ) ([ x ] [ x ] ) V( x, x ) ([ x ] [ x ] ) ad if, the Theorem L x e IS ASYMPTOTICALLY STABLE

37 Example 5. AGAIN d dt f q x1( t) x1( t) x( t) f 3 x( t) x( t) q[ x1( t)] 3 x( t) x1 x 3 x q[ x1 ] 3x H R is a ope set V ( x) V ( x, x ) [ x ] [ x ] V ( ) ad V ( x) V ( x, x ) [ x ] [ x ] if x Hece, V( x) q is positive defiite q x e x1 x

38 Example 5. AGAIN. V ( x) V ( x, x ) [ x ] [ x ] V ( x, x ) V ( x, x ) V x x f x x f x x 1 1 ( 1, ) 1( 1, ) ( 1, ) x x 1 V ( x1, x) x 1 qx [ ] V ( x1, x) x q x f x1 x f1( x1, x) 3 x q[ x1 ] 3 x f( x1, x). V ( x, x ) V ( x, x ) ( 1, ) qx [ 1] 1( x 1, ) x q x ( 1 1, x ) x x 1 V x x f x x ( f[ x] x3 )

39 Example 5. AGAIN. V ( x, x ) V ( x, x ) ( 1, ) qx [ 1] 1( x 1, ) x q x ( 1 1, x ) x x 1 V x x f x x ( f[ x] x3 ). V( x, x ) q[ x ] x q[ x ] x 6[ x ] 6[ x ] Hece,. V( x) 6[ x ] for all xh R ad Theorem L1 implies that x x e is stable?? IS ASYMPTOTICALLY STABLE?? e

40 Example 5. AGAIN x?? IS ASYMPTOTICALLY STABLE?? e.. ( 1, ) 6[ x] V x x. BUT. V(1,) 6[] SO V(,) 6[] V( x) for all x H, x Theorem L does ot apply NEED A BETTER THEOREM

41 Example 5.3 d dt x1( t) x1( t) x( t) f 3 x( t) x( t) x1( t) [ x( t)] f x1 x 3 x x1 [ x] x e x1 x x?? IS STABLE?? Try the liearizatio Theorem e J f ( ) x [ ] 1 x x, x 1

42 Example det I J( ) det( ) det i Re( i ) for i 1, Liearizatio Theorems do ot apply BUT

43 Example 5.3 ZOOM IN

44 Example 5.3 LOOKS LIKE A CENTER

45 Example 5.3 d dt x1( t) x1( t) x( t) f 3 x( t) x( t) x1( t) [ x( t)] f x1 x 3 x x1 [ x] x e H R is a ope set V( x) V( x, x ) [ x ] [ x ] 1 1 x1 x V( ) ad V( x) V( x, x ) [ x ] [ x ] if x 1 1 Hece, V( x) is positive defiite

46 Example 5.3. V( x) V( x, x ) [ x ] [ x ] 1 1 V ( x, x ) V ( x, x ) V x x f x x f x x 1 1 ( 1, ) 1( 1, ) ( 1, ) x x 1 V ( x1, x) x 1 x 1 V ( x1, x) x x f x1 x f1( x1, x) 3 x x1 [ x] f( x1, x). V ( x, x ) V ( x, x ) V x x f x x f x x ( 1, ) x 1( x 1 1, ) x ( x ( 1[ 1, x] ) x x 1

47 Example 5.3. V ( x, x ) V ( x, x ) V x x f x x f x x ( 1, ) x 1( x 1 1, ) x ( x ( 1[ 1, x] ) x x 1 Hece,.. V x x x x x x x 3 ( 1, ) 1 ( 1 [ ] ) V( x, x ) x x x x [ x ] [ x ] V( x) [ x ] for all xh R ad Theorem L1 implies that 4 x e is stable x?? IS ASYMPTOTICALLY STABLE?? e

48 Example 5.3 x?? IS ASYMPTOTICALLY STABLE?? e. 4. ( 1, ) [ x] 4 V x x. BUT. 4 V(1,) [] SO V(,) [] V( x) for all x H, x Theorem L does ot apply NEED A BETTER THEOREM

49 Positively Ivariat Sets (Σ) x( t) f t, x( t) x 3 R (IC) x( t) x R x x 1 x M x M x( t) x( t; x ) M for all t t

50 SIR Models d S ( t ) S ( t ) I ( t ) dt d I ( t ) S ( t ) I ( t ) I ( t ) I ( t ) S ( t ) dt If St ( ), the d I ( t ) I ( t ) S ( t ) dt I e Equilibrium, S NOT ISOLATED e N

51 SIR Models S(t) + I(t) N = 1 I(t) LOTS OF (POSITIVELY) INVARIANT SETS M M S(t)

52

53 Trajectories x t t x ( ;, ) R x R x R x t t x ( ;, ) R

54 Trajectories & Limit Sets Give x R, the (positive) trajectory through x Is the set x( t; t, x ) R : t t Give x R, the (egative) trajectory through x Is the set x( t; t, x ) R : t t Give x R, the trajectory through x Is the set x x ( t; t, ) R : t (, )

55 -limit Sets x R Give, a poit p belogs to the omega limit set (-limit set) of x t t x ( ;, ) R if for each ad every there is a such that T x( t; t, x ) p < t t T ( x ) p : p is a -limit poit of x R

56 -limit Sets ( x ) p : p is a -limit poit of x ( x ) p : there is a sequece t with lim x( t ; t, x ) p R k k k Theorem LIM1. If t t, the x t t x ( ;, ) R ( x ) is bouded for is a o-empty, compact ad coected positively ivariat set. xˆ ( x ) xˆ( t; tˆ, xˆ ) ( x ) for all t tˆ

57 Covergece to a Set x 3 R x( t) x( t; x ) M as t M x 1 For ay there is a T T ( ) t such that if t T( ), the there is a poit p M with x( t; x ) p <

58 Covergece to a Set x 3 x( ;, ) t t x R p M p 1 M x( ;, ) t1 t x M x 1 x t t x ( ;, ) R x( T( );, ) t x x

59 Example NS

60 Example NS M

61 Example agai a =1 > x( t) y( t) y( t) x( t) ([ x( t)] [ y( t)] 1) y( t) d dt x1( t) x( t) x( t) x1 ( t) ([ x1 ( t)] [ x( t)] 1) x( t) f x1 x x x1 ([ x1 ] [ x] 1) x x x 1

62 Example agai a =1 >

63 Example agai a =1 > M

64 Example agai a =1 > M

65 Example agai a =1 > LIMIT CYCLE M

66 LaSalle Theorems Theorem LIM. If t t i.e., the x( tt ;, x ) x t t x ( ;, ) R x( tt ;, x ) ( x ) approaches its -limit set. is bouded for Theorem LIM3. If t t ( ) M x x t t x ( ;, ), ad the R is bouded for x t t x ( ;, ) M

67 Theorem LIM: Example NS x x t t x ( ;, ) R M ( x )

68 LaSalle s Ivariace Theorem Let Hˆ R be a bouded closed positively ivariat set ( ) : ˆ V x H R R ( i) V ( x) v >, for all x Hˆ ad. mi ( ii) V ( x) for all x Hˆ E x H ˆ : V ( x). E Ĥ M E Hˆ is LARGEST ivariat subset of E

69 LaSalle s Ivariace Theorem Theorem LaSalle IP: If fuctio satisfyig (i) ad (ii) above ad M E is the largest ivariat subset of ˆ. E x H : V ( x), the for each x x( ;, ) tt x Hˆ Ĥ the trajectory approaches M. ( ) : ˆ V x H R R x( tt ;, x ) M M is a E Ĥ x( tt ;, x ) M x Lets apply this to some previous examples

70 Example 5. AGAIN d dt f q x1( t) x1( t) x( t) f 3 x( t) x( t) q[ x1( t)] 3 x( t) x1 x 3 x q[ x1 ] 3x H R is a ope set V ( x) V ( x, x ) [ x ] [ x ] V ( ) ad V ( x) V ( x, x ) [ x ] [ x ] if x Hece, V( x) q is positive defiite q x e x1 x

71 Example 5. AGAIN. V ( x) V ( x, x ) [ x ] [ x ] V ( x, x ) V ( x, x ) V x x f x x f x x 1 1 ( 1, ) 1( 1, ) ( 1, ) x x 1 V ( x1, x) x 1 qx [ ] V ( x1, x) x q x f x1 x f1( x1, x) 3 x q[ x1 ] 3 x f( x1, x). V ( x, x ) V ( x, x ) ( 1, ) qx [ 1] 1( x 1, ) x q x ( 1 1, x ) x x 1 V x x f x x ( f[ x] x3 )

72 Example 5. AGAIN. V ( x, x ) V ( x, x ) ( 1, ) qx [ 1] 1( x 1, ) x q x ( 1 1, x ) x x 1 V x x f x x ( f[ x] x3 ). V( x, x ) q[ x ] x q[ x ] x 6[ x ] 6[ x ] Hece,. V( x) 6[ x ] for all xh R ad Theorem L1 implies that x x e is stable?? IS ASYMPTOTICALLY STABLE?? e

73 Example 5. AGAIN x?? IS ASYMPTOTICALLY STABLE?? e.. ( 1, ) 6[ x] V x x. BUT. V(1,) 6[] SO V(,) 6[] V( x) for all x H, x Theorem L does ot apply APPLY LaSALLE s Theorem

74 Example 5. AGAIN q= -.5 V ( x) V ( x, x ) [ x ] [ x ] q H ˆ ( x, x ) : V( x).4 1

75 Example 5. AGAIN q= -.5 E x H ˆ : V ( x). E x ( x, x ) Hˆ : x 1

76 Ivariat Sets i E d dt x1( t) x( t) 3 x( t) q[ x1 ( t)] 3 x( t) x e x1 x x E x ( x1, x) : 6[ x] M { x } e x 1 3 IF x( t), the x ( t) q[ x ( t)] 3 x ( t) 1 x ( t) 1

77 Example 5. AGAIN x E x ( x1, x) : 6[ x] M { x } e x 1 M EHˆ is LARGEST ivariat subset of E Hece LaSalle s Ivariace Theorem Implies x( tt ;, x ) M = x = x e e IS ASYMPTOTICALLY STABLE

78 Example 5.3 AGAIN d dt x1( t) x1( t) x( t) f 3 x( t) x( t) x1( t) [ x( t)] f x1 x 3 x x1 [ x] x e x1 x x?? IS STABLE?? Try the liearizatio Theorem e J f ( ) x [ ] 1 x x, x 1

79 Example 5.3 AGAIN 1 1 det I J( ) det( ) det i Re( i ) for i 1, Liearizatio Theorems do ot apply BUT

80 Example 5.3 AGAIN ZOOM IN

81 Example 5.3 AGAIN LOOKS LIKE A CENTER

82 Example 5.3 AGAIN d dt x1( t) x1( t) x( t) f 3 x( t) x( t) x1( t) [ x( t)] f x1 x 3 x x1 [ x] x e H R is a ope set V( x) V( x, x ) [ x ] [ x ] 1 1 x1 x V( ) ad V( x) V( x, x ) [ x ] [ x ] if x 1 1 Hece, V( x) is positive defiite

83 Example 5.3 AGAIN. V( x) V( x, x ) [ x ] [ x ] 1 1 V ( x, x ) V ( x, x ) V x x f x x f x x 1 1 ( 1, ) 1( 1, ) ( 1, ) x x 1 V ( x1, x) x 1 x 1 V ( x1, x) x x f x1 x f1( x1, x) 3 x x1 [ x] f( x1, x). V ( x, x ) V ( x, x ) V x x f x x f x x ( 1, ) x 1( x 1 1, ) x ( x ( 1[ 1, x] ) x x 1

84 Example 5.3 AGAIN. V ( x, x ) V ( x, x ) V x x f x x f x x ( 1, ) x 1( x 1 1, ) x ( x ( 1[ 1, x] ) x x 1 Hece,.. V x x x x x x x 3 ( 1, ) 1 ( 1 [ ] ) V( x, x ) x x x x [ x ] [ x ] V( x) [ x ] for all xh R ad Theorem L1 implies that 4 x e is stable x?? IS ASYMPTOTICALLY STABLE?? e

85 Example 5.3 AGAIN x?? IS ASYMPTOTICALLY STABLE?? e. 4. ( 1, ) [ x] 4 V x x. BUT. 4 V(1,) [] SO V(,) [] V( x) for all x H, x Theorem L does ot apply APPLY LaSALLE s Theorem

86 Example 5. AGAIN q= -.5 V( x) V( x, x ) [ x ] [ x ] 1 1 H ˆ ( x, x ) : V ( x). 1 1

87 Example 5.3 AGAIN Ĥ E x H ˆ : V( x) E x. 4 ( x1, x) : [ x]

88 Example 5.3 AGAIN d dt x1( t) x( t) 3 x( t) x1 ( t) 3[ x( t)] x e x1 x x E x 4 ( x1, x) : [ x] M { x } e x 1 IF x( t), the x ( t) x ( t) [ x ( t)] 1 x ( t) 1 3

89 Example 5.3 AGAIN x E x 4 ( x1, x) : [ x] M { x } e x 1 M EHˆ is LARGEST ivariat subset of E Hece LaSalle s Ivariace Theorem Implies x( tt ;, x ) M = x = x e e IS ASYMPTOTICALLY STABLE

90 Example 4. AGAIN x ( t) x ( t) 1 x ( t) qx ( t) x ( t) [ x ( t)] f x x 1 [ ] 3 x qx x x 1 1 x 1 1 x ( q [ x ] ) x or ( q [ x ] ) 1 1

91 Example 4. AGAIN f x 1 [ ] 3 x qx x x 1 1 x J x f ( x) J x f x 1 1 3[ ] x q x 1 1 q x e x 1 x

92 Example 4.: q AGAIN J x f ( x) J x f x 1 1 3[ ] x q x 1 1 x e x 1 x J f( x ) J x e x f 1 q 1

93 Example 4.: q < AGAIN 1 1 J ( f ( ) x q 1 q 1 q IN ALL CASES WHEN q real( ) ad real( ) 1 x e Theorem S1 IMPLIES is asymptotically stable

94 Example 4.: q > Also, we foud that IN ALL CASES WHEN q Theorem S1 IMPLIES xe is asymptotically stable

95 Example 4.: q = f x x x 1 x qx x [ x ] 3 x [ x ] x e x1 x J x f ( ) x e 1 1 Jxf q 1 1

96 Example 4.: q = J x 1 f( x ) Jxf e det ( 1)

97 Example 4.: q = f x x x 1 x qx x [ x ] 3 x [ x ] H R is a ope set V ( x) V ( x, x ) [ x ] [ x ] V ( ) ad V ( x) V ( x, x ) [ x ] [ x ] if x Hece, V( x) is positive defiite

98 Example 4.: q =. V ( x) V ( x, x ) [ x ] [ x ] V ( x, x ) V ( x, x ) V x x f x x f x x 1 1 ( 1, ) 1( 1, ) ( 1, ) x x 1 V ( x1, x) x 1 [ x ] V ( x1, x) x 1 x f x1 x f1( x1, x) 3 x [ x1 ] x f( x1, x). V (, x ) V ( x, x ) ( 1, ) [ x1 ] 1( x 1, ) x ( x1 1, x ) x x 1 V x x f x x ( f [ x] x )

99 Example 4.: q =. V (, x ) V ( x, x ) ( 1, ) [ x1 ] 1( x 1, ) x ( x1 1, x ) x x 1 V x x f x x ( f [ x] x ). V( x, x ) [ x ] x [ x ] x [ x ] [ x ] Hece,. V( x) [ x ] for all xh R ad Theorem L1 implies that x e is stable x?? IS ASYMPTOTICALLY STABLE?? e

100 Example 4.: q = d dt x1( t) x( t) 3 x( t) [ x1 ( t)] x( t) x e x1 x x E x ( x1, x) : [ x] M { x } e x 1 IF x( t), the x ( t) [ x ( t)] x ( t) 3 1 x ( t) 1

101 Example 4.: q = x E x 4 ( x1, x) : [ x] M { x } e x 1 M EHˆ is LARGEST ivariat subset of E Hece LaSalle s Ivariace Theorem Implies x( tt ;, x ) M = x = x e e IS ASYMPTOTICALLY STABLE

102 Bifurcatio Diagram: Example 4. R x 1 q STABLE x STABLE LaSalle s Ivariace Theorem Implies x STABLE x UNSTABLE q q? EXPONENTIALLY??STABLE?

103 Example 4.1 q < x( t) y( t) y( t) qx( t) ([ x( t)] [ y( t)] 1) y( t) d dt x1( t) x( t) x( t) qx1 ( t) ([ x1 ( t)] [ x( t)] 1) x( t) f x1 x x qx1 ([ x1 ] [ x] 1) x x x 1

104 Example 4.1 q = -1 q 1 f1( x, y) y ( xy, ) : f( x, y) qx ( x y 1) y ( x, y) : f ( x, y) y 1 ( x, y) : f( x, y) qx ( x y 1) y

105 Example 4.1 q = -1 q 1 x1 () t si( t) x() t cos( t) ([ x ( t)] [ x ( t)] 1) 1 PERIODIC SOLUTION

106 Example 4.1 q = -1 x1 () t si( t) x() t cos( t) q 1

107 Example 4.1 q = -1 q 1 LIMIT CYCLE

108 Example 4.1 q = -1 d dt x1( t) x( t) x( t) x1 ( t) ([ x1 ( t)] [ x( t)] 1) x( t) f x1 x x x1 ([ x1 ] [ x] 1) x x x 1 V ( x) V ( x, x ) ([ x ] [ x ] 1)

109 Example 4.1 q = -1 V ( x1, x) x 1 V ( x) V ( x, x ) ([ x ] [ x ] 1) x ([ x ] [ x ] 1) V ( x1, x) x x ([ x ] [ x ] 1) 1 f x1 x x x1 ([ x1 ] [ x] 1) x. V ( x, x ) V ( x, x ) V ( x, x ) f ( x, x ) ( f ( x,([ x )] [ ] 1) ) x1 ([ x1 ] [ x] 1) x 1 1 x x 1 x 1 1 x x ([ x1 ] [ x] 1) x x 1 V( x, x ) x x ([ x ] [ x ] 1) x x ([ x ] [ x ] 1) ([ x ] [ x ] 1) x

110 . Example 4.1 q = -1 V( x, x ) x x ([ x ] [ x ] 1) x x ([ x ] [ x ] 1) ([ x ] [ x ] 1) x V( x, x ) ([ x ] [ x ] 1) x 1 1 E x H ˆ : V( x) E x ( x, x ) Hˆ : ([ x ] [ x ] 1) x. 1 1 E x ( x 1, x ˆ ˆ ) H : ([ x1 ] [ x] 1) x ( x1, x) H : x E x ( x, x ) H ˆ : ([ x ] [ x ] 1) E x ( x, x ) Hˆ : x 1 E E1 E WHAT IS Ĥ

111 Example 4.1 q = -1 V ( x) V ( x, x ) ([ x ] [ x ] 1) E 1 M { xe } E M { x= [ x, x ] :([ x ] [ x ] 1) } 1 1 T 1 H ˆ ( x, x ) : V ( x) 1

112 Example 4.1 q = -1 q 1 LIMIT CYCLE H ˆ ( x, x ) : V ( x) 1

113 Bifurcatio Theory: 1D 3 f ( t, x, q) qx [ x] x( t) qx( t) [ x( t)] 3 x e = q (1/) x e = -q (1/) q < q = q > x e = x e = x e = Supercritical Pitchfork Bifurcatio

114 Bifurcatio Theory: 1D R 1 x 1 1/ [ q] STABLE x STABLE x UNSTABLE q STABLE x [ q] 1/

115 Bifurcatio: 1D 3 5 x( t) qx( t) [ x( t)] [ x( t)] 3 5 f ( x, q) qx [ x] [ x] x 1 x4 x 1 x

116 Bifurcatio: 1D 3 5 x( t) qx( t) [ x( t)] [ x( t)] 3 5 f ( x, q) qx [ x] [ x] x 5 x x4 x 1 x 3 x 5 x x x 4 1 x 3

117 Bifurcatio: 1D 3 5 x( t) qx( t) [ x( t)] [ x( t)] 3 5 f ( x, q) qx [ x] [ x] x x 5 1 x 3 x 5 x3 x 1

118 Bifurcatio: 1D Subcritical Pitchfork Bifurcatio x 3 R 1 STABLE x x UNSTABLE STABLE x UNSTABLE q = -.5 UNSTABLE q x 4 x 5 STABLE

119 Bifurcatio: 1D q =

120 Bifurcatio: 1D q =

121 Bifurcatio: 1D R 1 STABLE x 3 x UNSTABLE x STABLE x UNSTABLE q = -.5 x UNSTABLE 4 q x 5 STABLE Subcritical Pitchfork Bifurcatio = BIG JUMP!!!

122 Typical Hopf Bifurcatio, b ad q x t q x t y t x t y t x t ( ) ( [ ( )] [ ( )] ([ ( )] [ ( )] ) ) ( ) ( b([ x( t)] [ y( t)] )) y( t) y t q x t y t x t y t y t ( ) ( [ ( )] [ ( )] ([ ( )] [ ( )] ) ) ( ) +( b([ x( t)] [ y( t)] )) x( t) f x ( q x y ( x y ) ) x ( b( x y )) y y ( q x y ( x y ) ) y+( b( x y )) x OR

123 Typical Hopf Bifurcatio ( q x y ( x y ) ) ( b( x y )) x ( b( x y )) ( q x y ( x y ) ) y det ( q x y ( x y ) ) ( b( x y )) ( b( x y )) ( q x y ( x y ) ) ( q x y ( x y ) ) ( b( x y )), b ad q ( q x y ( x y ) ) ( b( x y )) x e x y

124 Typical Hopf Bifurcatio f x ( q x y ( x y ) ) x ( b( x y )) y y ( q x y ( x y ) ) y+( b( x y )) x J x f ( ) x e J x f q q 1 q q 1 q q q q det ( q)

125 Typical Hopf Bifurcatio q q det ( q) ( q) 1 qi qi Re( ) i q q q xe IS STABLE xe IS UNSTABLE

126 Polar Coordiates x t q x t y t x t y t x t ( ) ( [ ( )] [ ( )] ([ ( )] [ ( )] ) ) ( ) ( b([ x( t)] [ y( t)] )) y( t) y t q x t y t x t y t y t ( ) ( [ ( )] [ ( )] ([ ( )] [ ( )] ) ) ( ) +( b([ x( t)] [ y( t)] )) x( t) x( t) r( t)cos( ( t)) y( t) r( t)si( ( t)) x( t) r( t)cos( ( t)) r( t)si( ( t)) ( t) y( t) r( t)si( ( t)) r( t)cos( ( t)) ( t)

127 Polar Coordiates rt () x t q x t y t x t [ ( )] y t x t 4 ( ) ( [ ( )] [ ( )] ([ ( )] [ ( )] ) ) ( ) ( b([ x( t)] [ y( t)] )) y( t) rt r( t)cos( ( t)) rt () r( t)si( ( t)) y t q x t y t x t [ rt ( )] y t y t 4 ( ) ( [ ( )] [ ( )] ([ ( )] [ ( )] ) ) ( ) r t q r t r t r t rt () rt () ( b([ x( t)] [ y( t)] )) x( t) 4 ( ) ( [ ( )] ([ ( )] ) ( ) ( t) ( b[ r( t)] ) r( t)cos( ( t))? HOW? WORK BACKWARDS r( t)si( ( t))

128 Polar Coordiates r t q r t r t r t 4 ( ) ( [ ( )] ([ ( )] ) ( ) ( t) ( b[ r( t)] ) x( t) r( t)cos( ( t)) r( t)si( ( t)) ( t) x t q r t r t r t t 4 ( ) ( [ ( )] ([ ( )] ) ( )cos( ( )) r( t)si( ( t)) ( t) x t q r t r t r t t 4 ( ) ( [ ( )] ([ ( )] ) ( )cos( ( )) r t t b r t ( )si( ( ))( [ ( )] )

129 Polar Coordiates x t q r t r t r t t 4 ( ) ( [ ( )] ([ ( )] ) ( )cos( ( )) r t t b r t ( )si( ( ))( [ ( )] ) r x y x r y r ( ) cos( ) si( ) x t q x t y t x t y t r t t ( ) ( ([ ( )] [ ( )] ) (([ ( )] [ ( )] ) ) ( )cos( ( )) r t t b x t y t ( )si( ( ))( ([ ( )] [ ( )] )) x t q x t y t x t y t x t ( ) ( ([ ( )] [ ( )] ) (([ ( )] [ ( )] ) ) ( ) y t b x t y t ( )( ([ ( )] [ ( )] ))

130 Polar Coordiates x t q x t y t x t y t x t ( ) ( ([ ( )] [ ( )] ) (([ ( )] [ ( )] ) ) ( ) y t b x t y t ( )( ([ ( )] [ ( )] )) SIMILARLY y t q x t y t x t y t y t ( ) ( [ ( )] [ ( )] ([ ( )] [ ( )] ) ) ( ) +( b([ x( t)] [ y( t)] )) x( t) r t q r t r t r t 4 ( ) ( [ ( )] ([ ( )] ) ( ) ( t) ( b[ r( t)] )

131 Polar Coordiates r t q r t r t r t 4 ( ) ( [ ( )] ([ ( )] ) ( ) ( t) ( b[ r( t)] ) r( t) qr( t) [ r( t)] [ r( t)] ( t) ( b[ r( t)] ) qr r r qr r r 3 5

132 Recall 1D example 3 5 x( t) qx( t) [ x( t)] [ x( t)] 3 5 f ( x, q) qx [ x] [ x]

133 Polar Coordiates r( t) qr( t) [ r( t)] [ r( t)] ( t) ( b[ r( t)] ) 3 5 qr r r 3 5 qr r r 3 5

134 q = -.5

135 q = -.5

136 q = -.

137 q = -.5

138 q =

139 q =.

140 Hopf Bifuricatio R x s_lc () t STABLE LIMIT CYCLE x q = -.5 x us_lc STABLE x 4 () t UNSTABLE LIMIT CYCLE x UNSTABLE q Subcritical Hopf Bifurcatio = BIG JUMP!!!

141 Itroductio to Dyamical Systems Basic Ideas

142 Dyamical Systems (Σ) x( t) f x( t) (IC) x() zr x( t) x( t, z) R for all t DEFINE S( t) : R R BY S( t) z x( t, z) ( i) S() I : R R ( ii) S( t ) zs( t) S( ) z ( iii) lim S( t) z t z

143 Dyamical Systems A FAMILY OF CONTINUOUS OPERATORS S() z x(, z) z = Iz lim S ( t ) z lim x ( t, z) x (, z) z S( t) : R R, t SATISFYING (i), (ii) ad (iii) ABOVE IS CALLED A DYNAMICAL SYSTEM ON t t R WHAT ABOUT (ii)? ( ii) S( t ) zs( t) S( ) z

144 Dyamical Systems time t x( t, z) x( t, x(, z)) z x(, z) S( t ) z x( t, z) x( t, x(, z)) S( t) S( ) z

145 Semi-groups A FAMILY OF CONTINUOUS OPERATORS SATISFYING S( t) : R R, t ( i) S() I : R R ( ii) S( t ) zs( t) S( ) z ( iii) lim S( t) z t z IS CALLED A SEMI-GROUP ON R

146 Ivariat Sets (Σ) St ( ) : x 3 R z Positively Ivariat Set z M M z x 1 S( t) z x( t; z) M for all t

147 Ivariat Sets (Σ) St ( ) : x 3 R z Negatively Ivariat Set z M M z x 1 S( t) z x( t; z) M for all t

148 Ivariat Sets (Σ) St ( ) : x 3 R z Ivariat Set z M M z x 1 S( t) z x( t; z) M for all t ad S( t) z x( t; z) M for all t

149 Covergece to a Set x 3 R S( t) z x( t, z) M as t M x 1 For ay T if T t T there is a such that, the there is a poit p M with S( t) z p <

150 Covergece to a Set x 3 p St ( ) z M p 1 M R St ( 1) z M S() t z R x 1 ST ( ) z x

151 Orbits & Limit Sets z R Give, the (positive) orbit through is the set ( z) S( t) zr : t z IF Stz () EXISTS FOR ALL t the (egative) orbit through z is the set ( z) S( t) zr : t ad the orbit through z is the set ( z) S( t) z R : t (, )

152 Give z R -Limit Sets, a poit p is a omega limit poit of () z T t T if for each ad every there is a such that S()z t p < - limit set of z ( z) p: p is a -limit poit of zr EQUIVALENTLY ( z) St ( ) z s ts

153 Give z R -Limit Sets, a poit p is a alpha limit poit of () z T if for each ad every there is a such that S()z t p < - limit set of z ( z) p: p is a -limit poit of zr t T EQUIVALENTLY 1 ( z) [ S( t)] z s ts

154 Limit Sets Give Z R, - limit set of Z ( Z) St ( ) Z s ts - limit set of Z S t 1 Z s ts ( Z) [ ( )]

155 Ivariat Set & Attractors Z R A set semi-group, is (fuctioal) ivariat for the S( t) : R R, t if S( t) ZZ, t A R B R St () ( AB, ) A set attracts a set uder the semi-group if, for ay there exists a t t such that S( t) BN( A, ) for all t t

156 Attractors B N( A, ) A StB () St ( ) B A attractor is a set (i) ad A A A R is compact ad ivariat satisfyig (ii) attracts a ope eighborhood of A U

157 Global Attractor A global attractor is a set (i) ad (ii) A A is a attractor A R attracts all bouded sets A semi-group satisfyig B R S( t) : R R, t said to be dissipative if there is a bouded positively ivariat set bouded sets B R W R is that attracts all

158 (Σ) Dissipative System x( t) f x( t) (IC) x() x( t) x( t, z) R for all t zr DEFINE S( t) : R R BY S( t) z x( t, z) Theorem D1. If costats, such that the the dyamical system defied by with absorbig set f : R R is smooth ad there exist f ( x), x x for all x R, St W B(, / ) x : x / is dissipative

159 Dissipative System Theorem D: If S( t) : R R, t is a dissipative dyamical system with absorbig set W, the is a global attractor. ( ) St ( ) s ts A W W A. M. Stuart ad A. R. Humphries, Dyamical Systems ad Numerical Aalysis, Cambridge Uiversity Press, ad Jack K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys 5, AMS Publicatios,Providece, RI 1989.