Intermediate Differential Equations. Stability and Bifurcation II. John A. Burns

Transcription

1 Intermediate Differential Equations Stability and Bifurcation II John A. Burns Center for Optimal Design And Control Interdisciplinary Center for Applied Mathematics Virginia Polytechnic Institute and State University Blacksburg, Virginia MATH FALL 1

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

3 Autonomous Systems (IVP) { (Σ) (IC) x( t) f x( t), q x( t) x R n n m n f ( xq, ) : D R R R Let x e = x e (q) be an 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 an 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 n R x ( ) 4 q x ( ) 3 q NON-ISOLATED EQUILIBRIUM CAN NOT BE ASYMPTOTICALLY STABLE

5 First Order Linear 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 > : Unstable 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 Infected Removed

13 Epidemic Models SIR Models (Kermak McKendrick, 197) Susceptible Infected 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 constant

14 SIR Models d R ( t ) I ( t ) and S ( t ) I ( t ) R ( t ) N constant 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) and 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 hence x 1 x x 1 can be any 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 Fundamental Stability Theorem Let 1,, 3, n be the eigenvalues 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,,. n, then x e is an asymptotically stable equilibrium for the non-linear system x( t) f x( t). In particular, there exist > such that if x() x, then lim ( t). e t x x e

20 Non-Stability Theorem Theorem S: If there is one eigenvalue p such that Re( p ) >, then x e is an unstable equilibrium for the non-linear system x t f x t ( ) ( ). The two theorems above may be found in: Richard K. Miller and Anthony N. Michel, Ordinary Differential Equations, Academic Press, 198. (see pages 58 53) and Earl A. Coddington and Norman Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, (see pages )

21 Critical Case If there is one eigenvalue p of [ J f x ( x, q e )] such that Re( p ) =, then x e the linearization theorems do not apply and other methods must be used to determine the stability properties of the equilibrium for the nonlinear 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 and [ 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 linearization 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 and Theorem S do not 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 linearization Theorem J f ( ) x [ ] 3 3 qx 1 x, x 1

26 Example det I J( ) det( ) det ( 3) ( 3) and 3 1 Re( ) Linearization Theorems do not apply BUT

27 Example 5. ZOOM IN

28 Example 5. LOOKS ASYMPTOTICALLY STABLE

29 Isolated Equilibrium H R n x 3 x e n R an open set x e x 1 xˆe n H R and x H, for x V( x) : H R R e n e

30 Lyapunov Functions n H R e an open set x e x x V( x) : H R R n V( x) V( x, x,... x ) 1 n If V and when ( ) V ( x), x, then V ( x) is said to be positive definite

31 Lyapunov Functions d dt x() t f f f 1 n ( x) ( x) ( x) R n x x x x R 1 n T n x n 1 V( ) : H R R V( x) V( x, x,... x ) n We define the function. V( x) : H R R n by. V ( x) V ( x) V ( x) V ( x) f1( x) f( x)... fn( x) x x x 1 n

32 Lyapunov Functions V( x) : H R R n is called a Lyapunov function for the equilibrium (Σ) x e of the system x( t) f x( t) if and ( i) V ( x) is positive definite in H. ( ii) V ( x) for all x H

33 Lyapunov Theorems Theorem L1. If there exists a Lyapunov function for the equilibrium x of the system then the equilibrium e (Σ) x( t) f x( t), x e is stable. Theorem L. If there exists a Lyapunov function for the equilibrium x of the system and. e (Σ). x( t) f x( t), V( ) and V( x) for all x H, x, then 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 an open set V( x) V( x, x ) [ x ] [ x ] 1 1 V( ) and V( x) V( x, x ) [ x ] [ x ] if x 1 1 Hence, V( x) is positive definite

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 ] ) and if, then 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 an open set V ( x) V ( x, x ) [ x ] [ x ] V ( ) and V ( x) V ( x, x ) [ x ] [ x ] if x Hence, V( x) q is positive definite 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 ] Hence,. V( x) 6[ x ] for all xh R and 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 not 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 linearization Theorem e J f ( ) x [ ] 1 x x, x 1

42 Example det I J( ) det( ) det i Re( i ) for i 1, Linearization Theorems do not 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 an open set V( x) V( x, x ) [ x ] [ x ] 1 1 x1 x V( ) and V( x) V( x, x ) [ x ] [ x ] if x 1 1 Hence, V( x) is positive definite

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 Hence,.. 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 and 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 not apply NEED A BETTER THEOREM

49 Positively Invariant Sets (Σ) x( t) f t, x( t) x 3 n R (IC) x( t) x R n 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 ( ), then 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 n x n R x n R x t t x ( ;, ) R n

54 Trajectories & Limit Sets n Given x R, the (positive) trajectory through x Is the set x( t; t, x ) R : t t n n Given x R, the (negative) trajectory through x Is the set x( t; t, x ) R : t t n n Given x R, the trajectory through x Is the set x x n ( t; t, ) R : t (, )

55 -limit Sets x R n Given, a point p belongs to the omega limit set (-limit set) of x t t x ( ;, ) R if for each and every there is a such that T x( t; t, x ) p < n t t T ( x ) p : p is an -limit point of x n R

56 -limit Sets ( x ) p : p is an -limit point of x ( x ) p : there is a sequence t with lim x( t ; t, x ) p n R k k k Theorem LIM1. If t t, then x t t x ( ;, ) R ( x ) is bounded for is a non-empty, compact and connected positively invariant set. xˆ ( x ) n xˆ( t; tˆ, xˆ ) ( x ) for all t tˆ

57 Convergence to a Set x 3 n R x( t) x( t; x ) M as t M x 1 For any there is a T T ( ) t such that if t T( ), then there is a point p M with x( t; x ) p <

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

59 Example NS

60 Example NS M

61 Example again 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 again a =1 >

63 Example again a =1 > M

64 Example again a =1 > M

65 Example again a =1 > LIMIT CYCLE M

66 LaSalle Theorems Theorem LIM. If t t i.e., then x( tt ;, x ) x t t x ( ;, ) R x( tt ;, x ) ( x ) approaches its -limit set. n is bounded for Theorem LIM3. If t t ( ) M x x t t x ( ;, ), and then R n is bounded for x t t x ( ;, ) M

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

68 LaSalle s Invariance Theorem Let Hˆ R n be a bounded closed positively invariant set ( ) : ˆ n V x H R R ( i) V ( x) v >, for all x Hˆ and. min ( ii) V ( x) for all x Hˆ E x H ˆ : V ( x). E Ĥ M E Hˆ is LARGEST invariant subset of E

69 LaSalle s Invariance Theorem Theorem LaSalle IP: If function satisfying (i) and (ii) above and M E is the largest invariant subset of ˆ. E x H : V ( x), then for each x x( ;, ) tt x Hˆ Ĥ the trajectory approaches M. ( ) : ˆ n 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 an open set V ( x) V ( x, x ) [ x ] [ x ] V ( ) and V ( x) V ( x, x ) [ x ] [ x ] if x Hence, V( x) q is positive definite 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 ] Hence,. V( x) 6[ x ] for all xh R and 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 not 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 Invariant Sets in 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), then 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 invariant subset of E Hence LaSalle s Invariance 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 linearization 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, Linearization Theorems do not 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 an open set V( x) V( x, x ) [ x ] [ x ] 1 1 x1 x V( ) and V( x) V( x, x ) [ x ] [ x ] if x 1 1 Hence, V( x) is positive definite

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 Hence,.. 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 and 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 not 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), then 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 invariant subset of E Hence LaSalle s Invariance 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( ) and real( ) 1 x e Theorem S1 IMPLIES is asymptotically stable

94 Example 4.: q > Also, we found 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 an open set V ( x) V ( x, x ) [ x ] [ x ] V ( ) and V ( x) V ( x, x ) [ x ] [ x ] if x Hence, V( x) is positive definite

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 ] Hence,. V( x) [ x ] for all xh R and 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), then 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 invariant subset of E Hence LaSalle s Invariance Theorem Implies x( tt ;, x ) M = x = x e e IS ASYMPTOTICALLY STABLE

102 Bifurcation Diagram: Example 4. R x 1 q STABLE x STABLE LaSalle s Invariance 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 sin( t) x() t cos( t) ([ x ( t)] [ x ( t)] 1) 1 PERIODIC SOLUTION

106 Example 4.1 q = -1 x1 () t sin( 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 Return to Bifurcation Theory

114 Bifurcation 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 Bifurcation

115 Bifurcation Theory: 1D R 1 x 1 1/ [ q] STABLE x STABLE x UNSTABLE q STABLE x [ q] 1/

116 Bifurcation: 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

117 Bifurcation: 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

118 Bifurcation: 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

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

120 Bifurcation: 1D q =

121 Bifurcation: 1D q =

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

123 Typical Hopf Bifurcation, b and 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

124 Typical Hopf Bifurcation ( 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 and q ( q x y ( x y ) ) ( b( x y )) x e x y

125 Typical Hopf Bifurcation 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)

126 Typical Hopf Bifurcation q q det ( q) ( q) 1 qi qi Re( ) i q q q xe IS STABLE xe IS UNSTABLE

127 Polar Coordinates 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)sin( ( t)) x( t) r( t)cos( ( t)) r( t)sin( ( t)) ( t) y( t) r( t)sin( ( t)) r( t)cos( ( t)) ( t)

128 Polar Coordinates 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)sin( ( 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)sin( ( t))

129 Polar Coordinates r t q r t r t r t 4 ( ) ( [ ( )] ([ ( )] ) ( ) ( t) ( b[ r( t)] ) x( t) r( t)cos( ( t)) r( t)sin( ( t)) ( t) x t q r t r t r t t 4 ( ) ( [ ( )] ([ ( )] ) ( )cos( ( )) r( t)sin( ( t)) ( t) x t q r t r t r t t 4 ( ) ( [ ( )] ([ ( )] ) ( )cos( ( )) r t t b r t ( )sin( ( ))( [ ( )] )

130 Polar Coordinates x t q r t r t r t t 4 ( ) ( [ ( )] ([ ( )] ) ( )cos( ( )) r t t b r t ( )sin( ( ))( [ ( )] ) r x y x r y r ( ) cos( ) sin( ) x t q x t y t x t y t r t t ( ) ( ([ ( )] [ ( )] ) (([ ( )] [ ( )] ) ) ( )cos( ( )) r t t b x t y t ( )sin( ( ))( ([ ( )] [ ( )] )) x t q x t y t x t y t x t ( ) ( ([ ( )] [ ( )] ) (([ ( )] [ ( )] ) ) ( ) y t b x t y t ( )( ([ ( )] [ ( )] ))

131 Polar Coordinates 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)] )

132 Polar Coordinates 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

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

134 Polar Coordinates r( t) qr( t) [ r( t)] [ r( t)] ( t) ( b[ r( t)] ) 3 5 qr r r 3 5 qr r r 3 5

135 q = -.5

136 q = -.5

137 q = -.

138 q = -.5

139 q =

140 q =.

141 Hopf Bifurication 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 Bifurcation = BIG JUMP!!!