Theory of Ordinary Differential Equations Stability and Bifurcation I John A. Burns Center for Optimal Design And Control Interdisciplinary Center for Applied Mathematics Virginia Polytechnic Institute and State University Blacksburg, Virginia 46-53 MATH 545 - FALL
Initial Value Problem (IVP) { (Σ) (IC) ( t) f t, ( t), q ( t ) f ( t,,q) : D A solution to the initial value problem (IVP) is a vector valued differentiable function ( t) : ( a, b) defined on a connected interval (a, b) such that () t ( a, b), () ( t ), n n n m n (3) ( t, ( t)) D, for all t ( a, b), (3) ( t) satisfies ( ) for all t ( a, b).
Consider Global in Time Solutions t > d ( t ) ( t ), () dt d ( t ) ( t ), () dt t () t t () t l ( t), t r ( t), t t t
Consider Global in Time Solutions t > l () 5 l ( t), t t 5 r () -5 - r ( t), t t -5 - -3 - - 3 4
Stability of Solutions and Equilibrium
Stability of Solutions (Σ) ( t) f t, ( t), q f( t,, q) :[, ) n n m n ASSUME SOLUTIONS EXIST FOR ALL t > t
Equilibrium (SOLUTIONS) (Σ) ( t) f t, ( t), q ( t) ( t) = ( q) a constant vector (depends on q) e e e e ( t) f t,, q) e TO FIND EQUILIBRIUM: FIND ALL VECTORS f t,, ) e q ( t) = ( q) e e e e SATISFYING
( t) 3 ( t) Eamples f t, 3 e ( t) t( t) f t, t e t t t ( ) ( )[4 [ ( )] ] f t, [4 ],, 3 e e e
Eample 4. a d dt ( t) ( t) ( t) a( t) ([ ( t)] [ ( t)] ) ( t) f a ([ ] [ ] ) e
Eample 4. ( t) ( t) ( t) a ( t) ( t) [ ( t)] 3 f a [ ] 3 ( a [ ] ) or ( a [ ] )
Eample 4. ( a [ ] ) or ( a [ ] ) a a e a a e a e e
Eample 4. f(, y) y ( y, ) : 3 f(, y) a y (, y) : f (, y) y 3 (, y) : f(, y) a y (, y) : y a 3
Eample 4.3 ( t) ( t) ( t) ( t)( a [ ( t)] ) f ( a [ ] ) ( a [ ] ) or ( a [ ] )
Eample 4.3 ( a [ ] ) a a > a
Eample NS d t t y t t y t y t dt ( ) ( ) 3 ( ) ( [ ( )] [ ( )] ) ( ) d y t y t t y t t dt ( ) ( ) ( [ ( )] [ ( )] ) ( ) FIND ALL OF THE EQUILIBRIUM HOMEWORK
) ( R p p ) ( ) ( ) ( t p t p K r t p dt d LE ), ), ( ( ) ( ) ( ) ( K r t p f t p t p K r t p dt d p p K r K r p f ),, ( Logistics Equation
Analytical Solution 9 Kp p ( t, r, K) p K p e r t K 8 7 6 5 4 3 5 5
Initial p : < p <, r.963, K 9, 5 4.8 p, p 5,.6.4..8.6.4. p p 5 75 p p p 5 5 5 K
Equilibrium States LE d dt p( t) r p( t) p( t) K EQUILIBRIUM STATES ARE CONSTANT SOLUTIONS p ( t) p( t) p e a constant p ( t) r K p p e e p ( t) p e or p e K p e ( t) p e p e ( t) p e K
Equilibrium States p e or UNSTABLE p e K STABLE 4.8.6.4. K.8.6.4. 5 5 5
First Order Linear ( t) a( t) at () t e 5 ( t) a( t) f ( ( t), a) 4.5 4 3.5 3 a =. a =.5.5.5.5 a = a = -.5 a = -..5.5.5
Equilibrium e =, a < : Stable a f (, a) a < e.8.6.4. -. e -.4 -.6 -.8 -.5.5.5 3 3.5 4 4.5 5
Equilibrium e =, a > : Unstable a f (, a) a > e 5 5 e -5 - -5.5.5.5
Equilibrium e =, a = : Stable a f (, a) a = e.8.6.4. -. -.4 -.6 e e.5. e..5 e e -.8 -.5.5.5 3 3.5 4 4.5 5
Let φ( ) be a given solution to the differential equation. The solution φ( ) is called stable in the sense of Lyapunov (or simply stable) if for any any ε > and any t [, + ), there is a (ε, t ) > such that if (t) is any solution with initial condition satisfying then We say that φ( ) is uniformly stable if φ( ) is stable and (ε, t )= (ε ) can be taken to be independent of t. The solution φ( ) is said to be asymptotically stable if. The solution φ( ) is stable and lim t () ( t).. T t Stability ( t) f t, ( t) ( t ) ( t ) (, t ), t () ( t), t t.
Stable Solution (t) (t) t t
Asymptotically Stable Solution STABLE and lim ( t) ( t) t (t) (t) t t
Stable and Asymptotically Stable (t) (t) t t (t) (t) t t
Stability WHY SUCH A GENERAL DEFINITION? WHY NOT JUST LIMIT STABILITY TO EQUILIBRIUM? d dt ( t) ( t) ( t) ( t) EQUILIBRIUM e PERIODIC SOLUTIONS () t sin( t) r () t cos( t)
Predator - Prey Models Vito Volterra Model (95) Alfred Lotka Model (96) t () y() t ab,, c, d d ( t ) ( t ) ab y ( t ) dt d dt THINK OF SHARKS AND SHARK FOOD Numbers of prey Numbers of predators Parameters y( t) y( t) c d( t) t ()
Symplectic Methods > WRONG Eplicit Euler Implicit Euler Symplectic WRONG CORRECT > >
Eample 4. a >... AGAIN ( t) y( t) y( t) a( t) ([ ( t)] [ y( t)] ) y( t) d dt ( t) ( t) ( t) a( t) ([ ( t)] [ ( t)] ) ( t) f a ([ ] [ ] )
Eample 4. a = a ( y, ) : f(, y) y f(, y) a ( y ) y (, y) : f (, y) y (, y) : f(, y) a ( y ) y
Eample 4. a =... AGAIN a () t sin( t) () t cos( t) ([ ( t)] [ ( t)] ) PERIODIC SOLUTION
Eample 4. a =... AGAIN () t sin( t) () t cos( t) a
Eample 4. a =... AGAIN a LIMIT CYCLE
Lorenz System d ( t ) ( ( t ) ( t )) dt d q ( t ) ( t ) ( t )( 3( t )) dt d 3( t ) 3( t ) ( t ) ( t ) dt f(,, 3, q, q, q3) ( ) f (,,, q, q, q ) ( ) f ( ( t), q) 3 3 3 f3(,, 3, q, q, q3) 3 f(,, 3, q) ( ) f ( ( t), q) f (,,, q) ( ) 3 3 f3(,, 3, q) 3
Lorenz System f(,, 3, q) ( ) f (, q) f (,,, q) ( ) 3 3 f3(,, 3, q) 3 f (,,, q) ( ) 3 f (,,, q) ( ) 3 3 f(,, 3, q) f(,, 3, q) f (,,, q) 3 f(,, 3, q) 3 f(,, 3, q) 3 f (,,, q) 3 3
f3(,, 3, ) Lorenz System f (,,, q) 3 3 3 q f 3 (,,, ) f (,,, q) q 3 3 f(, q) f(, q) f(, q) 3 J (, q) f(, q) f(, q) f(, q) 3 f3(, q) f3(, q) f3(, q) 3 J(, q) 3
Lorenz System f(, q) f(, q) f(, q) 3 J(, q) f(, q) f(, q) f(, q) 3 3 f3(, q) f3(, q) f3(, q) 3 ALL ENTRIES ARE CONTINUOUS FOR ALL (,,, q, q, q ) 3 3 Theorem (n). IS OK
Lorenz System f(,, 3, q) ( ) f (, q) f (,,, q) ( ) Equilibrium 3 3 f3(,, 3, q) 3 ( ) ( ) 3 3 ( ) ( ) ( ) ( )
A 3D Eample ( t) Re ( t) d ( t ). Re ( t ) dt 3( t). 3( t) EQUILIBRIUM 3 e a b ( t) [ ( t)] [ ( t)] [ 3( t)] a c ( t) b c ( t) 3 THERE ARE MORE BUT MORE IMPORTANTLY.
Eample 4.4 ( t) [ ( t)] 3 f ( t, ) [ ] 3 e 3 ( t) [ ( t)], () t () t [ ]
Eample 4.4 t () t t () t SLOW DECAY
Eponential Decay ( t) a( t) () t e t t () t
Eponential Asymptotic Stable ( t) f t, ( t) Let φ( ) be a given solution to the differential equation. The solution φ( ) is said to be eponentially asymptotically stable if. The solution is asymptotically stable, and. There is a > such that for any >, there is a M() > such that if then ( t ) ( t ) t t M e t t ( t t ( ) ( ) ( ) ),. Uniform eponentially asymptotically stable is the same with. replaced by uniform asymptotically stable.
Stability of Equilibrium ( t) f ( t) f () t e e e ( t; ) e () t e e t? HOW DO WE KNOW IF e (t) IS ASYMPTOTICALLY STABLE?
Linear Constant Coefficient System ( t) A( t) A a a a a a a a a a,,, n,,, n n, n, n, n e n e When is the equilibrium e (t)= stable?
Constant Coefficient System (L) ( t) A( t) Let,, 3, n be the eigenvalues of A i.e. det( ) k I A i (Re( ), Im( ) ) k k k k k k k The following two theorems may be found in: Richard K. Miller and Anthony N. Michel, Ordinary Differential Equations, Academic Press, 98. (see pages 79 86)
Linear Stability Results Theorem: If Re( k ) < for all k=,,. n, then the zero equilibrium e (t) = of (L) is eponentially asymptotically stable. In particular, there eist M and > such that t ( t) Me, for all t. Theorem: If there is one eigenvalue p of A such that Re( p ) >, then the zero equilibrium e (t)= of (L) is unstable. In particular, for any > there eists and initial condition satisfying < such that the solution (t; ) to (L) satisfies () = and lim ( t; ) t
Linearization About Equilibrium ( t) f ( t) f ( ) : n n f f(,,... n ) f(,,... n ) n fn(,,... n ) is a C function d dt ( t) ( t) f( ( t), ( t),... n( t)) ( t) ( t) f( ( t), ( t),... n( t)) f n ( t) n ( t) fn( ( t), ( t),... n( t))
Linearization About Equilibrium f e e n T J f ( ) e f (,,... ) f (,,... ) f (,,... ) n f (,,... ) f (,,... ) f (,,... ) n f (,,... ) f (,,... ) f (,,... ) n n n n n n n n n n n n n J f ( ) e is called the Jacobian matri of f( ) evaluated at e
Linearization About Equilibrium J f ( ) J f( ) e e f ( ) f ( ) f ( ) n f ( ) f ( ) f ( ) n fn( ) fn( ) f ( ) n n e BASIC STABILITY RESULTS
Fundamental Stability Theorem Let,, 3, n be the eigenvalues of J f ( ) e, i.e. det( I J ( )) k f e i (Re( ), Im( ) ) k k k k k k k Theorem S: If Re( k ) < for all k=,,. n, then the equilibrium solution e() t e is an asymptotically stable equilibrium for the non-linear system ( t) f ( t). In particular, there eist > such that if (), then lim ( t). e t e
Non-Stability Theorem Theorem S: If there is one eigenvalue p such that Re( p ) >, then e() t e is an unstable equilibrium for the non-linear system ( t) f ( t). The two theorems above may be found in: Richard K. Miller and Anthony N. Michel, Ordinary Differential Equations, Academic Press, 98. (see pages 58 53) and Earl A. Coddington and Norman Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, 955. (see pages 34 3)
Eample 4.5 d dt ( t) ( t) 3 ( t) ( t) ( t) ( t) ( t) ( t) ( t) 3 f d dt f ( t) 3 ( t) ( t) ( t) ( t) ( t) ( t) ( t) 3
Eample 4.5 d dt ( t) 3 ( t) ( t) ( t) ( t) ( t) ( t) ( t) 3 f ( ) 3 or
Eample 4.5 J J f( ) J 3 f( ) f J 3 f J 3 f( ) f J 3.566.566 Theorem S IMPLIES is asymptotically stable Theorem S IMPLIES is not stable
Eample 4. a ( t) y( t) y( t) a( t) ([ ( t)] [ y( t)] ) y( t) d dt ( t) ( t) ( t) a( t) ([ ( t)] [ ( t)] ) ( t) f a ([ ] [ ] )
Eample 4. a f a ([ ] [ ] ) J f ( ) J f a [ ] 3[ ] J f ( ) J f a EIGENVALUES ARE GIVEN BY a det a
Eample 4. a 4a a implies 4 a is real and 4a 4a 4a 4 a implies 4 a is real and 4a 4a 4a
Eample 4. a a implies 4 a 4 i 4 i Re( ) Re( ) / Re( ) IN ALL CASES Theorem S IMPLIES is not stable
Eample 4.6 d ( t) [ ( t)] ( t) 3( t) ( t ) ( t ) ( t ) dt 3( t) [ ( t)] 3( t) f(,, 3) [ ] 3 f f(,, 3) 3 f3(,, 3) [ ] 3 and [ ] [ ] 3 [ ] [ ] 3 [ ]
Eample 4.6 and 3 [ ] [] 3 [ ] [ ] 8 6 EQUILIBRIUM 3 e EQUILIBRIUM 3 6 e
Eample 4.6 J f 3 4 EQUILIBRIUM 3 e EQUILIBRIUM 3 6 e f J 4 f J 4 6 8
Eample 4.6 4 f f J J 4 6 8.4856.48.75i.48.75i 3.7664.999i.7664.999i.538 Theorem S IMPLIES 3 e are both unstable 3 3 6 e
Eample 4. AGAIN ( t) ( t) ( t) a ( t) ( t) [ ( t)] 3 f a [ ] 3 ( a [ ] ) or ( a [ ] )
Eample 4. AGAIN or ( a [ ] ) a ( a [ ] ) a a a a
Eample 4. AGAIN f a [ ] 3 J f ( ) J f a 3[ ] a
Eample 4. AGAIN: a J f ( ) J f a 3[ ] J f ( ) J f a
Eample 4. AGAIN: a > f a [ ] 3 J f ( ) J f a 3[ ] a a
Eample 4. AGAIN: a > J f ( ) J f a 3[ ] J f ( ) J f a
Eample 4. AGAIN: a > J f ( ) J f a 3[ ] a J ( ) J f a f a
Eample 4. AGAIN: a > J f ( ) J f a 3[ ] a J f ( ) J f a a
Eample 4.: a <, =[ ] T J f ( ) J f a a a det ( ) a a a
Eample 4.: a <, =[ ] T 4( a) 4a 4 a implies 4 a is real and 4a 4a 4a Theorem S IMPLIES is asymptotically stable
Eample 4.: a <, =[ ] T a 4 implies 4a 4a 4a Theorem S IMPLIES is asymptotically stable
Eample 4.: a <, =[ ] T 4( a) 4a a 4 implies ( 4 a) and hence 4 a is comple ( 4 a) ( 4 a) ii i i i i real( ) real( )
Eample 4.: a <, =[ ] T IN ALL CASES WHEN a real( ) and real( ) Theorem S IMPLIES is asymptotically stable
Eample 4.: a >, =[ ] T ( t) ( t) ( t) a ( t) ( t) [ ( t)] 3 f a [ ] 3 a a
Eample 4.: a >, =[ ] T J f ( ) J f a a a det ( ) a a a
Eample 4.: a >, =[ ] T a 4( a) 4a 4a 4a 4a Theorem S IMPLIES is unstable
Eample 4.: a >, =[ a ] T J f ( ) J f a 3[ ] a J f a a a a
Eample 4.: a >, =[ a ] T a det ( ) a a 4( a) 8a 8 a implies 8 a is real and 8a 8a 8a
Eample 4.: a >, =[ a ] T 4( a) 8a a implies 8 a = 8 8a 8a 8 a implies ( 8 a) 8 a is comple so ( 8 a) ( 8 a) ii
Eample 4.: a >, =[ i i i i real( ) real( ) IN ALL CASES WHEN a real( ) and real( ) Theorem S IMPLIES a is asymptotically stable a a ] T SAME
Eample 4.: a =, =[ ] T 4( a) 4a a Theorems S and S FAIL Stability of is not known
Bifurcation Diagram: Eample 4. a < R a STABLE STABLE UNSTABLE a > a WHEN a = WE STILL DO NOT KNOW ABOUT THE STABAILITY OF STABLE a
Enzyme Kinetics k k S E S E P E k LAW of Mass Action: The rate of a reaction is proportional to the product of the concentrations of the reactants. s [ S], e [ E], c [ S E], p [ P] s( t) k e( t) s( t) k c( t), e( t) k e( t) s( t) ( k k ) c( t) c( t) k e( t) s( t) ( k k ) c( t), p( t) k c( t) s() s, e() e, c(), p()
Enzyme Kinetics s( t) k e( t) s( t) k c( t), e( t) k e( t) s( t) ( k k ) c( t) c( t) k e( t) s( t) ( k k ) c( t), p( t) k c( t) s() s, e() e, c(), p() Conserved quantity: e( t) c( t) e() c() e c( t) e( t) e( t) e c( t) p( t) k c( t) p( t) k c( ) d t
Enzyme Kinetics s( t) k e( t) s( t) k c( t), e( t) k e( t) s( t) ( k k ) c( t) c( t) k e( t) s( t) ( k k ) c( t), p( t) k c( t) s() s, e() e, c(), p() s( t) k e( t) s( t) k c( t), s() s e( t) [ e c( t)] s( t) k [ e c( t)] s( t) k c( t), s() s s( t) k e s( t) [ k s( t) k ] c( t), s() s
Enzyme Kinetics c( t) k e( t) s( t) ( k k ) c( t), c() e( t) [ e c( t)] c( t) k [ e c( t)] s( t) ( k k ) c( t), c() c( t) k e s( t) [ k s( t) k k ] c( t), c() s( t) k e s( t) [ k s( t) k ] c( t), s() s c( t) k e s( t) [ k s( t) k k ] c( t), c()
Dimensionless Version s( t) k e s( t) [ k s( t) k ] c( t), s() s c( t) k e s( t) [ k s( t) k k ] c( t), c() k e t or t / k e u() st () s s( / k e ) s v() ct () e c( / k e ) e DERIVE DIFFERENTIAL EQUATIONS FOR u( ) and v( )
Dimensionless Version u() st () s s( / k e ) s d d s( / ke ) s( / ke ) s() t u() d d s k e s k e s d d c( / ke ) c( / ke ) c() t v() d d e k e e k e e
Dimensionless Version d d s( / ke ) s( / ke ) s() t u() d d s k e s k e s s( t) k e s( t) [ k s( t) k ] c( t) st () kes () t [ ks( t) k ] c( t) k s e k s e k s e s( t) s( t) s( t) k c( t) [ ] k s e s s k s e u( ) v( ) st ( ) s ct ( ) e s( / ke s c( / ke e ) ) d s( t) s( t) s( t) k c( t) u( ) [ ] d k e s s s k s e u u k v ks ( ) [ ( ) ] ( )
Dimensionless Version LIKEWISE d k u u u v d k s ( ) ( ) [ ( ) ] ( ) e d k k d v( ) u( ) [ u( ) ] v( ) s k s DEFINE k k k k k K e k s k s k s s s m, K, K k k k k k s k s k s
Dimensionless Version d d u ( ) u ( ) [ u ( ) K ] v ( ), u () d d v ( ) u ( ) [ u ( ) K ] v ( ), v () k e t or t / k e k k k k k K e k s k s k s s s m, K, K m k k k Michaelis constant
Dimensionless Version d d u ( ) u ( ) [ u ( ) K ] v ( ), u () d d v ( t ) u ( ) [ u ( ) K ] v ( ), v () d d u( ) K u( ) u( ) v( ) v( ) K v( ) u( ) v( ) e s
Dimensionless Version d d u ( ) u ( ) [ u ( ) K ] v ( ), u () d d v ( t ) u ( ) [ u ( ) K ] v ( ), v () d d u ( ) u ( ) [ u ( ) K ] v ( ), u () d d v ( t ) u ( ) [ u ( ) K ] v ( ), v () SIMULATE USING PPLANE7
Dimensionless Version
Dimensionless Version
? STABILITY?
Stability: Enzyme Kinetics d d u ( ) u ( ) [ u ( ) K ] v ( ), u () d d v ( t ) u ( ) [ u ( ) K ] v ( ), v () d d u( ) K u( ) u( ) v( ) v( ) K v( ) u( ) v( ) K u uv K v uv
Stability: Enzyme Kinetics K u uv K v uv u ( K ) v uv u Kv uv v k ks u v ONLY EQUILIBRIUM
Stability: Enzyme Kinetics d d u( ) K u( ) u( ) v( ) v( ) K v( ) u( ) v( ) d d ( ) K ( ) ( ) ( ) ( ) / K / ( ) ( ) ( ) / f ( K ) K
Stability: Enzyme Kinetics f ( K ) K J ( K ) f( ) Jf K J ( K ) f( ) f K o J
Stability: Enzyme Kinetics J ( K ) f( ) f K o J EIGENVALUES ARE GIVEN BY ( K ) K det ( K / ) / ( K/ ) [( K/ )] 4 / / k k k k k k, K k s k s k s k s k s
Stability: Enzyme Kinetics ( K/ ) [( K/ )] 4 / / k ks [( K / )] 4 K / [ K / )] 4( K ) / K/ [ K/ )] 4 / [ K/ )] 4 / [ K/ )] [( K / )] 4 / [( K / )] [( K / )]
Stability: Enzyme Kinetics ( K/ ) [( K/ )] 4 / / ( K/ ) [( K/ )] 4 / / Theorem S IMPLIES is asymptotically stable