L ECTURE 5: D YNAMICAL S YSTEMS 4 I NSTRUCTOR : G IANNI A. D I C ARO - PowerPoint PPT Presentation

15-382 C OLLECTIVE I NTELLIGENCE – S18 L ECTURE 5: D YNAMICAL S YSTEMS 4 I NSTRUCTOR : G IANNI A. D I C ARO

L INEAR M ULTI -D IMENSIONAL M ODELS For the case of linear (one dimensional) growth model, 𝑦̇ = 𝑏𝑦, solutions § were in the form: 𝑦 𝑢 = 𝑦 4 𝑓 67 § The sign of a would affect stability and asymptotic behavior: x = 0 is an asymptotically stable solution if a < 0, while x = 0 is an unstable solution if a > 0, since other solutions depart from x = 0 in this case. Does a multi-dimensional generalization of the form 𝒚 𝑢 = 𝒚 4 𝑓 𝑩7 hold? § What about operator 𝑩 ? § A two-dimensional example: 𝒚 = 𝑦 $ −4 −3 𝑦̇ $ = −4𝑦 $ − 3𝑦 ) 𝐵 = 𝒚 (0) = (1,1) 2 3 𝑦̇ ) = 2𝑦 $ + 3𝑦 ) 𝑦 ) Eigenvalues and Eigenvectors of 𝐵 : § 1 3 𝜇 $ = 2, 𝒗 $ = 𝜇 ) = −3, 𝒗 ) = −2 −1 (real, negative) (real, positive) 2

S OLUTION ( EIGENVALUES , EIGENVECTORS ) The eigenvector equation: 𝐵𝒗 = 𝜇𝒗 § Let’s set the solution to be 𝒚 𝑢 = 𝑓 97 𝒗 and lets’ verify that it § satisfies the relation 𝒚̇ 𝑢 = 𝐵𝒚 Multiplying by 𝐵 : 𝐵𝒚(𝑢) = 𝑓 97 𝐵𝒗 , but since 𝒗 is an eigenvector: § 𝐵𝒚 𝑢 = 𝑓 97 𝐵𝒗 = 𝑓 97 (𝜇𝒗 ) 𝒗 is a fixed vector, that doesn’t depend on 𝑢 → if we take 𝒚 𝑢 = 𝑓 97 𝒗 § and differentiate it: 𝒚̇ 𝑢 = 𝜇𝑓 97 𝒗 , which is the same as 𝐵𝒚 𝑢 above Each eigenvalue-eigenvector pair ( 𝜇 , 𝒗 ) of 𝐵 leads to a solution of 𝒚̇ 𝑢 = 𝐵𝒚 , taking the form: 𝒚 𝑢 = 𝑓 97 𝒗 § The general solution to the linear ODE is obtained by the linear combination of the 𝒚 𝑢 = 𝑑 $ 𝑓 9 > 7 𝒗 $ + 𝑑 ) 𝑓 9 ? 7 𝒗 ) individual eigenvalue solutions (since 𝜇 $ ≠ 𝜇 ), 𝒗 𝟐 and 𝒗 𝟑 are linearly independent) 3

S OLUTION ( EIGENVALUES , EIGENVECTORS ) 𝒚 𝑢 = 𝑑 $ 𝑓 9 > 7 𝒗 $ + 𝑑 ) 𝑓 9 ? 7 𝒗 ) 𝑦 ) 𝒚 0 = (1,1) (1,1) 1,1 = 𝑑 $ (1,−2) + 𝑑 ) (3,−1) 𝒗 𝟑 à 𝑑 $ = −4/5 𝑑 ) = 3/5 𝑦 $ 𝒗 $ 𝒚 𝑢 = −4/5𝑓 )7 𝒗 $ + 3/5𝑓 FG7 𝒗 ) 𝑦 $ 𝑢 = − 4 5 𝑓 )7 + 9 5 𝑓 FG7 𝑦 ) 𝑢 = 8 5 𝑓 )7 − 3 5 𝑓 FG7 Saddle equilibrium (unstable) Except for two solutions that approach the origin along the direction of the § eigenvector 𝒗 ) = (3 , - 1), solutions diverge toward ∞ , although not in finite time Solutions approach to the origin from different direction, to after diverge from it § 4

T WO REAL EIGENVALUES , OPPOSITE SIGNS § The straight lines corresponding 𝑦 ) to 𝒗 $ and 𝒗 𝟑 are the trajectories corresponding to all multiples of (1,1) individual eigenvector solutions 𝐷𝑓 97 𝒗 : 𝒗 𝟑 𝑦 $ 𝒗 $ 𝒗 $ : 𝑦 $ 𝑢 1 = 𝑑 $ 𝑓 )7 𝑦 ) 𝑢 −2 𝒗 ) : 𝑦 $ 𝑢 3 = 𝑑 ) 𝑓 FG7 𝑦 ) 𝑢 −1 § The slope of a trajectory corresponding to one eigenvalue is constant in N ? O >? ( 𝑦 $ ,𝑦 ) ) à It’s a line in the phase space (e.g., for 𝒗 $ : N > 𝑢 = O >> = −2 ) The eigenvectors corresponding to the same eigenvalue 𝜇 , together with § the origin (0,0) (which is part of the solution for each individual eigenvalue), form a linear subspace , called the eigenspace of λ The two straight lines are the two eigenspaces, that, as 𝑢 → ∞, play the § role of “separators” for the different behaviors of the system 5

T WO REAL EIGENVALUES , S AME S IGN 𝒚 𝑢 = 𝑑 $ 𝑓 9 > 7 𝒗 $ + 𝑑 ) 𝑓 9 ? 7 𝒗 ) 𝑦̇ $ = −2𝑦 $ 𝑦̇ ) = 𝑦 $ − 4𝑦 ) Asymptotically Stable or unstable § behavior depending on the sign of 𝜇 $) Node Trajectories either moving away from the § equilibrium to infinite-distant away (when 𝜇 > 0), or moving directly toward, and converge to equilibrium (when 𝜇 < 0). The trajectories that are the eigenvectors § move in straight lines. 6

O NE REAL , REPEATED EIGENVALUE § Case with two linearly independent eigenvectors 𝒚 𝑢 = 𝑓 9 > 7 (𝑑 $ 𝒗 $ + 𝑑 ) 𝒗 ) ) § Every nonzero solution traces a straight-line trajectory: constant slope, direction given by the linear combination of the eigenvectors It is unstable if the eigenvalue is positive; asymptotically stable if the eigenvalue is negative. Focus Proper node (star point) 7

O NE REAL , REPEATED EIGENVALUE § Case with two linearly independent eigenvectors 𝒚 𝑢 = 𝑓 9 > 7 (𝑑 $ 𝒗 $ + 𝑑 ) 𝒗 ) ) 8

O NE REAL , REPEATED EIGENVALUE § Case with one linearly independent eigenvectors 𝒚 𝑢 = 𝑑 $ 𝒗 $ 𝑓 9 > 7 + 𝑑 ) (𝒗 $ 𝑢𝑓 9 > 7 + 𝒗𝑓 9 > 7 ) Positive eigenvalue 𝑦 ) One eigenvalue solution: 𝑓 9 > 7 𝒗 $ § § Need to find another solution, linearly independent From solutions of the form: 𝑢𝑓 9 > 7 𝒗 $ + 𝒗 𝑓 9 > 7 § 𝑦 $ 𝒗 is a generalized eigenvector that can be § determined from 𝐵 − 𝜇 $ 𝐽 𝒗 $ = 𝒗 ~ (Node + Spiral) Repulsive focus (Improper node) All solutions except for the equilibrium diverge to infinity Negative eigenvalue 9

I MAGINARY EIGENVALUES § Roots of the characteristic equation are complex numbers: Λ ± = 𝜇 ± 𝑗𝜈 S is also an eigenvalue. § If Λ is a complex eigenvalue , then its conjugate Λ § If 𝒗 is a complex eigenvector of Λ, then 𝒗 S , the complex conjugate of its S entries, is an eigenvector associated to Λ § The solutions for the two conjugate eigenvalues: § 𝑓 (9TUV)7 𝒗 = 𝑓 97 𝑓 UV7 𝒗 = 𝑓 97 (cos𝜈𝑢 + 𝑗 sin𝜈𝑢)𝒗 § 𝑓 (9FUV)7 𝒗 S = 𝑓 97 𝑓 FUV7 𝒗 S = 𝑓 97 (cos 𝜈𝑢 − 𝑗 sin𝜈𝑢) 𝒗 S 𝒗 𝟐 𝒗 S 𝟐 Let 𝒗 = 𝒗 𝟑 , 𝒗 S = S 𝟑 , 𝑣 $ = 𝛽 $ + 𝑗𝛾 $ , 𝑣 ) = 𝛽 ) + 𝑗𝛾 ) § 𝒗 𝑓 (9TUV)7 𝑣 $ = 𝑓 97 (cos𝜈𝑢 + 𝑗 sin𝜈𝑢) (𝛽 $ + 𝑗𝛾 $ ) 𝑓 (9TUV)7 𝒗 = 𝑓 (9TUV)7 𝑣 ) = 𝑓 97 (cos 𝜈𝑢 + 𝑗 sin𝜈𝑢) (𝛽 ) + 𝑗𝛾 ) ) 𝑓 (9FUV)7 𝑣 b $ = 𝑓 97 (cos𝜈𝑢 − 𝑗 sin𝜈𝑢) (𝛽 $ − 𝑗𝛾 $ ) 𝑓 (9FUV)7 𝒗 S = 𝑓 (9FUV)7 𝑣 b ) = 𝑓 97 (cos 𝜈𝑢 − 𝑗 sin𝜈𝑢) (𝛽 ) − 𝑗𝛾 ) ) 10

I MAGINARY EIGENVALUES 𝑓 97 [(𝛽 $ cos𝜈𝑢 − 𝛾 $ sin𝜈𝑢) + 𝑗(𝛾 $ cos 𝜈𝑢 + 𝛽 $ sin𝜈𝑢)] 𝑓 (9TUV)7 𝒗 = 𝑓 97 [(𝛽 ) cos𝜈𝑢 − 𝛾 ) sin𝜈𝑢) + 𝑗(𝛾 ) cos𝜈𝑢 + 𝛽 ) sin𝜈𝑢)] 𝑓 97 [(𝛽 $ cos𝜈𝑢 − 𝛾 $ sin𝜈𝑢) − 𝑗(𝛾 $ cos 𝜈𝑢 + 𝛽 $ sin𝜈𝑢)] 𝑓 (9FUV)7 𝒗 S = v 𝑓 97 [(𝛽 ) cos𝜈𝑢 − 𝛾 ) sin𝜈𝑢) − 𝑗(𝛾 ) cos𝜈𝑢 + 𝛽 ) sin𝜈𝑢)] 𝒁 ) = 𝑓 97 𝛾 $ cos 𝜈𝑢 + 𝛽 $ sin𝜈𝑢 𝒁 $ = 𝑓 97 𝛽 $ cos𝜈𝑢 − 𝛾 $ sin𝜈𝑢 𝛾 ) cos 𝜈𝑢 + 𝛽 ) sin𝜈𝑢 𝛽 ) cos𝜈𝑢 − 𝛾 ) sin𝜈𝑢 𝑓 (9FUV)7 𝒗 S = 𝒁 $ − 𝑗𝒁 ) 𝑓 (9TUV)7 𝒗 = 𝒁 $ + 𝑗𝒁 ) Since both sum and difference of 𝒁 $ and 𝒁 ) are solutions, also 𝒁 $ and 𝒁 ) § are solutions; moreover, it can be proved that they are linearly independent 𝒚 𝑢 = 𝑑 $ 𝒁 $ + 𝑑 ) 𝒁 ) is a general solution (real, combining real solutions) 𝑦 $ 𝑢 = 𝑑 $ 𝑓 97 𝛽 $ cos𝜈𝑢 − 𝛾 $ sin𝜈𝑢 + 𝑑 ) 𝑓 97 (𝛾 $ cos𝜈𝑢 + 𝛽 $ sin𝜈𝑢 ) 𝑦 ) 𝑢 = 𝑑 $ 𝑓 97 𝛽 ) cos𝜈𝑢 − 𝛾 ) sin𝜈𝑢 + 𝑑 ) 𝑓 97 (𝛾 ) cos𝜈𝑢 + 𝛽 ) sin𝜈𝑢) 11

(P URE ) I MAGINARY EIGENVALUES 𝑦̇ $ = 𝑦 ) 𝒚 = 𝑦 $ 0 1 𝑦̇ ) = −𝑦 $ 𝐵 = 𝑦 ) −1 0 Eigenvalues 𝐵 : § 𝑦 ) Λ $ = 𝑗, Λ ) = −𝑗 ( 𝜇 = 0, 𝜈 = 1) (pure imaginary, conjugate) Eigenvectors : § 𝑦 $ 𝑗 𝒗 = 0 + 𝑗(−1) −𝑗 𝒗 S = = 1 1 + 𝑗(0) 1 𝛽 $ = 0, 𝛾 $ = −1,𝛽 ) = 1,𝛾 ) = 0 𝑦 $ 𝑢 = 𝑑 $ 𝑓 97 𝛽 $ cos 𝜈𝑢 − 𝛾 $ sin 𝜈𝑢 + 𝑑 ) 𝑓 97 (𝛾 $ cos𝜈𝑢 + 𝛽 $ sin 𝜈𝑢 ) 𝑦 ) 𝑢 = 𝑑 $ 𝑓 97 𝛽 ) cos𝜈𝑢 − 𝛾 ) sin 𝜈𝑢 + 𝑑 ) 𝑓 97 (𝛾 ) cos𝜈𝑢 + 𝛽 ) sin 𝜈𝑢) Center § Periodic solutions General solutions: § Some points initially move farther § 𝑦 $ 𝑢 = 𝑑 $ sin𝑢 − 𝑑 ) cos𝑢 away, but not too far away. 𝑦 ) 𝑢 = 𝑑 $ cos𝑢 − 𝑑 sin𝑢 The origin is stable but not attracting. § 12

T HE TWO NEIGHBORHOODS OF LYAPUNOUV STABILITY 𝜇 ± = ±𝑗 𝜇 ± = ±3𝑗 𝑦 ) 𝑦 ) 𝜀(𝜁) v v 𝑦 $ 𝜁 𝑦 $ Lyapunov stability needs two neighborhoods, 𝜁,𝜀(𝜁) : In order to have solutions § stay within a neighborhood 𝜻 whose radius is the larger axis of an ellipse, initial conditions must be restricted to a neighborhood 𝜀(𝜁) whose radius is no larger than the smaller axis of the solution If the neighborhood 𝜀 𝜁 is equal to the larger axis, then an initial point could be § placed on another, larger orbit, that would not satisfy the requirement to stay within the original neighborhood 𝜁 13