LECTURE 6: DYNAMICAL SYSTEMS 5

TEACHER:GIANNI A. DI CARO

15-382 COLLECTIVE INTELLIGENCE – S19

2

STABILITY OF LINEAR MODELS

§ Let’s find equilibrium points in linear dynamical systems …

§ The general form for a linear ODE:

"̇ = $", " ∈ ℝ(, $ an )×) coefficient matrix

§ Equilibrium points are the points of the Null space / Kernel of matrix $$" = +, )×) homogeneous system

§ Invertible Matrix Theorem, equivalent facts:§ $ is invertible ⟷ det $ ≠ 0§ The only solution to $" = + is the trivial solution, " = +§ Matrix $ has full rank

§ det $ = ∏012( 30, all eigenvalues are non null

§ …

§ In a linear dynamical system, solutions and stability of the origin, " = 0,depends on the eigenvalues (and eigenvectors) of the matrix $

3

RECAP ON EIGENVECTORS AND EIGENVALUES

Geometry:

§ Eigenvectors: Directions ! that the linear transformation " (our vector field!) doesn’t change

§ The eigenvalue # is the scaling factor of the transformation along ! (the direction that stretches the most)

Algebra:§ Roots of the characteristic equation§ $ # = #& − " ! = 0 → det #& − " = 0§ For 2×2 matrices: det #& − " = #/ − # tr " + det "§ Algebraic multiplicity 2: each eigenvalue can be repeated 3 ≥ 1 times

(e.g., (# − 3)/, 3 = 2)§ Geometric multiplicity 9: Each eigenvalue has at least one or : ≥ 1

eigenvectors, and only 1 ≤ < ≤ : could be linearly independent§ An eigenvalue can be 0, as well as can be a real or a complex number

4

RECAP ON EIGENVECTORS AND EIGENVALUES

§ The value of the determinant determines the scaling factor (contraction or

expansion) of the volumes in (in the phase space!). The sign relates to the

orientation of the volume: negative sign means that the orientation is inverted

5

LINEAR MULTI-DIMENSIONAL MODELS

§ A two-dimensional example:

"̇# = −4"# − 3"("̇( = 2"# + 3"( +(0) = (1,1) + = "#

"(, = −4 −3

2 3

§ Eigenvalues and Eigenvectors of ,:

-# = 2, /# =1−2 -( = −3, /( =

3−1

§ For the case of linear (one dimensional) growth model, "̇ = 1", solutions were in the form: " 2 = "3456

§ The sign of a would affect stability and asymptotic behavior: " = 0 is an asymptotically stable equilibrium point if 1 < 0, while " = 0 is an unstable equilibrium point if a > 0, since other solutions depart from " = 0 in this case.

§ Does a multi-dimensional generalization of the form + 2 = +3496 hold? What about operator 9?

(real, positive) (real, negative)

6

SOLUTION (EIGENVALUES, EIGENVECTORS)

§ The eigenvector equation: !" = $"§ Let’s set the solution to be % & = '()" and lets’ verify that it satisfies the

relation %̇ & = !%§ Multiplying by !: !%(&) = '()!" , but since " is an eigenvector: !% & =

'()!" = '()($")

§ " is a fixed vector, that doesn’t depend on & → if we take % & = '()" and differentiate it: %̇ & = $'()", which is the same as !% & above

Each eigenvalue-eigenvector pair ($, ") of ! leads to a solution of %̇ & = !% ,taking the form: % & = '()"

% & = /0'(1)"0 + /3'(4)"3§ The general solution to the linear ODEis obtained by the linear combination of the individual eigenvalue solutions (since $0 ≠$3, "6 and "7 are linearly independent)

7

SOLUTION (EIGENVALUES, EIGENVECTORS)

! " = $%&'()*% + $,&'-)*,

! 0 = (1,1)1,1 = $%(1, −2) + $,(3, −1)à $% = −4/5 $, = 3/5

! " = −4/5&,)*% + 3/5&9:)*,;% " = −45 &

,)+ 95 &9:)

;, " = 85 &

,)− 35 &9:)

;,

;%*%*>

(1,1)

§ Except for two solutions that approach the origin along the direction of the eigenvector *, =(3, -1), solutions diverge toward ∞, although not in finite time

§ Solutions approach to the origin from different direction, to after diverge from it

§ Trajectories are hyperbolas

Saddle equilibrium(unstable)

8

TWO REAL EIGENVALUES, OPPOSITE SIGNS

!"

!#$#$%

(1,1)

§ The straight lines corresponding to $# and $% are the trajectories corresponding to all multiples of individual eigenvector solutions &'()$:

$#:!# +!" +

= -# '")1−2

$":!# +!" +

= -" '12)3−1

§ The eigenvectors corresponding to the same eigenvalue 4, 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 + → ∞, play the role of “separators” for the different behaviors of the system

§ The slope of a trajectory corresponding to one eigenvalue is constant in (!#, !")à It’s a line in the phase space (e.g., for $#: :;

:<+ = =<;

=<<= −2)