LECTURE 4:DYNAMICAL SYSTEMS 3

INSTRUCTOR:GIANNI A.DI CARO

15-382COLLECTIVE INTELLIGENCE – S18

2

EQUILIBRIUM

§ A state 𝒙" is said an equilibrium state of a dynamical system �̇� = 𝒇(𝒙),if and only if

𝒙" = 𝒙 𝑡;𝒙";𝒖 𝑡 = 0 , ∀𝑡 ≥ 0

§ If a trajectory reaches an equilibrium state (and if no input is applied) the trajectory will stay at the equilibrium state forever: internal system’s dynamics doesn’t move the system away from the equilibrium point, velocity is null: 𝒇 𝒙" = 0

3

IS THE EQUILIBRIUM STABLE?

Stable equilibrium Unstable equilibrium Neutral equilibrium

When a displacement (a force) is applied to an equilibrium condition:

Metastable equilibrium§ Why are equilibrium properties so important?

§ For the same definition of an abstract model of a (complex) real-world scenario

4

SANDPILES, SNOW AVALANCHES AND META-STABILITY

Abelian sandpile model (starting with one billion grains pile in the center)

5

LYAPUNOUV VS. STRUCTURAL EQUILIBRIUM𝑝 𝑝 𝑝

§ Lyapunouv equilibrium: stability of an equilibrium with respect to a small deviation from the equilibrium point

§ Structural equilibrium: is the equilibrium persistent to (small) variations in the structure of the systems? à Sensitivity to the value of the parameters of the vector field 𝒇

6

IS THE EQUILIBRIUM (LYAPUNOUV) STABLE?

§ An equilibrium state 𝒙" is said to be Lyapunouv stable if and only if

foranyε > 0,there exists a positive number 𝛿 𝜀 such that the inequality

𝒙 0 − 𝒙" ≤ 𝛿

implies that 𝒙 𝑡; 𝒙 0 ,𝒖 𝑡 = 0 − 𝒙" ≤ ε∀𝑡 ≥ 0

§ An equilibrium state 𝒙" is stable (in the Lyapunouv sense) if the response following after starting at any initial state 𝒙 0 that is sufficiently near 𝒙"will not move the state far away from 𝒙"

𝑡

7

IS THE EQUILIBRIUM (LYAPUNOUV) STABLE?

What is the difference between a stable and an asymptotically stable equilibrium?

8

IS THE EQUILIBRIUM ASYMPTOTICALLY STABLE?

§ If an equilibrium state 𝒙" is Lyapunouv stable and every motion starting

sufficiently near to 𝒙" converges (goes back) to 𝒙" as 𝑡 → ∞ , the

equilibrium is said asymptotically stable

𝑡

𝜀, 𝛿 𝜀 →0 as 𝑡 → ∞

9

SOLUTION OF LINEAR ODES

§ The general form for a linear ODE:

�̇� = 𝐴𝒙, 𝒙 ∈ ℝ<, 𝐴an 𝑛×𝑛 coefficient matrix

§ A solution is a differentiable function 𝑿 𝑡 that satisfies the vector field

§ Theorem: Linear combination of solutions of a linear ODEIf the vector functions 𝒙(@) and 𝒙(A) are solutions of the linear system �̇� =𝒇(𝒙), then the linear combination 𝑐@𝒙(@) + 𝑐A𝒙(A) is also a solution for any real constants 𝑐@ and 𝑐A

§ Corollary: Any linear combination of solutions is a solutionBy repeatedly applying the result of the theorem, it can be seen that every finite linear combination 𝒙 𝑡 = 𝑐@𝒙 @ (𝑡) + 𝑐A𝒙 A (𝑡) +…𝑐E𝒙 E (𝑡)of solutions 𝒙 @ , 𝒙 A ,… , 𝒙 E is itself a solution to �̇� = 𝒇(𝒙)

§ The general form for an ODE: �̇� = 𝒇(𝒙), where 𝒇 is a 𝑛-dim vector field

10

FUNDAMENTAL AND GENERAL SOLUTION OF LINEAR ODES

§ Theorem: Linearly independent solutions

If the vector functions 𝒙 @ , 𝒙 A ,… , 𝒙 < are linearly independent solutions of the 𝑛-dim linear system �̇� = 𝒇(𝒙), then, each solution 𝒙(𝑡)can be expressed uniquely in the form: 𝒙 𝑡 = 𝑐@𝒙 @ (𝑡) + 𝑐A𝒙 A (𝑡) +…𝑐<𝒙 < (𝑡)

§ Corollary: Fundamental and general solution of a linear system

If solutions 𝒙 @ , 𝒙 A ,… , 𝒙 < are linearly independent (for each point in the time domain), they are fundamental solutions on the domain, and the general solution to a linear �̇� = 𝒇(𝒙), is given by:

𝒙 𝑡 = 𝑐@𝒙 @ (𝑡) + 𝑐A𝒙 A (𝑡) +…𝑐<𝒙 < (𝑡)

11

GENERAL SOLUTIONS FOR LINEAR ODES

§ Corollary: Non-null Wronskian as condition for linear independenceThe proof of the theorem uses the fact that if 𝒙 @ , 𝒙 A ,… , 𝒙 < are linearly independent (on the domain), then det𝑿 𝑡 ≠ 0

𝑿(𝑡) =𝑥@@(𝑡) ⋯ 𝑥@<(𝑡)⋮ ⋱ ⋮

𝑥<@(𝑡) ⋯ 𝑥<<(𝑡)

Therefore, 𝒙 @ , 𝒙 A ,… , 𝒙 < are linearly independent if and only if

W[𝒙 @ , 𝒙 A ,… , 𝒙 < ](𝑡) ≠ 0

Wronskian

§ Theorem: Use of the Wronskian to check fundamental solutionsIf 𝒙 @ , 𝒙 A ,… , 𝒙 < are solutions, then the Wroskian is either identically to zero or else is never zero for all 𝑡

§ Corollary: To determine whether a given set of solutions are fundamental solutions it suffices to evaluate W[𝒙 @ , 𝒙 A ,… , 𝒙 < ](𝑡) at any point 𝑡

12

STABILITY OF LINEAR MODELS

§ Let’s start by studying stability in linear dynamical systems …§ The general form for a linear ODE:

�̇� = 𝐴𝒙, 𝒙 ∈ ℝ<, 𝐴an 𝑛×𝑛 coefficient matrix

§ Equilibrium points are the points of the Null space / Kernel of matrix 𝐴𝐴𝒙 = 𝟎, 𝑛×𝑛 homogeneous system

§ Invertible Matrix Theorem, equivalent facts:§ 𝐴 is invertible ⟷ det 𝐴 ≠ 0

§ The only solution is the trivial solution, 𝒙 = 𝟎

§ Matrix 𝐴 has full rank

§ det𝐴 = ∏ 𝜆U<UV@ , all eigenvalues are non null

§ …§ In a linear dynamical system, solutions and stability of the origin

depends on the eigenvalues (and eigenvectors) of the matrix 𝐴

13

RECAP ON EIGENVECTORS AND EIGENVALUES

Geometry:

§ Eigenvectors: Directions 𝒙that the linear transformation 𝐴doesn’t change.

§ The eigenvalue 𝜆is the scaling factor of the transformation along 𝒙 (the direction that stretches the most)

Algebra:§ Roots of the characteristic equation§ 𝑃 𝜆 = 𝜆𝑰 − 𝐴 𝒙 = 0 → det 𝜆𝑰 − 𝐴 = 0§ For 2×2 matrices: det 𝜆𝑰 − 𝐴 = 𝜆A − 𝜆tr𝐴 + det𝐴§ Algebraic multiplicity 𝒏: each eigenvalue can be repeated 𝑛 ≥ 1 times

(e.g., (𝜆 − 3)A, 𝑛 = 2)§ Geometric multiplicity 𝒎: Each eigenvalue has at least one or 𝑚 ≥ 1

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

14

RECAP ON EIGENVECTORS AND EIGENVALUES

15

LINEAR MULTI-DIMENSIONAL MODELS

§ A two-dimensional example:�̇�@= −4𝑥@ − 3𝑥A�̇�A = 2𝑥@ + 3𝑥A 𝒙(0) = (1,1) 𝒙 =

𝑥@𝑥A

𝐴 = −4 −32 3

§ Eigenvalues and Eigenvectors of 𝐴:

𝜆@ = 2, 𝒖@ =1−2 𝜆A = −3, 𝒖A =

3−1

§ For the case of linear (one dimensional) growth model, �̇� = 𝑎𝑥, solutions were in the form: 𝑥 𝑡 = 𝑥c𝑒ef

§ 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 𝒙 𝑡 = 𝒙c𝑒𝑨f hold? What about operator 𝑨?

(real, positive) (real, negative)

16

SOLUTION (EIGENVALUES, EIGENVECTORS)

§ The eigenvector equation: 𝐴𝒖 = 𝜆𝒖

§ Let’s set the solution to be 𝒙 𝑡 = 𝑒hf𝒖 and lets’ verify that it satisfies the relation �̇� 𝑡 = 𝐴𝒙

§ Multiplying by 𝐴: 𝐴𝒙(𝑡) = 𝑒hf𝐴𝒖, but since 𝒖 is an eigenvector: 𝐴𝒙 𝑡 = 𝑒hf𝐴𝒖 = 𝑒hf(𝜆𝒖)

§ 𝒖 is a fixed vector, that doesn’t depend on 𝑡 → if we take 𝒙 𝑡 = 𝑒hf𝒖and differentiate it: �̇� 𝑡 = 𝜆𝑒hf𝒖, which is the same as 𝐴𝒙 𝑡 above

Each eigenvalue-eigenvector pair (𝜆, 𝒖) of 𝐴 leads to a solution of �̇� 𝑡 = 𝐴𝒙,taking the form: 𝒙 𝑡 = 𝑒hf𝒖

𝒙 𝑡 = 𝑐@𝑒hif𝒖@ + 𝑐A𝑒hjf𝒖A

§ The general solution to the linear ODEis obtained by the linear combination of the individual eigenvalue solutions (since 𝜆@ ≠𝜆A,𝒖𝟏 and 𝒖𝟐 are linearly independent)

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SOLUTION (EIGENVALUES, EIGENVECTORS)

𝒙 𝑡 = 𝑐@𝑒hif𝒖@ + 𝑐A𝑒hjf𝒖A

𝒙 0 = (1,1)

1,1 = 𝑐@(1,−2) + 𝑐A(3,−1)à 𝑐@ = −4/5 𝑐A = 3/5

𝒙 𝑡 = −4/5𝑒Af𝒖@ + 3/5𝑒opf𝒖A𝑥@ 𝑡 = −

45𝑒

Af+95𝑒

opf

𝑥A 𝑡 =85 𝑒

Af−35𝑒

opf

𝑥A

𝑥@𝒖@𝒖𝟐

(1,1)

§ Exceptfortwosolutionsthatapproachtheoriginalongthedirectionoftheeigenvector𝒖A =(3,-1), solutionsdivergetoward∞,althoughnotinfinitetime

§ Solutionsapproachtotheoriginfromdifferentdirection,toafterdivergefromit

Saddle equilibrium(unstable)

18

TWO REAL EIGENVALUES, OPPOSITE SIGNS

𝑥A

𝑥@𝒖@𝒖𝟐

(1,1)

§ The straight lines corresponding to 𝒖@ and 𝒖𝟐 are the trajectories corresponding to all multiples of individual eigenvector solutions 𝐶𝑒hf𝒖:

𝒖@:𝑥@ 𝑡𝑥A 𝑡

= 𝑐@ 𝑒Af1−2

𝒖A:𝑥@ 𝑡𝑥A 𝑡

= 𝑐A 𝑒opf3−1

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

§ The slope of a trajectory corresponding to one eigenvalue is constant in (𝑥@, 𝑥A)à It’s a line in the phase space (e.g., for 𝒖@:

vjvi

𝑡 = wijwii

= −2)