15-382 COLLECTIVE INTELLIGENCE – S18

LECTURE 4: DYNAMICAL SYSTEMS 3

INSTRUCTOR: GIANNI A. DI CARO EQUILIBRIUM

§ A state � is said an equilibrium state of a dynamical system �̇ = �(�), if and only if � = � �; �; � � = 0 , ∀ � ≥ 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: � � = 0

2 IS THE EQUILIBRIUM STABLE?

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

Stable equilibrium Unstable equilibrium Neutral equilibrium

Metastable equilibrium § Why are equilibrium properties so important? § For the same definition of an abstract model of a (complex) real-world scenario

3 SANDPILES, SNOW AVALANCHES AND META-STABILITY

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

4 LYAPUNOUV VS. STRUCTURAL EQUILIBRIUM

� � �

§ 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 �

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

5 IS THE EQUILIBRIUM (LYAPUNOUV) STABLE?

§ An equilibrium state � is said to be Lyapunouv stable if and only if for any ε > 0, there exists a positive number � � such that the inequality � 0 − � ≤ � implies that � �; � 0 , � � = 0 − � ≤ ε ∀ � ≥ 0

�

§ An equilibrium state � is stable (in the Lyapunouv sense) if the response following after starting at any initial state � 0 that is sufficiently near � will not move the state far away from � 6 IS THE EQUILIBRIUM (LYAPUNOUV) STABLE?

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

7 IS THE EQUILIBRIUM ASYMPTOTICALLY STABLE?

§ If an equilibrium state � is Lyapunouv stable and every motion starting sufficiently near to � converges (goes back) to � as � → ∞ , the equilibrium is said asymptotically stable

�

�, � � →0 as � → ∞

8 SOLUTION OF LINEAR ODES

§ The general form for an ODE: �̇ = �(�), where � is a � -dim vector field § The general form for a linear ODE: �̇ = ��, � ∈ ℝ, � an �×� coefficient matrix

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

§ Theorem: Linear combination of solutions of a linear ODE If the vector functions �() and �() are solutions of the linear system �̇ = () () �(�), then the linear combination �� + �� is also a solution for any real constants � and �

§ Corollary: Any linear combination of solutions is a solution By repeatedly applying the result of the theorem, it can be seen that every finite linear combination � � = �� (�) + �� (�) + … �� (�) of solutions � , � , … , � is itself a solution to �̇ = �(�)

9 FUNDAMENTAL AND GENERAL SOLUTION OF LINEAR ODES

§ Theorem: Linearly independent solutions

If the vector functions � , � ,… , � are linearly independent solutions of the �-dim linear system �̇ = �(�), then, each solution �(�) can be expressed uniquely in the form: � � = �� (�) + �� (�) + … �� (�)

§ Corollary: Fundamental and general solution of a linear system

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

� � = �� (�) + �� (�) + … �� (�)

10 GENERAL SOLUTIONS FOR LINEAR ODES

§ Corollary: Non-null Wronskian as condition for linear independence The proof of the theorem uses the fact that if � , � ,… , � are linearly independent (on the domain), then det � � ≠ 0

�(�) ⋯ �(�) �(�) = ⋮ ⋱ ⋮ Wronskian �(�) ⋯ �(�)

Therefore, � , � ,… , � are linearly independent if and only if

W[� , � , … , � ](�) ≠ 0

§ Theorem: Use of the Wronskian to check fundamental solutions If � , � , … , � are solutions, then the Wroskian is either identically to zero or else is never zero for all �

§ Corollary: To determine whether a given set of solutions are fundamental solutions it suffices to evaluate W[� , � , … , � ](�) at any point � 11 STABILITY OF LINEAR MODELS

§ Let’s start by studying stability 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 is the trivial solution, � = � § Matrix � has full rank § det � = ∏ �, 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 � 12 RECAP ON EIGENVECTORS AND EIGENVALUES

Geometry: § Eigenvectors: Directions � that the linear transformation � 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 �: each eigenvalue can be repeated � ≥ 1 times (e.g., (� − 3), � = 2) § Geometric multiplicity �: Each eigenvalue has at least one or � ≥ 1 eigenvectors, and only 1 ≤ � ≤ � can be linearly independent § An eigenvalue can be 0, as well as can be a real or a complex number 13 RECAP ON EIGENVECTORS AND EIGENVALUES

14 LINEAR MULTI-DIMENSIONAL MODELS

§ For the case of linear (one dimensional) growth model, �̇ = ��, solutions were in the form: � � = �� § 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 � � = �� hold? What about operator �?

§ A two-dimensional example:

�̇ = −4� − 3� � −4 −3 �(0) = (1,1) � = � = �̇ = 2� + 3� � 2 3

§ Eigenvalues and Eigenvectors of �: 1 3 � = 2, � = � = −3, � = −2 −1 (real, positive) (real, negative) 15 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: � � = � �

§ The general solution to the linear ODE is obtained by the linear combination of the � � = �� � + �� � individual eigenvalue solutions (since � ≠ �, �� and �� are linearly independent)

16 SOLUTION (EIGENVALUES, EIGENVECTORS)

� � = �� � + �� � �

� 0 = (1,1) (1,1) 1,1 = �(1, −2) + �(3, −1) �� à � = −4/5 � = 3/5 � � � � = −4/5� � + 3/5� � 4 9 � � = − � + � 5 5 8 3 � � = � − � 5 5 Saddle equilibrium (unstable) § 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

17 TWO REAL EIGENVALUES, OPPOSITE SIGNS

§ The straight lines corresponding � to � and �� are the trajectories corresponding to all multiples of individual eigenvector solutions (1,1) ��� : �� � � � � 1 �: = � � � � −2

� � 3 �: = � � � � −1 § 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) § The eigenvectors corresponding to the same eigenvalue �, 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 18