LECTURE 6:DYNAMICAL SYSTEMS 5

INSTRUCTOR:GIANNI A.DI CARO

15-382COLLECTIVE INTELLIGENCE – S18

2

NON LINEAR SYSTEMS?

Four equilibria: 0, ±2 , (± 3, 1)Countably infinite equilibria: (𝑛𝜋, 0)

3

AN EXAMPLE OF NON-LINEAR SYSTEM

§ Autonomous non-linear system:𝑑𝑥-𝑑𝑡 = 4 − 2𝑥2

𝑑𝑥2𝑑𝑡 = 12 − 3𝑥-2

Critical points: 4 − 2𝑥2 = 0

12 − 3𝑥-2 = 0→ −2,2 , (2,2)

§ Trajectories:567568

= 567 59⁄568 59⁄ = ;(68,67)

<(68,67)which is a first order equation in 𝑥-, 𝑥2

If the equation can be solved (based on 𝐹 and 𝐺), solution trajectories can

be written implicitly in the form: 𝐻 𝑥-,𝑥2 = 𝑐

→Trajectories lie on the level curves of the implicit function 𝐻

𝑑𝑥2𝑑𝑥-

=12 − 3𝑥-2

4 − 2𝑥2Variableseparation:

𝑑𝑥212 − 3𝑥-2

=𝑑𝑥-

4 − 2𝑥2

𝐻 𝑥-,𝑥2 = 4𝑥2 − 12𝑥22 − 12𝑥- + 𝑥-O = 𝑐

4

AN EXAMPLE OF NON-LINEAR SYSTEM

§ Autonomous non-linear system:𝑑𝑥-𝑑𝑡 = 4 − 2𝑥2

𝑑𝑥2𝑑𝑡 = 12 − 3𝑥-2

Critical points: 4 − 2𝑥2 = 0

12 − 3𝑥-2 = 0→ −2,2 , (2,2)

§ Trajectories: 𝐻 𝑥-,𝑥2 = 4𝑥2 − 12𝑥22 − 12𝑥- + 𝑥-O = 𝑐

Easy case, usually it’s not possible to obtain the implicit function 𝐻

5

STABILITY OF LINEAR SYSTEMS

6

STABILITY OF LINEAR SYSTEMS

Critical point

Linear system Stability ?

Coefficients of matrix 𝐴

Problem model

Measurements,Statistical estimates

Uncertainties

Perturbation of coefficients?

Perturbation of eigenvalues

7

“STABLE” PERTURBATIONS?

§ Small perturbations in matrix 𝐴 ⇒ Small perturbations in the eigenvalues

§ When perturbations do disrupt and/or radically change stability properties?

Pure imaginary eigenvalues

Stability change, also characteristics of the orbits change significantly

8

“STABLE” PERTURBATIONS?

Repeated eigenvalues

Linearly independent eigenvectors

Linearly dependent eigenvectors

Stability doesn’t change, but trajectories change significantly

No: Perturbation àNo change in sign

9

FROM PERTURBATION TO LINEARIZATION

§ In all the other configurations of the eigenvalues stability or instability is not changed, nor is the type of critical point altered by a perturbation

§ E.g., if eigenvalues are unequal and both real and negative, then a small change in the coefficients will neither change their sign now allow them to become exactly equal à the critical point remains asymptotically stable

§ How can we use these observations from linear systems to assess stability of a critical point for a general non linear system �̇� = 𝒇 𝒙 ?

§ Perturbation ~Approximating the vector field near critical points§ Easiest / More manageable way of approximating? à Linearize!

10

IS LINEARIZATION ENOUGH FOR STABILITY?§ Non-linear system: �̇� = 𝒇 𝒙

§ 𝒙𝒄 is a critical point corresponding to the origin. If 𝒙𝒄 ≠ 𝟎, then the coordinate transformation 𝒖 = 𝒙 − 𝒙𝒄 will transforms the system into a system with a critical point in the origin

§ Let’s assume we can write the non linear system about the critical point as a quasi linear system, meaning that is close to a linear one:

§ �̇� = 𝐴𝒙+ 𝒈 𝒙

§ 𝒈 𝒙 is a perturbation about the critical point: it’s smooth and goes to zero faster than linear ∥𝒈 𝒙 ∥

∥𝒙∥ \ → 0 as 𝒙 → 𝟎

§ Let’s also assume that 𝒙𝒄 is an isolated critical point of the system: no other critical points are in its neighborhood à It’s also an isolated equilibrium for the linear part

§ Since 𝒈 𝒙 plays the role of a perturbation to 𝐴𝒙 (around 𝒙𝒄), can we expect that studying the stability of 𝒙𝒄 only using the linear part of the system, 𝐴𝒙,is sufficient to derive sound conclusions?

11

CORE QUESTION: LINEARIZATION

§ Linearized version about the critical point 𝟎: �̇� ≈ 𝐽𝒇 𝒙_ 𝒙 = 𝐴𝒙

§ Under which conditions the study of the linearized system about a critical point is predictive about stability of the point in the original non linear system?

§ If we use �̇� ≈ 𝐽𝒇 𝒙_ 𝒙 = 𝐴𝒙, what we say about the critical point, will hold in the original non linear system?

§ Related question:

How do we linearize the system ↔ The vector function 𝒇 ?

12

(SCALAR) FUNCTION LINEARIZATION

§ A scalar function of one variable, 𝑓 𝑥 , 𝑛 times differentiable, can be expanded in a function series about a point 𝑥 = 𝑥_ (Taylor series of order 𝒏)

𝑓 𝑥 = 𝑓 𝑥_ + 𝑓c 𝑥_ 𝑥 − 𝑥_ +𝑓cc 𝑥_2! (𝑥 − 𝑥_)2+…

𝑓(f) 𝑥_𝑛! (𝑥 − 𝑥_)f

§ Taylor series of 1st order about point 𝑥 = 𝑥_𝑓 𝑥 = 𝑓 𝑥_ + 𝑓c 𝑥_ 𝑥 − 𝑥_ + 𝑜(𝑥 − 𝑥_)

§ Linear approximation of the function about point 𝑥_𝑓 𝑥 ≈ 𝑓 𝑥_ + 𝑓c 𝑥_ 𝑥 − 𝑥_ = 𝑚(𝑥 − 𝑥_)+ 𝑞

Equation of the tangent line to 𝑓 in 𝑥_, the first derivative gives the slope

13

(SCALAR) FUNCTION LINEARIZATION

§ Taylor series for a scalar function of two variables 𝑓 𝑥-, 𝑥2 , 𝑛 times differentiable about a point 𝑥-, 𝑥2 = 𝑥_-, 𝑥_2 (Taylor series of order 𝟐)

§ Linear approximation of the function about point 𝑥_

𝑓 𝒙 = 𝑓 𝒙_ +𝜕𝑓𝜕𝑥-

𝒙_ 𝑥- − 𝑥_- +𝜕𝑓𝜕𝑥2

𝒙_ 𝑥2 − 𝑥_2 +

-2!

l7ml687

𝒙_ (𝑥- − 𝑥_-)2+l7ml677

𝒙_ (𝑥2− 𝑥_2)2+2l7m

l68l67𝒙_ 𝑥- − 𝑥_- 𝑥2− 𝑥_2 + 𝑜 𝒙 − 𝒙_ 2

Equation of the tangent plane to 𝑓in𝒙_, the gradient vector is orthogonal to the tangent plane (i.e., partial derivative determines the slope of the plane)

𝑓 𝒙 ≈ 𝑓 𝒙_ + lml68

𝒙_ 𝑥-− 𝑥_- + lml67

𝒙_ 𝑥2− 𝑥_2

𝑓 𝒙 ≈ 𝑓 𝒙_ + 𝛻o𝑓(𝒙_)p(𝒙 − 𝒙_)

14

(VECTOR) FUNCTION LINEARIZATION: JACOBIAN

§ The vector field 𝒇 of a generic dynamical system in the form of an ODE is a vector function of the 𝑛-dimensional state vector 𝒙:�̇� = 𝒇 𝒙 , 𝒇: ℝf → ℝf

§ Linear approximation of function component 𝑓r, 𝑖 = 1,… 𝑛, about point 𝑥_

§ Linear approximation of the vector function (vector field): Derivative (𝑓: ℝ → ℝ) à Gradient (𝑓: ℝ → ℝf) à Jacobian (𝒇:ℝt → ℝf)

§ �̇� = 𝒇 𝒙 ≈ 𝒇 𝒙_ + 𝐽𝒇(𝒙_) 𝒙 − 𝒙_ , 𝐽𝒇 (𝒙_) : 𝑛×𝑛 matrix, evaluated in 𝒙_

à 𝑛scalar functions of 𝑛-dim variables

�̇�- = 𝑓-(𝑥-, 𝑥2,… , 𝑥f)�̇�2 = 𝑓2(𝑥-,𝑥2,… , 𝑥f)

……�̇�f = 𝑓f(𝑥-,𝑥2,… , 𝑥f)

𝑓r 𝒙 ≈ 𝑓r 𝒙_ + ∑ lmwl6x

𝒙_ 𝑥y − 𝑥_yfyz- = 𝑓r 𝒙_ + 𝛻o𝑓r(𝒙_) p 𝒙 − 𝒙_

𝐽𝒇 =

𝜕𝑓-𝜕𝑥-

𝜕𝑓-𝜕𝑥2

⋯𝜕𝑓-𝜕𝑥f

⋮ ⋱ ⋮𝜕𝑓f𝜕𝑥-

𝜕𝑓f𝜕𝑥2

⋯𝜕𝑓f𝜕𝑥f

=

𝛻o𝑓-∶∶∶

𝛻o𝑓f

�̇� ≈ 𝒇 𝒙_ + 𝐽𝒇(𝒙_) 𝒙 − 𝒙_Assuming (𝒙_ = 𝟎, 𝒇 𝒙_ = 0):

𝐽𝒇 = Linear approximation near equilibrium 𝒙_

�̇� ≈ 𝐽𝒇 𝒙_ 𝒙 = 𝐴𝒙

15

RESULTS FROM LINEARIZATION

§ Theorem (Stability of critical points of non-linear systems):§ Let 𝑟- and 𝑟2 be the eigenvalues of the linear system �̇� ≈

𝐽𝒇 𝒙_ 𝒙 = 𝐴𝒙 resulting from the linearization of an original non-linear system about the critical point 𝒙 = 𝟎 (via the definition of a quasi-linear system)

§ The type and stability of the critical point on the linear and non linear system are the following:

16

EXAMPLE: DAMPED PENDULUM

Change of angular momentum about the origin = Moment of (Gravitational force + Damping force)

Second order ODE à Convert to a system of two 1st order equations:

https://www.youtube.com/watch?v=oWiuSp6qAPk

17

EXAMPLE: DAMPED PENDULUM

Critical points

𝜃 = 0Stable

𝜃 = 𝜋Unstable γ(𝑐) = 0

oscillatory

18

EXAMPLE: DAMPED PENDULUM

Critical points

𝐽𝒇 =

𝜕𝑓6𝜕𝑥

𝜕𝑓6𝜕𝑦

𝜕𝑓�𝜕𝑥

𝜕𝑓�𝜕𝑦

= 0 1−𝜔2 cos𝑥 −𝛾

(0,0)0 1

−𝜔2 −𝛾

(𝜋, 0)0 1𝜔2 −𝛾

§ In(0,0): 𝛾2 − 4𝜔2 < 𝛾

§ 𝛾2 − 4𝜔2 > 0: Damping is strong, eigenvalues are real, unequal, negative à (0,0) is an asymptotically stable node of the linear, as well as, non linear system

§ 𝛾2 − 4𝜔2 = 0: Eigenvalues are real, equal, negative à (0,0) is an asymptotically stable (proper or improper) node of the linear system. It may be either an asymptotically stable node or spiral point of the non linear system

§ 𝛾2 − 4𝜔2 < 0: Damping is weak, eigenvalues are complex with negative real part à(0,0) is an asymptotically stable spiral of both linear and non linear systems

19

DAMPED PENDULUM

§ à 0,0 is a spiral point of the system if the damping is small and a node if the damping is large enough. In either case, the origin is asymptotically stable.

§ Can we directly derive the direction of motion on the spirals near (0,0) with small damping, 𝛾2 − 4𝜔2 < 0? § Being a spiral about the origin, the trajectory will intersect the positive 𝑦-

-axis (𝑥 = 0 and 𝑦 > 0). At such a point, from equation 𝑑𝑥 𝑑𝑡⁄ = 𝑦 > 0it follows that the 𝑥-velocity is positive, meaning that the direction of motion is clockwise

§ Equilibrium at points (±𝑛𝜋, 0) with 𝑛 even is the same as in (0,0), these are in fact all corresponding to the same configuration of the downward equilibrium position of the pendulum

20

DAMPED PENDULUM

§ At equilibrium point (𝜋, 0), the eigenvalues of the Jacobian are:

Since 𝛾2 + 4𝜔2 > 𝛾, the eigenvalues always have opposite sign, 𝑟- > 0, 𝑟2 < 0,making the point a saddle à Regardless of the damping, the equilibrium is an unstable saddle (for both the linear and the original system)

§ The same applies to all other equilibrium points (𝑛𝜋, 0), with 𝑛 odd

§ How do we derive the direction of motion near the equilibrium?

21

DAMPED PENDULUM

§ General solution near the equilibrium:

§ Becauseof𝑟- > 0, 𝑟2 < 0, the linearized solution that approaches zero (i.e., the equilibrium point) as𝑡 → ∞, must correspond to 𝐶- = 0(otherwise either 𝑢 and/or v,would grow exponentially)

§ For this solution, the slope of entering trajectories is �� = 𝑟2 < 0, one lies in the first quadrant, the other in the fourth, as shown in the figure

§ The pair of (linearized) trajectories exiting from the saddle point correspond to 𝐶2 = 0, that have a constant slope 𝑟- > 0, and lies in 1st and 4th quadrant

22

BASIN OF ATTRACTION: A GLOBAL NOTION

§ Basin of attraction: The set of all initial points from which trajectories approach a given asymptotically stable critical point (region of asymptotic stability for that a critical point)

§ Eachasymptoticallystablecriticalpointhasitsownbasinofattraction,whichisboundedbytheseparatrices throughtheneighboringunstablesaddlepoints.

Separatrix

An instance of the pendulum model:§ The trajectories that enter the saddle points separate the phase plane into regions.

Such a trajectory is called a separatrix. Each region contains exactly one of the asymptotically stable spiral points.

Basin of attraction

23

NULLCLINES

§ Isoline: set of points where a function takes same value à𝑥-, 𝑥2 𝐻 𝑥-,𝑥2 = 𝑐}

§ Isocline: set of points where a function has the same slope, along some given coordinate directions à 𝑥-, 𝑥2

l� 68,67l6w

= 𝑐} § For a differential equation of the form �̇� = 𝑓 𝑥 , an isocline corresponds to

the set 𝑥 𝑓 𝑥 = 𝑐} (the slope / derivative is constant) à An isocline is an isoline of the vector field 𝑓 𝑥

§ Nullcline: set of points where a function has the same, null, slope. For a �̇� =𝑓 𝑥 , a nullcline corresponds to the set 𝑥 𝑓 𝑥 = 0}

§ For a system of differential equations �̇� = 𝒇(𝒙) , a nullcline is considered with respect to each coordinate direction: a system of two ODEs:

��̇�- = 𝑓-(𝑥-, 𝑥2)�̇�2 = 𝑓2(𝑥-,𝑥2)

has two nullclines sets, corresponding respectively to 𝑓-(𝑥-, 𝑥2) and 𝑓2(𝑥-,𝑥2)à The nullclines are the curves where either �̇�- = 0,or �̇�2 = 0

24

NULLCLINES

§ The nullclines can help to construct the phase potrait

§ Let’s consider a 2x2 system, the following properties hold:

§ The nullclines cross at the critical / equilibrium points

§ Trajectories cross vertically the nullcline 𝑓- 𝑥-, 𝑥2 = 0, since for this nullcline, �̇�- = 0, all flow variations happen along 𝑥2

§ Trajectories cross horizontally the nullcline 𝑓2 𝑥-, 𝑥2 = 0, since for this nullcline, �̇�2 = 0, all flows variations happen along 𝑥-

§ In regions enclosed by the nullclines, the ratio 568567has constant sign:

trajectories are either going upward to downward

§ Trajectories can only go flat or vertical across nullclines

25

NULLCLINES FOR A SIMPLE POPULATION MODEL

Nullclines:

= 𝐻 𝑁-,𝑁2 → Variableseparationandintegration:Solution: