15-382 collective intelligence – s18
TRANSCRIPT
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LECTURE 6:DYNAMICAL SYSTEMS 5
INSTRUCTOR:GIANNI A.DI CARO
15-382COLLECTIVE INTELLIGENCE – S18
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NON LINEAR SYSTEMS?
Four equilibria: 0, ±2 , (± 3, 1)Countably infinite equilibria: (𝑛𝜋, 0)
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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
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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
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STABILITY OF LINEAR SYSTEMS
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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
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“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
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“STABLE” PERTURBATIONS?
Repeated eigenvalues
Linearly independent eigenvectors
Linearly dependent eigenvectors
Stability doesn’t change, but trajectories change significantly
No: Perturbation àNo change in sign
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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!
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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?
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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 ?
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(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
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(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_)
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(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
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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:
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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
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EXAMPLE: DAMPED PENDULUM
Critical points
𝜃 = 0Stable
𝜃 = 𝜋Unstable γ(𝑐) = 0
oscillatory
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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
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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
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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?
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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
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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
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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
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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
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NULLCLINES FOR A SIMPLE POPULATION MODEL
Nullclines:
= 𝐻 𝑁-,𝑁2 → Variableseparationandintegration:Solution: