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THE DIFFERENTIAL EQUATION THAT SOLVES EVERY PROBLEM OR HOW TO LIE WITH UNIVERSAL EQUATIONS

Tamás Kalmár-Nagy

Budapest University of Technology and Economics

Balázs Sándor

Department of Hydraulic and Water Resources Engineering Budapest University of Technology and Economics Griffith University, School of Engineering, Gold Coast, Australia

II I III IV

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SOLVING EVERY PROBLEM

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THANK YOU!

QUESTIONS?

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SOLVING EVERY PROBLEM

Theorem (Rubel, 1981) : There exists a fourth-

• rder Algebraic Differential Equation (ADE)

Py,y,y,y  0

that has a solution arbitrarily close to any smooth function for all t

Def: Let F be a class of real-valued functions defined on D⊆R. An ADE is called universal with respect to F(D), if every continuous function on F can be uniformly approximated by F-solutions of this ADE on D.

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CONSTRUCTING UNIVERSAL EQUATIONS

Universal Equation Recipe:

• 1. Take an ounce of sigmoid
• 2. Differentiate, then differentiate again
• 3. Eliminate constants
• 4. Shake, not stir

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• 1. TAKE AN OUNCE OF SIGMOID

finite sigmoid f on [0,1] f0  0, f1  1 f0  f1  0 ft  0, t  0,1. # # #

yt  Aft    B

The constants A, B, α, β serve to shift and scale the sigmoid

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WHY A SIGMOID?

f is a sigmoid on [0,1]

yt  Aft    B

Any continuous function can be arbitrarily approximated by concatenating scaled and shifted sigmoids

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• 2. DIFFERENTIATE, THEN DIFFERENTIATE AGAIN

yt  Aft  

yt  A2ft  ,  ynt  Anfnt  . #

The constant B can simply be eliminated by differentiating More constants? More differentiation…

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• 3. ELIMINATE CONSTANTS

Py,,yn  PAf,,Anfn  0

that are independent of the constants A and α. Look for equations of the form

gt  ft  

PAg,A2g,A3g,A4g,   0

Introduce the “kernel”

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• 4. SHAKE, NOT STIR (RUBEL’S EXAMPLE)

Assume a given form for the kernel

gt  e

1 t21

1  t  1

• therwise

3y4yy2  4y4y2y  6y3y2yy  24y2y4y  12y3yy3  29y2y3y2  12y7  0. #

PAg,A2g,A3g,A4g  0

Shake with some algebra

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OTHER UNIVERSAL EQUATIONS

2n2  3n  1y3  3n  1nyyy  n2y2y  0

Duffin 1981

gt  1  t2n

yy2  3yyy  21 

1 n y3  0 Briggs 2002

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CAN WE CONSTRUCT ALL UNIVERSAL EQUATIONS?

To "eliminate" the constants A and α we need P be a homogeneous polynomial of degree D

PAg,A2g,A3g,A4g,   AgDP  g

g ,2 g g ,3 g g ,

 0

Introduce the logarithmic derivative

ht 

gt gt

g  hg, g  hg  hg  h  h2g, g  h  3hh  h3g, g4  h  4hh  3h2  6h2h  h4g # # # #

Ph,2h  h2,3h  3hh  h3,4h  4hh  3h2  6h2h  h4,  



i

cib12b23b34b4hb1h  h2b2h  3hh  h3b3h  4hh  3h2  6h2h  h4b4 0. # #

b1  2b2  3b3  4b4  W

weight

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CONSTRUCTING UNIVERSAL EQUATIONS

The derivatives of h in terms of the derivatives of y

h  y y , h  yy  y2 y2 , h  2y3  3yyy  y2y y3 , h  6y4  12yy2y  3y2y2  4y2yy4  y3y5 y4 . #

h  h2  y y , h  3hh  h3  y4 y , h  4hh  3h2  6h2h  h4  y5 y ,



i b12b23b34b4W

ci

y y b1 y y b2 y4 y b3 y5 y b4 0

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CONSTRUCTING UNIVERSAL EQUATIONS

W  2

b1  2b2  2

b1 b2 W  bi 1 1 2



i,b12b2W

ciy

W biyb1yb2  c1yy  c2y2

The simplest Universal Differential Equation?

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CONSTRUCTING UNIVERSAL EQUATIONS

W  3

b1  2b2  3b3  3

b1 b2 b3 3 1 1 1 W  bi 1 2



i,b12b23b3W

ciy

W biyb1yb2yb3  c1y3  c2yyy  c3y2y

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CONSTRUCTING UNIVERSAL EQUATIONS

W  5

b1  2b2  3b3  4b4  5b5  5

b1 b2 b3 b4 b5 W  bi 5 3 1 1 1 2 2 2 1 2 1 1 3 1 1 3 1 4



i,b12b23b34b45b5W

ciy

W biyb1yb2yb3y5 b4y6 b5 

c1y5  c2yy3y  c3y2y2y4  c4y2yy2  c5y3yy5  c6y3yy4  c7y4y6 # #

WHAT ARE THE CORRESPONDING KERNELS/SIGMOIDS?

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ANOTHER DIRECTION: LIE GROUPS

The theory of Lie-groups contains powerful tools to analysing differential

• equations. It has been developed by

Marius Sophus Lie at the end of the XIX.-th century.

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LIE-POINT TRANSFORMATION GROUPS AS SYMMETRY GROUPS OF DIFFERENTIAL EQUATIONS

Let G be a Lie-group acting on the M manifold. Let M be locally the total space of the independent (m) and dependent (n) variables of a differential equation. G is a symmetry group of the differential equation if it transforms its solutions into another solutions on M. Example 1.: Let G be the one parameter group of the shifts of the independent variable (t) of an ODE (with y(t) as unknown). The group elements act on the (t,y) plane as:

= ≃ ℝ × ℝ ∗ = + ε, ∗ = → , , ε = + ε, , , ε = ∈ , , ∈ ≃ ℝ × ℝ

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AFFINE GROUP OF THE SOLUTION PLANE OF UNIVERSAL DIFFERENTIAL EQUATIONS

ft  Aft    B

To be universal, a differential equation must have a solution with which an arbitrary function can be uniformly approximated. Hence a universal differential equation must be form invariant under translations and scaling. Working definition An algebraic differential equation is universal if i.) it has a nonconstant solution whose derivatives vanish at the endpoints of a finite interval, ii.) it has the four parameter symmetry group of the affine transformations of the plane of the independent and dependent variable.

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AFFINE GROUP OF THE SOLUTION PLANE OF UNIVERSAL DIFFERENTIAL EQUATIONS

We introduce the affine group of the plane of the independent (t) and dependent variable (y) with its Lie algebra in operator form:

g  v1,v2,v3,v4

v1 

 t , v2  t  t , v3   y , v4  y  y

translation scaling The affine group of the plane is a symmetry group of the differential algebraic expression

Py,,yn : Rn  R

if it is invariant under the n-th extension (prolongation) of the infinitesimal generators

v1 nP  0, v2 nP  0, v3 nP  0, v4 nP  0

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AFFINE GROUP OF THE SOLUTION PLANE OF UNIVERSAL DIFFERENTIAL EQUATIONS

The n-th prolongation of the group action (with the infinitesimal operator representation ) describes how the derivatives of y ( ) behave under the group

• action. The invariance condition is stated in the n-th jet

space Jⁿ:(t, ) of the total space of the independent, and dependent variable (the (t,y) plane).

vn

yt ,ytt ,,yn

v  t,y 

t  t,y  y

vn   

t    y  1  yt  2  ytt    n  yn

1t,y,yt  Dt  ytDt, 2t,y,yt,ytt  Dt1  yttDt,  nt,y,yt,ytt,,yn  Dtn1  ynDt, #

Dt 

 t  yt  y  ytt  yt  

yt ,ytt ,,yn

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EXAMPLE: RUBEL’S EQUATION v1 n  v1 , v2 n  v2  yt  yt  2ytt  ytt    nyn  yn , v3 n  v3 , v4 n  v4  yt  yt  ytt  ytt    yn  yn . # PRubel  Py,y,y,y  3y4yy2  4y4y2y  6y3y2yy  24y2y4y  12y3yy3  29y2y3y2  12y7, #

v1 4 PRubel  0, v2 4 PRubel  0, v3 4 PRubel  0, v4 4 PRubel  0

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SOLVING EVERY PROBLEM

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THE MINIMAL UNIVERSAL EQUATION

The simplest universal equation

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FIRST INTEGRALS

First integrals are conserved quantities of the underlying dynamical system

• provides qualitative information about behavior
• can be used to reduce the order of the differential

equation

• useful in stability analysis
• can many times be related to the symmetries

Construction: many times ad hoc Here: an intuitive approach, backed by Lie methods

x   fx,x 

 : R2  R; x  x

First integral

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THE TOTAL DERIVATIVE

Integrating factor

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GEOMETRIC VIEW

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SECOND ORDER EQUATIONS

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USING LIE

Symmetry condition Parabolic PDE!

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EXAMPLES

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SYMMETRY EQUATION

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DAMPED OSCILLATOR

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NONLINEAR OSCILLATOR

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THANK YOU!

QUESTIONS?