Advances in Mathematical Modeling:Dynamical Equations on Time Scales

Ian A. Gravagne

School of Engineering and Computer ScienceBaylor University, Waco, TX

Outline

• Background and Motivation• Intro to Time Scales

• Mathematical Basics• Software and Simulation

• Wrap Up

Background

“ A major task of mathematics today is to harmonize the continuous and the discrete, to include them in one comprehensive mathematics, and to eliminate obscurity from both.” E.T. Bell, 1937

Discrete + Continuous = …

… Time Scales!

Cantor sets, limit points, etc!

R

hZ

Pab

H0

h

ba

Where 99.9 % of engineering has taken place up to now…

• Body of theory springs from Ph.D. dissertation of S. Hilger in 1988.

• Captured interest of math community in 1993. First comprehensive monograph on subject published in 2002.

• Definition: a time scale is a closed subset of the real numbers: special case of a measure chain.

TerminologyForward Jump Operator:Backward Jump Operator:Graininess: ttt

tsTst

tsTst

)(:)(

}:sup{:)(

}:inf{:)(

)(t)(t t

)(t1t 2t 3t 4t

t1 is isolated

t2 is left-scattered (right-dense)

t3 is dense

t4 is right-scattered (left-dense))()(

)()(

)()(

)()(

ttt

ttt

ttt

ttt

Operators

• Derivatives:

)(

)())((:)(

t

tftftf

CT:f

(The delta-derivative only exists for and . This offer expires 11/21/03.) }{max: TTT t rdCf

f

dt

df

0

1

• Integrals:

t

tftF

0

)()( )()(

)()(

)(

0

0

i

t

i

t

t

tftF

dftF

0

1

(Hilger integrals only exist if and over .)rdCf regulated is f Tt

Diff/Int Rules

• Product Rule for differentiation

fggfgfgffg )( ))((: tff

• Chain Rule for differentiation

1

0 )()()(')()( dhtgthtgftggf

• No more “rules of thumb” for differentiation!!

• Very few closed-form indefinite integrals known.

tt 2)( 2

b

a

b

attgtfafgbfgttgtf

)()()())(()()(

• Integration by Parts

Derivatives and Integrals are linear and homogeneous.

“Differential” Equations

• The first (and arguably most important) dynamical equation to examine is

T ttxtxtptx ,1)( ),()()( 0

The solution is

t

tp ttetx

00 )(

))()p(log(1exp:),()(

)(0

0),( ttpp ette )( ,)1(),( 0

10 ttordkptte k

p

constp ,0 constp ,1

The TS exponential exists iff If then

T ttttRtp }0)()(1:)({:)( RTp )( TtTx for 0)(

Properties of TS exp

),()(),(

),(

),(),(),(

),(),(),(

),(

1),( and 1),(

),(

),(

),(1

0

stetpste

ste

stesteste

rterseste

ste

tteste

pp

qpste

ste

qpqp

ppp

pste

p

q

p

p

Why do we need ?

Operators form a Lie Group on the Regressive Set with identity

))(()(:))((

)()()()()(:))((

)()(1

)(:)(

tqtptqp

tqtpttqtptqp

tpt

tptp

,

},{ R0I

Higher Order Systems

• As expected, solutions to higher order linear systems are sums of ),( 0ttep

0)(...)()()( 11

1

xtatxtatx

nn

n

00 )( ),()()( xxxAx tttt

),()( 0ttet A0xx

n

ip ttetx

i1

0),()(

)...,(sinh),,(cosh

),)((:),(cos

),)((:),(sin

00

021

0

021

0

tttt

tteett

tteett

pp

jjj

jj

Leads to logical definitions

• Alternatively, systems of linear equations are also well-defined:

• Need tttIt nn }0)()(:{:)( RA

Properties of TS sin, cos…

2

2

22

22

sinhcosh

coshsinh

sinhcosh

sincos

cossin

sincos

ppp

pp

pp

e

p

p

e

Thought of the day: the “natural” trig functions (i.e. above) are defined as the solutions to a 2nd (or 4th) order undamped diff. eqs. They cannot alias no matter how high the “frequency”!

Notes:

later. Morediverge.or convergemay

points scattered of # infinite iff diverges always

0 iff 1

2

2

2

e

e

e

Other TS work

We have only scratched the surface of existing work in Time Scales:

• Nabla derivatives:

• PDE’s:

• Generalized Laplace Tranform:

• Ricatti equations, Green’s functions, BVPs, Symplectic systems, nonlinear theory, generalized Fourier transforms.

)()( );()()( tfxtxtxtptx

)(),(),(),( sftsbxtskxtsxm

0 )0,()(:)}({ ttetxzxL z

OK, OK… But what do these things look like??

TS Toolbox

Worked with John Davis, Jeff Dacunha, Ding Ma over summer ’03 to develop first numerical routines to:• Construct and manipulate time scales• Perform basic arithmetic operations• Calculate• Solve arbitrary initial-value ODEs• Visualize functions on timescales

Routines were written in MATLAB.

...),(,cosh,sinh,cos,sin, etcte ppppp

Time Scale Objects

It quickly became apparent that we would need to use MATLAB’s object-oriented capabilities:• A time scale cannot be effectively stored as a simple vector or array.• Need to overload arithmetic functions, syntax

Is T=[0,1,2,3,4,5,6,7,8,9,10]• an isolated time scale?• a discretization of a continuous interval? • a mixture?

Need more information: where are the breaks between intervals, and what kind of intervals are they: discrete or continuous.

Package this info up into an object…

}10,9,...,2,1,0{T]10,0[T

}10,9,8,7,6{]5,0[ T

Time Scale Objects 2

Solution:

T.data=[0,0.1,0.2,0.3,0.4,0.5,1,1.5,2,2.1,2.3,2.4,2.5]

T.type=[6 ,0 8 ,1 13,0]

]5.0,0[ }5.1,1{ ]5.2,2[

Shows final ordinal for last point in intervalShows whether interval is discrete (1) or continuous (0)

OverloadsNow we can overload common functions, e.g. + - * / ^ as well as syntax, e.g. [ ], ( ), : etc…

Overloads 2

Graphics

The “tsplot” function plots time scale images, and colors the intervals differently.

The TS exponential

TS exponential on the time scale 5.0,5.0PT

)0,(1 te)0,(2 te)0,(4 te

teIf then

at

2p0)(1 tp

5.0t

More TS ExpTS exponential on the first 20 harmonics.

)0,(10 te

AD AC MC

Definition:The Hilger Circle is

}11:{ HC

-10

Im

Sin, Cos

)0,(sin4 t)0,(cos4 t

)0,(16 te

Sin, Cos on a logarithmic time scale.

Fin!

• Dynamical Equations on Time Scales == powerful tool to model systems with mixtures of continuous/discrete dynamics or discrete dynamics of non-uniform step size.

• Mathematics very advanced in some ways, but in other ways still in relative infancy.

• Need to overcome “rut thinking”