Advances in Mathematical Modeling: Dynamical Equations on Time Scales
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Advances in Mathematical Modeling:Dynamical Equations on Time Scales
Ian A. Gravagne
School of Engineering and Computer ScienceBaylor University, Waco, TX
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Outline
• Background and Motivation• Intro to Time Scales
• Mathematical Basics• Software and Simulation
• Wrap Up
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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
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Discrete + Continuous = …
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… 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.
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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
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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
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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.
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“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)(
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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
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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
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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
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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??
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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
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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
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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)
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OverloadsNow we can overload common functions, e.g. + - * / ^ as well as syntax, e.g. [ ], ( ), : etc…
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Overloads 2
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Graphics
The “tsplot” function plots time scale images, and colors the intervals differently.
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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
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More TS ExpTS exponential on the first 20 harmonics.
)0,(10 te
AD AC MC
Definition:The Hilger Circle is
}11:{ HC
-10
Im
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Sin, Cos
)0,(sin4 t)0,(cos4 t
)0,(16 te
Sin, Cos on a logarithmic time scale.
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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”