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Probabilistic Models for Dynamical Systems, 2nd Edition
book

Probabilistic Models for Dynamical Systems, 2nd Edition

by Haym Benaroya, Seon Mi Han, Mark Nagurka
May 2013
Intermediate to advanced content levelIntermediate to advanced
764 pages
6h 43m
English
CRC Press
Content preview from Probabilistic Models for Dynamical Systems, 2nd Edition
10.18. Solve Equation 10.54 for 1 using the approach suggested in the footnoted Equation 10.59.
Solution: Start with
θ

+
2
θ + εθ
3
= Γ cos τ.
Let
2
= 1 + εβ and Γ = εγ, expand θ as θ(τ ) = θ
0
+ εθ
1
, and sub into the original equation
θ

0
+ εθ

1
+ (1 + εβ)(θ
0
+ εθ
1
) + ε(θ
0
+ εθ
1
)
3
= εγ cos τ.
Separating by terms produces
θ

0
+ θ
0
= 0
θ

1
+ θ
1
+ βθ
0
+ θ
3
0
= γ cos τ.
The solution for θ
0
is given by
θ
0
(τ) = b
0
cos(τ).
Using the initial conditions θ(0) = b
0
and θ
(0) = 0. This simplifies the equation for θ
1
to
θ

1
+ θ
1
= γ cos τ β (b
0
cos(τ)) (b
0
cos(τ))
3
= [γ βb
0
3
4
b
3
0
] cos(τ)
1
4
b
3
0
cos(3τ).
To avoid secular terms, choose b
0
such that
γ βb
0
3
4
b
3
0
= 0.
The solution for ...
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Publisher Resources

ISBN: 9781439849897