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In-Depth Information
1. Duffing's chaotic system:
x
1
.t/ D x
2
.t/
x
2
.t/ D 1:1x
1
.t/ x
1
.t/ 0:4x
2
.t/ C 1:8cos.1:8t/
(9.1)
2. The Genesio-Tesi chaotic system:
x
1
.t/ D x
2
.t/
x
2
.t/ D x
3
.t/
x
3
.t/ Dcx
1
.t/ bx
2
.t/ ˛x
3
.t/ C x
1
.t/
(9.2)
where a, b, and c are real constants. When at least one of the system's Lyapunov
exponents is larger than zero the system is considered to be chaotic. For example,
when a D 1:2, b D 2:92, and c D 6 the system behaves chaotically.
3. The Chen chaotic system
x
1
.t/ Da.x
1
.t/ x
2
.t//
x
2
.t/ D .c a/x
1
.t/ C cx
2
.t/ x
1
.t/x
3
.t/
x
3
.t/ D x
1
.t/x
2
.t/ bx
3
.t/
(9.3)
where a, b, and c are positive parameters. When at least one of the system's
Lyapunov exponents is larger than zero the system is considered to be chaotic.
For example, when a D 40, b D 3, and c D 3 the system behaves chaotically.
The evolution in time of the phase diagram of the Duffing oscillator and of the
Genesio-Tesi chaotic system are shown in Fig.
9.1
. Other dynamical systems that
exhibit chaotic behavior are Lorenz's system, Chua's system, and Rössler's system.
It will be shown that for chaotic dynamical systems exact linearization is possible,
through the application of a diffeomorphism that enables their description in new
state space coordinates. As an example, the Duffing oscillator will be used, while it
is straightforward to apply the method to a large number of chaotic oscillators.
9.1.2
Differential Flatness of Chaotic Oscillators
1. Differential flatness of Duffing's oscillator:
By defining the flat output y D x
1
one has x
1
D y, x
2
Dy and
D 1:8cos.1:8t/ Dy 1:1y C y
3
u
C 0:4 y. Thus all state variables and the
control input of the Duffing oscillator are expressed as functions of the flat
output and its derivatives, and differential flatness of the oscillator is proven.
Moreover, by defining the new control input
v
Dy 1:1y C y
3
C 0:4 y or
D 1:1x
1
x
1
0:4x
2
C
u
, and the new state variables y
1
D y and y
2
Dy
v
