Periodic oscillations - Politecnico di Torino

Transcript

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An introduction to synchronization in complex systems
Fernando Corinto
Politecnico di Torino
A.A. 2014–2015
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F.Corinto (Politecnico di Torino)
Synchronization in complex systems
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Aim of the course
Study of the asymptotic oscillatory behavior in nonlinear dynamic
systems
Spectral methods (Harmonic Balance and Describing Function
technique)
Examples of continuous-time nonlinear dynamic systems (Van der Pol
circuit and Chua circuit)
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Part I
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Limit Cycles
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Limit Cycles
Nonlinear systems/circuits: limit cycles
dx
“ f px, tq x P R n , t P R `
dt
A solution xptq “ Φpt, x0 q is said to be periodic if there exists T such that:
@ t : Φpt ` T , x0 q “ Φpt, x0 q
The image of Φpt, x0 q in the state-space (or phase-space) R n is called
periodic trajectory or limit cycle γ of period T .
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Limit Cycles
Linearsystems/circuits:
systems/circuits:limit
limitcycles
cycles
Linear
+
i
v
C
dv
dt
di
dt
L
−
Eigenvalues: λ12 = ±jω
= −
i
C
v
L
=
1
ω=√
LC
The
circuit
presents
infinitely
many
non-isolated
cycles
with
thethe
same
The
circuit
presents
infinitely
many
non-isolated
cycles
with
frequency.
The
cycle
amplitude
depends
on
the
initial
conditions.
same frequency. The cycle amplitude depends on the initial
conditions.
A. Ascoli, M. Biey, F. Corinto and M. Righero
An Introduction to synchronization in complex systems Part II - Periodic oscillations .
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Limit Cycles
Linear systems/circuits: limit cycles
iL
vC
The energy in the circuit is:
w ptq “ 0.5 L iL2 ` 0.5 C vC2 , ñ
dw
dvC
diL
“ C vC
` L iL
“0
dt
dt
dt
The following trajectories satisfy the state equation
0.5 L iL2 ptq ` 0.5 C vC2 ptq “ 0.5 L iL2 p0q ` 0.5 C vC2 p0q “ costant
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Limit Cycles
+
i
v
C
L
Rn
!i (v ) = −v + k v 3
−
1
dv
Figure:
Pol
= Van
− der
[i + ı̂(v
)] circuit
dt
C
v ´v ` k v 3
di ı̂pv q “
=
dt
L
The circuit presents a single limit cycle, that attracts all the
1
dv
trajectories.
“ ´ ri ` ı̂pv qs
dt
C
di
v
“
dt
L
The circuit presents a single limit cycle, that attracts all the trajectories.
A. Ascoli, M. Biey, F. Corinto and M. Righero
An Introduction to synchronization in complex systems Part II - Periodic oscillations
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Limit Cycles
Equilibrium point analysis
Equilibrium point x “ pv , iq “ p0, 0q
λ12
#
¨ 1
1 ˛
´
˚ C
C ‹
J “˝
‚
1
0
L
1
1 a 2
“
˘
L ´ 4LC
2C
2LC
L ă 4C
unstable focus
L ą 4C
unstable node
The circuit has no stable equilibrium points. It can be shown that voltage
and current are bounded as t Ñ 8.
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Limit Cycles
Bounded state
Let us consider the state function
V pv , iq “ 0.5Cv 2 ` 0.5Li 2 ñ
dV
“ v pv ´ kv 3 q
dt
It follows that
.
v ptq is bounded
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1. if v ptq Ñ 8 then V pv , iq Ñ 8, but this is impossible as
@|v | ă ?1k
2. As a consequence, v ptq Û 8 as t Ñ 8, i.e.
dV
dt
ă 0,
. DM such that |v ptq| ă M, @t
3
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Limit Cycles
Bounded state
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iptq is bounded
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1. if iptq Ñ ˘8 then
2. if iptq Ñ ´8 then
bounded)
3. if iptq Ñ `8 then
dv
dt
dv
dt
„ ´ Ci , because v ptq is bounded.
ą 0, ñ v ptq Ñ `8 is impossible (as v ptq is
dv
dt
di
ă 0, ñ v ptq Ñ 0, ñ dt
“
impossible (due to the assumption iptq Ñ `8)
v
L
Ñ 0 is
. it follows that DM such that |iptq| ă M, @t
4
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Limit Cycles
Limit cycle
L“
9
2
C “1
k “1
0.8
0.6
0.4
Current i(t)
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
−1.2
−1.5
−1
−0.5
0
0.5
1
1.5
2
Voltage v(t)
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Limit Cycles
Time waveforms
1.5
voltage v(t)
1
0.5
0
−0.5
−1
−1.5
0
5
10
15
20
25
30
35
40
45
50
30
35
40
45
50
time
1
current i(t)
0.5
0
−0.5
−1
−1.5
0
5
10
15
20
25
time
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Limit Cycles
Computation of limit cycles
Determination of all the periodic limit cycles (either stable and
unstable) and their stability properties (Floquets multipliers FMs)
In large scale dynamical systems the sole time–domain numerical
simulation does not allow to identify all the limit cycles (either stable
and unstable)
It would require to consider infinitely many initial conditions
Unstable limit cycles cannot be detected through simulation
By means of Spectral methods, the computation of all the limit cycles
is reduced to non-differential (sometimes algebraic) problem.
Harmonic Balance Technique
Describing Function Technique
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Limit Cycles
Computation of limit cycles: Time-domain methods
If the system possesses a stable cycle γ, we can try to find it by numerical
integration (simulation). If the initial point for the integration belongs to the
basin of attraction of γ, the computed orbit will converge to γ in forward
time. Such a trick will fail to locate a saddle cycle, even if we reverse time.
there exist different time domain methods especially to directly locate
periodic orbits even if they are saddle or unstable cycles. The problem of
finding the steady state is converted into a boundary-value problem, to
which the standard approaches, such as shooting methods and
finite-difference methods, can be applied.
Φpt0 ` T , x0 q “ Φpt0 , x0 q “ x0 , where the minimum cycle period T is
usually unknown. An extra phase condition has to be added in order to
select a solution among all those corresponding to the cycle.
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Limit Cycles
Spectral methods
Computation of limit cycles: Spectral methods
. Harmonic Balance (HB) and Describing function (DF) techniques
permit
1
to determine the set of all stable and unstable limit cycles.
to provide an accurate characterization of each limit cycle
- A. I. Mees, Dynamics of feedback systems, John Wiley, New York,
1981.
. Floquet’s multipliers permit
2
to investigate limit cycle stability and bifurcations. They can be
determined by exploiting either a time-domain or a frequency domain
approach.
- F. Bonani and M. Gilli, “Analysis of stability and bifurcations of limit
cycles in Chua’s circuit through the harmonic balance approach,” IEEE
Transactions on Circuits and Systems: Part I, vol. 46, pp. 881-890,
1999.
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Limit Cycles
Spectral methods
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Fundamental concepts
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1) A periodic solution Φpt, x0 q “ xptq “ xpt ` T q P R n can be expanded
through the Fourier series
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8
ÿ
xptq “ A0 `
Ak cospkωtq ` Bk sinpkωtq
ω“
k“1
A0
Ak
Bk
“
“
“
1
T
2
T
2
T
żT
0
żT
0
żT
xptqdt P R n
xptq cospkωtqdt P R n
xptq sinpkωtqdt P R n
0
.
2π
T
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Limit Cycles
Spectral methods
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2) Approximated representation of xptq by means of a finite number of
N harmonics
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xptq « A0 `
N
ÿ
k“1
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Limit Cycles
Spectral methods
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3) Given a non autonomous nonlinear dynamical system dx
dt “ f px, tq,
the r.h.s f px, tq can be expanded through the Fourier series by using
N harmonics
.
f px, tq “ F0 `
N
ÿ
FAk cospkωtq ` FBk sinpkωtq
ω“
k“1
F0
FAk
FBk
“
“
“
1
T
2
T
2
T
żT
0
żT
0
żT
f px, tqdt P R n
f px, tq cospkωtqdt P R n
f px, tq sinpkωtqdt P R n
0
.
2π
T
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Limit Cycles
Spectral methods
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4) Substitute xptq and f px, tq in
.
d
dt
¨
˝A0 `
N
ÿ
dx
dt
“ f px, tq, i.e.
˛
Ak cospkωtq ` Bk sinpkωtq‚ “ F0 `
k“1
N
ÿ
k“1
FA cospkωtq ` FB sinpkωtq
k
k
A set of 2N ` 1 nonlinear algebraic equations is obtained, by equating
the coefficients of the constant term and of the harmonics cospkωtq,
sinpkωtq
F0 pA0 , A1 , . . . , AN , B1 , . . . , BN q “ 0
FBk pA0 , A1 , . . . , AN , B1 , . . . , BN q “ ´kωAk
FAk pA0 , A1 , . . . , AN , B1 , . . . , BN q “ kωBk
with 1 ď k ď N.
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Limit Cycles
Spectral methods
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Harmonic Balance (1 ď k ď N)
.
F0 pA0 , A1 , . . . , AN , B1 , . . . , BN q
“
0
FBk pA0 , A1 , . . . , AN , B1 , . . . , BN q
FAk pA0 , A1 , . . . , AN , B1 , . . . , BN q
“
“
´kωAk
kωBk
2N
. ` 1 equations in 2N ` 1 unknowns
.
Describing Function (N “ 1)
.
F0 pA0 , A1 , B1 q “
FB1 pA0 , A1 , B1 q “
FA1 pA0 , A1 , B1 q “
.
0
´ωA1
ωB1
2N ` 1 equations in 2N ` 1 unknowns
xptq “ A0 ` A1 cospωtq ` B1 sinpωtq
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Limit Cycles
Duffing equation
DF Technique: Duffing
d 2 y ptq
dy ptq
`α
` npy ptqq “ β cospωtq,
dt 2
dt
where nonlinearity npy ptqq “ y ptq3 .
- Let us approximate y ptq with one harmonic only:
y ptq « A1 cospωtq ` B1 sinpωtq
Note: A0 “ 0 from simulations.
Unknowns are A1 and B1 (ω is given).
- Express nonlinearity npy ptqq as a Fourier series:
npy ptqq « N1B sinpωtq ` N1A cospωtq ` N3B sinp3ωtq ` N3A cosp3ωtq,
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Limit Cycles
Duffing equation
Duffing
Coefficients N1A , N1B , N3A and N3B are given by
N1A “
N1B “
N3A “
2
T
żT
2
T
żT
2
T
2
N3B “
T
npy ptqq cospωtqdt “
3
A1 pA21 ` B12 q
4
npy ptqq sinpωtqdt “
3
B1 pA21 ` B12 q
4
0
0
żT
npy ptqq cosp3ωtqdt “
1
A1 pA21 ´ 3B12 q
4
npy ptqq sinp3ωtqdt “
1
B1 p3A21 ´ B12 q
4
0
żT
0
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Limit Cycles
Duffing equation
Duffing
- Using y ptq « A1 cospωtq ` B1 sinpωtq
dy ptq
“ Á1 ω sinpωtq ` B1 ω cospωtq,
dt
d 2 y ptq
“ Á1 ω 2 cospωtq ´ B1 ω 2 sinpωtq
dt 2
Neglect npy ptqq’s 3rd harmonic: npy ptqq « N1A cospωtq ` N1B sinpωtq.
Equating coefficients in
d 2 y ptq
dt 2
` α dydtptq ` npy ptqq “ β cospωtq yields
Á1 ω 2 ` αB1 ω ` N1A ´ β “ 0
´B1 ω 2 ´ αA1 ω ` N1B “ 0
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Limit Cycles
Duffing equation
Duffing
Use N1A “ 34 A1 pA21 ` B12 q, N1B “ 34 B1 pA21 ` B12 q:
3
3
A1 ω 2 ´ A31 ´ A1 B12 ´ αB1 ω ` β “ 0
4
4
3
3
B1 ω 2 ´ B13 ´ A21 B1 ` αA1 ω “ 0
4
4
Letting ω “ 1, α “ 0.08 and β “ 0.2 yields
A “ 1.07287
B “ 0.608554
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Limit Cycles
Lur’e systems
Computation of limit cycles for Lur’e systems
L
s (t)
-1
(D )
x (t)
n ( ·)
LpDqxptq ` nrxptqs “ sptq,
xptq P R
If the systems admits of a periodic solution of period T , then xptq can be
expanded through the Fourier series
xptq “ A0 `
8
ÿ
ω“
k“1
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2π
T
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Limit Cycles
Lur’e systems
Examples
Third order oscillator
LpDq “
D 3 ` p1 ` αqD 2 ` βD ` αβ
α pD 2 ` D ` βq
8
4
npxq “ ´ x ` x 3
7
63
Second order oscillator
LpDq “
LCD 2 ´ LD ` 1
kLD
npxq “ x 3
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Limit Cycles
Lur’e systems
The harmonic balance (HB) technique
1. The state is represented through a finite (N) number of harmonics
N
ÿ
xptq “ A0 `
k“1
2. The term LpDqxptq yields:
LpDqxptq
“
Lp0qA0
`
N
ÿ
tRerLpjkωqsAk ` ImrLpjkωqsBk u cospkωtq
k“1
`
N
ÿ
tRerLpjkωqsBk ´ ImrLpjkωqAk u sinpkωtq
k“1
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Limit Cycles
Lur’e systems
3. The term nrxptqs yields (by truncating the series to N harmonics):
nrxptqs “ C0 `
N
ÿ
Ck cospkωtq ` Dk sinpkωtq
k“1
C0
Ck
Dk
“
“
“
1
T
żT
2
T
żT
2
T
żT
«
n A0 `
0
ff
Ak sinpkωtq ` Bk cospkωtq
«
n A0 `
N
ÿ
ff
cospkωtq dt
k“1
«
n A0 `
N
ÿ
ff
sinpkωtq dt
k“1
.
dt
k“1
0
0
N
ÿ
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Limit Cycles
Lur’e systems
3. The term sptq yields (by truncating the series to N harmonics):
sptq “ P0 `
N
ÿ
Pk cospkωtq ` Qk sinpkωtq
k“1
P0
Pk
Qk
“
“
“
1
T
żT
2
T
żT
2
T
żT
sptq dt
0
sptq cospkωtq dt
0
sptq sinpkωtq dt
0
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Limit Cycles
Lur’e systems
4. A set of 2N ` 1 nonlinear equations is obtained, by equating the
coefficients of the constant term and of the harmonics cospkωtq,
sinpkωtq
Lp0qA0
`
C0 pA0 , ..., BN q
“
P0
RerLpjkωqsAk ´ ImrLpjkωqsBk
`
Ck pA0 , ..., BN q
“
Pk
1ďkďN
ImrLpjkωqsAk ` RerLpjkωqsBk
`
Dk pA0 , ..., BN q
“
Qk
1ďkďN
5. Autonomous systems: the term A1 is assumed to be equal to zero
(i.e. the phase of the first harmonic of xptq is arbitrarily fixed); since
ω is unknown, the system has an equal number [p2N ` 1q] of
equations and unknowns.
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Limit Cycles
Van der Pol oscillator
DF Technique: Van der Pol oscillator
The Van der Pol equation (see Fig. 1) is described by the following
autonomous system:
¯
1 ´x
´ k npxq ´ y
C R
1
y9 “ x,
L
x9 “
where nonlinearity npxq “ x 3 .
Step 1: Expression for Lpsq is
Lpsq “
kLRs
LCRs 2 ´ Ls ` R
Step 2: Approximate xptq as xptq « B1 sinpωtq.
A0 “ 0 from simulations, A1 “ 0 since phase may be chosen arbitrarily.
Unknowns are B1 and ω.
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Limit Cycles
Step 3: express npxptqq “ x 3 as a Fourier Series:
npxptqq “ pB1 sinpωtqq3 « N1 sinpωtq ` N3 sinp3ωtq,
where N1 and N3 are
N1 “
N3 “
2
T
żT
2
T
żT
3 3
B
4 1
npxptqq sinpωtqdt “
0
0
1
npxptqq sinp3ωtqdt “ ´ B13
4
Step 4: Write differential equation for Lur’e system:
xptq ` LpDqnpxptqq “ 0,
where xptq « B1 sinpωtq and npxptqq « N1 sinpωtq (3ω neglected, since Fourier
series for npxptq and xptq are of same order)
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Limit Cycles
Step 4 (contd): express LpDqnpxptqq as
LpDqnpxptqq “ LpDqN1 sinpωtq “ N1 RetLpjωqu sinpωtq ` N1 ImtLpjωqu cospωtq.
Insert this into B1 sinpωtq ` LpDqnpxptqq “ 0 and equate coefficients:
B1 ` N1 RetLpjωqu “ 0,
N1 ImtLpjωqu “ 0
Expressions for RetLpjωqu and ImtLpjωqu are
RetLpjωqu “
´kRL2 ω 2
,
R 2 p1 ´ LC ω 2 q2 ` pLωq2
ImtLpjωqu “ 0 yields ω “
ImtLpjωqu “
?1
LC
B1 ` 34 B13 RetLpjωqu “ 0 gives B1 “
?2
3kR
.
kLR 2 ωp1 ´ LC ω 2 q
R 2 p1 ´ LC ω 2 q2 ` pLωq2
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Limit Cycles
Chua’s circuit
The circuit
Figure: Chua’s circuit
dV1
V2 ´ V1
“
´ iR pV1 q
dt
R
dV2
V1 ´ V2
C2
“
` I3
dt
R
dI3
L
“ ´V2
dt
C1
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Limit Cycles
Chua’s circuit
The circuit model
Chua’s circuit is described by the following autonomous nonlinear dynamical
systems
dx1
“ α p´x1 ` x2 ´ npx1 qq
dτ
dx2
“ x1 ´ x2 ` x3
x9 2 “
dτ
dx3
x9 3 “
“ ´βx2 ,
dτ
x9 1 “
where τ “ tR ´1 C2´1 and
x1 “ V1 , x2 “ V2 , x3 “ RI3 ,
npx1 q “ RiR pV1 q “ γx1 ` δx13 .
α “ C2 C1´1 , β “ R 2 C2 L´1 ,
γ “ ´8{7, δ “ 4{63.
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Limit Cycles
Chua’s circuit
Equilibria
2
n(x1 )
−x1
1.5
n(x1 ), −x1
1
0.5
x1 = −1.5
0
x1 = 1.5
−0.5
x1 = 0
−1
−1.5
−2
−2
−1.5
−1
−0.5
0
x1
0.5
1
1.5
2
Figure: Equilibria: x1 values
There are three equilibria:
x̄ “ px̄1 , x̄2 , x̄3 q “ p´1.5, 0, 1, 5q,
x̃ “ px̃1 , x̃2 , x̃3 q “ p0, 0, 0q,
x̂ “ px̂1 , x̂2 , x̂3 q “ p1.5, 0, ´1, 5q.
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Limit Cycles
Chua’s circuit
Matlab code
function ry , ts=Chua
γ “ ´8{7; δ “ 4{63; α “ 5; β “ 15;
x0 “ r0.5, 0.1, ´0.5s;
tspan=r0 : 0.01 : 200s;
options = odeset(’RelTol’,1e ´ 12,’AbsTol’,1e ´ 12,’Jacobian’,@J);
rt, y s = ode15s(@f ,tspan,x0,options);
plot(y p:, 1q,y p:, 2q,’b’)
function dydt “ f pt, xq
n “ r´α ˚ g pxp1qq 0 0s1 ;
A “ r´α α 0 ; 1 ´ 1 1 ; 0 ´ β 0s;
dydt “ A ˚ x ` n;
end
function dfdx “ Jpt, xq
dfdx “ r´α ˚ p1 ` δ ` 3 ˚ δ ˚ xp1q.ˆ2q α 0 ; 1 ´ 1 1 ; 0 ´ β 0s;
end
function y “ g pxq
y “ γ ˚ x ` δ ˚ x.ˆ3;
end
end
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Limit Cycles
Chua’s circuit
Numerical simulations for α “ 5, β “ 15
0.1
α = 5, β = 15, (x1 (0), x2 (0), x3 (0)) = (0.5, 0.1, −0.5)
−0.4
0.08
−0.6
0.06
−0.8
x3
−1
x2
0.04
0.02
−1.2
0
−1.4
−0.02
−0.04
0.5
Figure:
x1
1.5
−1.8
−0.05
2
x2
0.1
3
2.5
0.1
2
0
1.5
−0.1
1
−0.2
0.5
−0.3
0
−0.4
−0.5
−1.5
x1
−1
−0.5
−1
−0.5
0
0.5
x2
Evolution towards x̄ “ p´1.5, 0, 1, 5q.
.
0.15
3.5
0.2
Figure:
0.05
α = 5, β = 15, (x1 (0), x2 (0), x3 (0)) = (−1, 0.1, −0.5)
0.3
−0.5
−2
0
Evolution towards x̂ “ p1.5, 0, ´1, 5q.
x3
x2
0.4
−1.6
1
.
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Limit Cycles
Chua’s circuit
Figure:
Evolution towards a stable limit cycle located at x1 ą 1 and x3 ă ´1.
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Limit Cycles
Chua’s circuit
Figure:
Evolution towards a stable limit cycle located at x1 ă ´1 and x3 ą 1.
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Limit Cycles
Chua’s circuit
α = 9, β = 15, (x1 (0), x2 (0), x3 (0)) = (1, 0.1, −0.5)
0.3
1
0.5
0.2
0
0.1
−0.5
x3
x2
0
−1
−0.1
−1.5
−0.2
−2
−0.3
−0.4
0
Figure:
−2.5
0.5
1
x1
1.5
−3
−0.4
2
−0.2
0
x2
0.2
0.4
Stable period-2 cycle mostly located at x1 ą 1 and x3 ă ´1.
0.4
α = 9, β = 15, (x1 (0), x2 (0), x3 (0)) = (−1, 0.1, −0.5)
3
2.5
0.3
2
0.2
1.5
x3
x2
0.1
1
0
0.5
−0.1
0
−0.2
−0.3
−2
Figure:
−0.5
−1.5
−1
x1
−0.5
0
−1
−0.4
−0.2
0
x2
0.2
0.4
Stable period-2 limit cycle mostly located at x1 ă ´1 and x3 ą 1.
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Limit Cycles
Chua’s circuit
Numerical simulations for α “ 9.1, β “ 15
Figure:
Stable single-scroll chaotic attractor mostly located at x1 ą 1 and x3 ă ´1.
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Limit Cycles
Chua’s circuit
Figure:
Stable single-scroll chaotic attractor mostly located at x1 ă ´1 and x3 ą 1.
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Limit Cycles
Chua’s circuit
Figure:
Stable double-scroll chaotic attractor located at |x1 | ď 2 and |x3 | ď 3.
.
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Limit Cycles
Chua’s circuit
α = 9.5, β = 15, (x1 (0), x2 (0), x3 (0)) = (−1, 0.1, 1)
2
3
1.5
2
1
1
x3
x1
0.5
0
−0.5
0
−1
−1
−2
−1.5
−2
750
Figure:
800
850
τ
900
950
1000
−3
750
800
850
τ
900
950
1000
x1 pτ q and x3 pτ q contributing to the development of the double-scroll
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Limit Cycles
Chua’s circuit
Chua’s circuit: Computation of limit cycles
The parameters α and β are chosen in such a way that the Chua’s circuit
exhibits (e.g. α “ 8 and β “ 15):
Three unstable equilibrium points (denoted by P ` , P ´ , P 0 and
corresponding to x “ ˘1.5 and to x “ 0 respectively).
Two stable asymmetric limit cycles (denoted by A` and A´ ) mainly
lying in the regions x ą 1 and x ă ´1 respectively.
One stable symmetric limit cycle (denoted by S s ).
One unstable symmetric limit cycle (denoted by S u ).
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Limit Cycles
Chua’s circuit
Chua’s circuit: Complex dynamic behavior
s
k
Stable symmetric LC (S )
Stable asymmetric LCs (A±)
10
k
k
z (t)
5
0
−5
−10
3
2
Unstable symmetric LC (Suk)
1
4
2
0
yk(t)
0
−1
−2
−2
−3
−4
.
xk(t)
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Limit Cycles
Chua’s circuit
Chua’s circuit: Lur’e model
$
’
’
&
’
’
%
dx
dt
dy
dt
dz
dt
“ ´αx ` αy ´ αnpxq
“ x ý `z
“ ´βy
State equations can be cast in a Lur’e model
LpDqxptq “ ´αnrxptqs
Lpsq “
s 3 ` p1 ` αqs 2 ` βs ` αβ
s2 ` s ` β
8
4
npxq “ ´ x ` x 3
7
63
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Limit Cycles
Chua’s circuit
Efficient HB implementations
1. Consider the time samples vectors
y ptq “ LpDqxptq
y “ ry pt0 q, y pt1 q, ..., y pt2N q, y pt2N`1 qs1
x “ rxpt0 q, xpt1 q, ..., xpt2N q, xpt2N`1 qs1
s “ rspt0 q, spt1 q, ..., spt2N q, spt2N`1 qs1
tp “
T
p
2N ` 1
p “ 1, . . . , 2N ` 1
.
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Limit Cycles
Chua’s circuit
2. Impose that the HB equation be satisfied for t “ tp
y ptp q ` nrxptp qs “ sptp q,
p “ 1, . . . , 2N ` 1
that in vector notation yields
y ` nrxs “ s
with
nrxs “ r nrxpt1 qs, nrxpt2 qs, ..., nrxpt2N`1 qs s1
.
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Limit Cycles
Chua’s circuit
x “ Γ´1 X ,
»
Γ´1
1
—1
—
“ —.
– ..
c
γ1,1
c
γ2,1
..
.
X “ rA0 , A1 , ..., AN , B1 , ..., BN s1
s
γ1,1
s
γ2,1
..
.
...
...
c
γ1,N
c
γ2,N
..
.
s
γ1,N
s
γ2,N
..
.
fi
ffi
ffi
ffi
fl
c
s
c
s
1 γ2N`1,1
γ2N`1,1
. . . γ2N`1,N
γ2N`1,N
ˆ
˙
q2πp
c
γp,q “ cospqωtp q “ cos
2N ` 1
ˆ
˙
q2πp
s
γp,q “ sinpqωtp q “ sin
2N ` 1
.
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Limit Cycles
Chua’s circuit
y “ Γ´1 Ωpωq X
»
Lp0q 0
0
— 0
R
I
1
1
—
— 0
Í1 R1
—
Ωpωq “ — .
..
..
— ..
.
.
—
– 0
0
0
0
0
0
Rk “ RetLpjkωqu,
...
...
...
0
0
0
..
.
0
0
0
..
.
fi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
IN fl
RN
. . . RN
. . . ÍN
Ik “ ImtLpjkωqu
nrxs “ nrΓ´1 X s
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Limit Cycles
Chua’s circuit
y ` nrxs “ s
Γ´1 Ωpωq X ` nrΓ´1 X s “ s
Ωpωq X ` Γ nrΓ´1 X s “ Γ s
The 2N ` 1 equations in the 2N ` 1 unknowns X can be solved without
performing any integrals.
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Limit Cycles
Chua’s circuit
Limit cycle stability
Limit cycles may present the same stability characteristics of
equilibrium points: they may be stable, unstable or behave as saddles.
The stability of limit cycles is studied through the Poincarè map, that
reduces the stability property of a limit cycle to those of a nonlinear
discrete system.
The stability can also be studied through spectral techniques.
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Limit Cycles
Chua’s circuit
Periodic limit cycle: final remarks
. The describing function technique is very effective for detecting the
existence of limit cycles (either stable or unstable) and also for a preliminary
study of their bifurcations.
1
. The harmonic balance technique allows to determine with a good accuracy
the main limit cycle characteristics.
2
. HB based technique can be exploited for computing FMs, even in large-scale
systems (Gilli et al.).
3
. Once the limit cycle has been detected through a spectral technique, the
Floquet’s multipliers can also be computed via a time-domain technique.
The application to large arrays of nonlinear oscillators (Chua’s circuits) has
allowed to determine all the significant limit cycle bifurcations (Gilli et al.).
4
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Periodic oscillations - Politecnico di Torino

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