PEF5737_ParametricExc

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PEF 5737 - Nonlinear dynamics and stability
Dynamics of parametrically excited systems: Floquet theory and
the use of the method of multiple scales
Prof. Carlos Eduardo Nigro Mazzilli
Associate Professor Guilherme Rosa Franzini
Escola Politécnica, University of São Paulo, Brazil
cenmazzi@usp.br
gfranzini@usp.br
05/04/2022
Mazzilli & Franzini (EPUSP) PEF5737 05/04/2022 1 / 40
PEF 5737 - Nonlinear dynamics and stability
Dynamics of parametrically excited systems: Floquet theory and
the use of the method of multiple scales
Prof. Carlos Eduardo Nigro Mazzilli
Associate Professor Guilherme Rosa Franzini
Escola Politécnica, University of São Paulo, Brazil
cenmazzi@usp.br
gfranzini@usp.br
05/04/2022
Mazzilli & Franzini (EPUSP) PEF5737 05/04/2022 2 / 40
Objectives and references
Outline
1 Objectives and references
2 Autonomous systems
3 Non-autonomous systems
4 Bifurcation of periodic orbits
5 Application to the Mathieu’s equation
6 Using the method of multiple scales (MMS)
Mazzilli & Franzini (EPUSP) PEF5737 05/04/2022 3 / 40
Objectives and references
Objectives and references
To introduce the study of parametrically excited systems, particular those governed by the
Mathieu’s equation;
To investigate the stability of periodic orbits using Floquet theory;
To derive the transition curves of the Strutt’s diagram using the method of multiple scales
(MMS).
References
1 Mailybaev, A. A. (2019) Introduction to the theory of parametric resonance.
2 Nayfeh, A. & Balachandran, B. (1995), Applied nonlinear dynamics - analytical,
computational and experimental methods;
3 Wiggins, S (1990). Introduction to applied nonlinear dynamical systems and chaos.
Mazzilli & Franzini (EPUSP) PEF5737 05/04/2022 4 / 40
Autonomous systems
Outline
1 Objectives and references
2 Autonomous systems
3 Non-autonomous systems
4 Bifurcation of periodic orbits
5 Application to the Mathieu’s equation
6 Using the method of multiple scales (MMS)
Mazzilli & Franzini (EPUSP) PEF5737 05/04/2022 5 / 40
Autonomous systems
Mathematical model

ẋ1
ẋ2
ẋ3
...
ẋn

=

f1(x1, x2, x3, . . . ,M)
f2(x1, x2, x3, . . . ,M)
f3(x1, x2, x3, . . . ,M)
...
fn(x1, x2, x3, . . . ,M)

→ ẋ = F (x,M) (1)
M being a m-dimensional vector with the parameters of the mathematical model.
Mazzilli & Franzini (EPUSP) PEF5737 05/04/2022 6 / 40
Autonomous systems
Perturbation
X0 =X0(t) is a periodic solution of Eq. 1 with minimal period T at M =M0. Let y = y(t)
a disturbance to be superimposed to X0 such that X0 + y = x(t) = x. Assuming F (C2), Eq.
1 can be linearized around X0, leading to
ẏ ≈ A(t,M0)y,A(t,M0) =DXF (X0,M0) (2)
where A(t,M0) is periodic with period T .
Mazzilli & Franzini (EPUSP) PEF5737 05/04/2022 7 / 40
Autonomous systems
Solution
Equation 2 is linear → It has n linearly independent (LI) solutions (fundamental set of
solutions) yi(t), i,= 1, 2, . . . , n;
We gather these solutions as the columns of a fundamental matrix solution
Y (t) = [y1(t) y2(t) y3(t) . . .yn(t)]→ Ẏ = A(t,M0)Y ;
Change of variables τ = t+ T → Eq. 2 becomes (notice that A(τ,M0) = A(τ − T,M0)
dY
dτ
= A(τ − T,M0)Y = A(τ,M0)Y (3)
We conclude that Y (t+ T ) = [y1(t+ T ) y2(t+ T ) y3(t+ T ) . . .yn(t+ T )] is
another fundamental matrix solution.
Mazzilli & Franzini (EPUSP) PEF5737 05/04/2022 8 / 40
Autonomous systems
Monodromy matrix
As Eq. 2 has n LI solutions, yi(t+ T ) is a linear combination of y1(t),y2(t) . . .yn(t);
In a more compact form Y (t+ T ) = Y (t)Φ, Φ unknown (up to now). Φ has dimension
n× n, depends on the choice of Y (t) and can be seen as a map from the initial condition
to T ;
If Y (0) = I, then Φ = Y (T ). Φ is the monodromy matrix.
Mazzilli & Franzini (EPUSP) PEF5737 05/04/2022 9 / 40
Autonomous systems
A transformation
Let P be a non-singular constant matrix of order n× n and we define Y (t) = V (t)P−1;
Y (t+ T ) = Y (t)Φ↔ V (t+ T )P−1 = V (t)P−1Φ↔
V (t+ T ) = V (t)P−1ΦP︸ ︷︷ ︸
J
(4)
We can conveniently choose J ...
Mazzilli & Franzini (EPUSP) PEF5737 05/04/2022 10 / 40
Autonomous systems
Case 1 - Φ has distinct eigenvalues ρm
In this case, we choose P = [p1 p2 . . .pn], pm being the eigenvectors of Φ →
Canonical Jordan form
J = P−1ΦP = P−1Φ[p1 p2 . . .pn] =
= P−1[Φp1 Φp2 . . .Φpn] = P−1[ρ1p1 ρ2p2 . . . ρnpn] =
= P−1PD =D =

ρ1 0 . . . 0
0 ρ2 . . . 0
...
...
. . .
...
0 0 . . . ρn
 (5)
Each ρm is named Floquet or characteristic multiplier → gives a measure of local
divergence or convergence of the orbit in a particular direction.
Mazzilli & Franzini (EPUSP) PEF5737 05/04/2022 11 / 40
Autonomous systems
Case 1 - Φ has distinct eigenvalues ρm
From Eqs. 5 and 4:
vm(t+ T ) = ρmvm(t),m = 1, 2, . . . , n↔
↔ vm(t+NT ) = ρNmvm(t),m = 1, 2, . . . , n and N integer (6)
Notice that N →∞ means t→∞. Eq. 6 indicates that the stability of the periodic
orbits can be assessed by the eigenvalues of the monodromy matrix;
vm(t)→ 0 if |ρm| < 1;
vm(t)→∞ if |ρm| > 1.
Mazzilli & Franzini (EPUSP) PEF5737 05/04/2022 12 / 40
Autonomous systems
Case 2 - Φ has some repeated eigenvalues ρm
In this case, the behavior of the system depends on the algebraic and geometric
multiplicities (am and gm, respectively) of the Floquet multipliers.
If am = gm > 1, we can use the approach aforementioned;
Now, we focus on the case with gm < am.
Mazzilli & Franzini (EPUSP) PEF5737 05/04/2022 13 / 40
Autonomous systems
Jordan chains
A Jordan chain can be defined as:
Φu1 = ρu1
Φu2 = ρu2 + u1
Φu3 = ρu3 + u2
... (7)
Φul = ρul + ul−1
Φul+1 6= ρul+1 + ul
u1 being the eigenvector and the u2 . . . ,ul the generalized eigenvectors.
The Jordan chain can be written as ΦP = PJ , with J =

ρ 1 . . . 0
0 ρ . . . 0
...
...
. . .
...
0 0 . . . ρ
 being a
Jordan block. The Jordan canonical form is composed of a number of Jordan blocks,
according to gm.
Mazzilli & Franzini (EPUSP) PEF5737 05/04/2022 14 / 40
Autonomous systems
A comment valid for autonomous systems
Taking the derivative of Eq. 1
ẍ =DXF (x,M)ẋ (8)
If x is a solution of Eq. 1, ẋ is solution of both Eqs. 8 and 2;
Provided X0(t) is solution of Eq. 1, Ẋ0(t) is solution of Eq. 2 and has period T . Hence
Ẋ0(t) = Ẋ0(t+ T );
We write Ẋ0(t) as a linear combination of y1(t),y2(t), . . . ,yn(t) as:
Ẋ0(t) = Y (t)α (9)
α being a vector of constants.
From Eq. 9: Ẋ0(0) = Y (0)α and Ẋ0(T ) = Y (T )α.
Mazzilli & Franzini (EPUSP) PEF5737 05/04/2022 15 / 40
Autonomous systems
A comment valid for autonomous systems
Provided X0(t) has period T , recalling that Y (0) = I and using the definition of the
monodromy matrix;
Φα = α (10)
From Eq. 10, one can notice that 1 is an eigenvalue of the monodromy matrix;
Definition (autonomous systems): A periodic solution of Eq. 1 is named as hyperbolic if
only one Floquet multiplier is located on the unit circle;
Definition (autonomous systems): A periodic solution of Eq. 1 is named as
non-hyperbolic if more than one Floquet multiplier are located on the unit circle;
The Hartman-Grobaman’s theorem is valid for hyperbolic periodic orbits.
Mazzilli & Franzini (EPUSP) PEF5737 05/04/2022 16 / 40
Autonomous systems
Definitions for autonomous systems
A hyperbolic period solution is asymptotically stable if no Floquet multiplier is outside the
unit circle → periodic attractor or stable limit cycle;
A hyperbolic period solution is unstable stable if at least on Floquet multiplier is outside
the unit circle → periodic repellor or unstable limit cycle;
Mazzilli & Franzini (EPUSP) PEF5737 05/04/2022 17 / 40
Non-autonomous systems
Outline
1 Objectives and references
2 Autonomous systems
3 Non-autonomous systems
4 Bifurcation of periodic orbits
5 Application to the Mathieu’s equation
6 Using the method of multiple scales (MMS)
Mazzilli & Franzini (EPUSP) PEF5737 05/04/2022 18 / 40
Non-autonomous systems
Non-autonomous systems
Non-autonomous system of first-order differential equationsẋ = F (x,M , t) (11)
Similarly to what was carried out for the autonomous systems, we study the stability of
the T -periodic solution X0 =X0(t) of Eq. 11 with M =M0;
We superimpose the perturbation z = z(t) to X0, obtaining x(t) = x =X0 + z;
Expanding Eq. 11 in Taylor series and neglecting higher-order terms, one obtains:
ż = A(t,M0)z (12)
Mazzilli & Franzini (EPUSP) PEF5737 05/04/2022 19 / 40
Non-autonomous systems
Definitions for non-autonomous systems
Practically all the discussions already made for the autonomous systems remain valid for
non-autonomous systems;
Notice, however, that 1 is not mandatory an eigenvalue of the monodromy matrix;
A periodic solution of 1 is named as hyperbolic if no Floquet multiplier is located on the
unit circle;
A periodic solution of 1 is named as non-hyperbolic if at least one Floquet multiplier is
located on the unit circle;
If all ρm are located within the unit circle → Periodic solution is a stable limit-cycle;
If some ρm is located outside the unit circle → Periodic solution is a unstable.
If all ρm are located outside the unit → Periodic solution is an unstable limit cycle;
If some but not all ρm are located outside the unit circle→ Periodic solution is a saddle;
The Hartman-Grobaman’s theorem is valid for hyperbolic periodic orbits.
Mazzilli & Franzini (EPUSP) PEF5737 05/04/2022 20 / 40
Bifurcation of periodic orbits
Outline
1 Objectives and references
2 Autonomous systems
3 Non-autonomous systems
4 Bifurcation of periodic orbits
5 Application to the Mathieu’s equation
6 Using the method of multiple scales (MMS)
Mazzilli & Franzini (EPUSP) PEF5737 05/04/2022 21 / 40
Bifurcation of periodic orbits
Bifurcation of periodic orbits
We are interested in investigating with the stability of a periodic orbit when the vector
gathering the parameters of the mathematical model (M) is varied;
We define the codimension of a bifurcation as the minimum number of independent
control parameters that must be varied in order to observe the bifurcation;
For periodic orbits, the bifurcations are observed when the Floquet multipliers cross the
unity circle. Three cases, below illustrated may occur.
Mazzilli & Franzini (EPUSP) PEF5737 05/04/2022 22 / 40
Bifurcation of periodic orbits
Transcritical, symmetry-breaking and fold bifurcations
Floquet multiplier crosses the unity circle through +1.
Mazzilli & Franzini (EPUSP) PEF5737 05/04/2022 23 / 40
Bifurcation of periodic orbits
Period-doubling or flip bifurcation
Floquet multiplier crosses the unity circle through -1.
Mazzilli & Franzini (EPUSP) PEF5737 05/04/2022 24 / 40
Bifurcation of periodic orbits
Neimark-Sackler (or secondary Hopf or torus) bifurcation
Floquet multiplier crosses the unity circle as complex conjugates.
Mazzilli & Franzini (EPUSP) PEF5737 05/04/2022 25 / 40
Bifurcation of periodic orbits
A comment on the period-doubling bifurcation
We recall that vm(t+ T ) = ρmvm(t),m = 1, 2, . . . , n
if ρm = −1 (period-doubling bifurcation) vm(t+ T ) = −vm(t)→ vm(t) has period 2T .
Mazzilli & Franzini (EPUSP) PEF5737 05/04/2022 26 / 40
Application to the Mathieu’s equation
Outline
1 Objectives and references
2 Autonomous systems
3 Non-autonomous systems
4 Bifurcation of periodic orbits
5 Application to the Mathieu’s equation
6 Using the method of multiple scales (MMS)
Mazzilli & Franzini (EPUSP) PEF5737 05/04/2022 27 / 40
Application to the Mathieu’s equation
Undamped Mathieu’s equation
ü+ (δ + 2ε cos 2τ)u = 0 (13)
We rewrite Eq. 13 as a system of first-order differential equations by defining x1 = u and
x2 = u̇:
ẋ1 = x2 (14)
ẋ2 = −(δ + 2ε cos 2τ)x1 (15)
The period of the non-autonomous linear system is T = π. We investigate the stability of
the periodic solution as a function of the parameters of the plane of control parameters
(δ; ε) (Strutt’s diagram).
We compute the monodromy matrix by numerically integrating Eqs. 14 and 15 from 0 to
T for different pairs (δ; ε). If at least one Floquet multiplier has modulus larger than 1.05,
we associate the blue color. If contrary, we associate the yellow color.
Notice that the Floquet theory indicates 1 as the threshold for the stability. In this
example, we adopt 1.05 for dealing with the errors intrinsically related to the numerical
methods.
Mazzilli & Franzini (EPUSP) PEF5737 05/04/2022 28 / 40
Application to the Mathieu’s equation
Strutt’s diagram
For the undamped Mathieu’s equation, the instability tongue arise from ε = 0. The adopted
discretization (600× 600) needs to be improved if this is an important aspect in the analysis. In
a standard notebook (i7, 10th gen, 8Gb RAM) with MATLAB®, the Strutt’s diagram was
obtained in 6.6 minutes.
Mazzilli & Franzini (EPUSP) PEF5737 05/04/2022 29 / 40
Application to the Mathieu’s equation
Floquet multipliers
Mazzilli & Franzini (EPUSP) PEF5737 05/04/2022 30 / 40
Application to the Mathieu’s equation
Floquet multipliers
Mazzilli & Franzini (EPUSP) PEF5737 05/04/2022 31 / 40
Application to the Mathieu’s equation
Floquet multipliers
Mazzilli & Franzini (EPUSP) PEF5737 05/04/2022 32 / 40
Using the method of multiple scales (MMS)
Outline
1 Objectives and references
2 Autonomous systems
3 Non-autonomous systems
4 Bifurcation of periodic orbits
5 Application to the Mathieu’s equation
6 Using the method of multiple scales (MMS)
Mazzilli & Franzini (EPUSP) PEF5737 05/04/2022 33 / 40
Using the method of multiple scales (MMS)
First steps
This derivation follows the one presented in Franzini & Vernizzi (2022) which, in turn, is
inspired by Nayfeh (1973), Perturbation methods;
Consider the undamped Mathieu’s equation given by Eq. 16. In this equation, all the
parameters are dimensionless, including the time τ . In this equation, ˙( ) =
d()
dτ
.
q̈ + (δ + 2ε cos(2τ))q = 0 (16)
Standard expansions employed in the method: Tk = εkτ and
q(τ) = q0(T0, T1, T2) + εq1(T0, T1, T2) + ε2q2(T0, T1, T2) (17)
Now, we define the family of differential operator Dpm( ) = ∂p
∂T
p
m
( ). With these
definitions, the quantities below are correct up to O(ε2)
˙( ) =
∂( )
∂T0
dT0
dτ
+
∂( )
∂T1
dT1
dτ
+
∂( )
∂T2
dT2
dτ
= (D0 + εD1 + ε2D2)( ) (18)
(̈ ) = (D2
0 + ε(2D1D0) + ε2(2D2D0 +D2
1))( ) (19)
q̈ = D2
0q0 + ε(2D1D0q0 +D2
0q1) + ε2(D2
0q2 +D2
1q0 + 2D0D2q0 + 2D0D1q1) (20)
Mazzilli & Franzini (EPUSP) PEF5737 05/04/2022 34 / 40
Using the method of multiple scales (MMS)
Another expansion
In this class, we are interested in the transition curves that define the principal instability
region of the Mathieu’s equation, which corresponds to δ = 1;
A second-order approximation is then proposed as δ = 1 + εδ1 + ε2δ2;
Using the above definition and collecting the terms of equal power in ε, we obtain the
following equation:
O(1) : D2
0q0 + q0 = 0 (21)
O(ε) : D2
0q1 + q1 = −(ei2T0 + e−i2T0 )q0 − δ1q0 − 2D1D0q0 (22)
O(ε2) : D2
0q2 + q2 = −δ2q0 − (ei2T0 + e−i2T0 )q1 − δ1q1−
−D2
1q0 − 2D2D0q0 − 2D1D0q1 (23)
Mazzilli & Franzini (EPUSP) PEF5737 05/04/2022 35 / 40
Using the method of multiple scales (MMS)
And here we go...
Solution of Eq. 21: q0 = AeiT0 +A∗e−iT0 = AeiT0 + c.c., with A = A(T1, T2);
Substituting the above result into Eq. 22 and after some algebraic manipulation, we have:
D2
0q1 + q1 = −Aei3T0 − eiT0 (δ1A−A∗ − i2D1A) + c.c. (24)
The secular terms of Eq. 24 are removed if δ1A−A∗ − i2D1A = 0 (solvability
condition). We separate the complex amplitude A into its real and imaginary parts as
A = Ar + iAi. The real and the imaginary parts of the solvability condition read:{
D1Ai
D1Ar
}
=
[
0 1
2
(1 + δ1)
1
2
(1− δ1) 0
]{
Ai
Ar
}
(25)
Solution of a linear system of ODEs with the form ż = Bz: z = ẑeλt. The existence of
non-trivial solutions implies that the values of λ are the eigenvalues of B;
Using the above information to solve Eq. 25, we obtain λ1 = 1
2
√
1− δ21 and λ2 = −λ1;
Mazzilli & Franzini (EPUSP) PEF5737 05/04/2022 36 / 40
Using the method of multiple scales (MMS)
First-order approximationfor the transition curves
Auxiliary quantity: β =
√
1−δ21
1+δ1
;
The solution of Eq. 25 is{
Ai(T1, T2)
Ar(T1, T2)
}
= a1(T2)v1e
λ1T1 + a2(T2)v2e
−λ1T1 (26)
v1 = {1 β}T and v2 = {1 − β}T being the eigenvectors of B. Notice that a1 and a2
are not constants, but functions of T2.
Explicitly, we have
Ai(T1, T2) = a1(T2)e
λ1T1 + a2(T2)e
−λ1T1 (27)
Ar(T1, T2) = a1(T2)βe
λ1T1 − a2(T2)βe−λ1T1 (28)
It is clear that the stability of Eqs. 27 and 28 depends on the values of λ1 and λ2.
|δ1| < 1, λ1 ∈ R+ → unbounded solutions;
|δ1| > 1, λ1 and λ2 = −λ1 = λ∗1 are pure imaginary numbers → bounded solutions;
Transition from bounded to unbounded solutions |δ1| = 1. Hence, a first-order
approximation for the transitions curves reads δ = 1 + εδ1 = 1± ε.
Mazzilli & Franzini (EPUSP) PEF5737 05/04/2022 37 / 40
Using the method of multiple scales (MMS)
And here we go (again)
After removing the secular term in Eq. 24. We need to compute the corresponding
particular solution (the homogeneous one has been already considered). This particular
solution reads:
q1 =
1
8
Aei3T0 + c.c. (29)
where the complex amplitude A = A(T1, T2) is the same already considered.
By substituting the above result into 23, one obtains (after some algebraic work):
D2
0q2 + q2 = −
[(
δ2 +
1
8
)
A+D2
1A+ i2D2A
]
eiT0 + c.c.+NST (30)
where NST indicates non-secular terms.
Since we need to remove secular terms:(
δ2 +
1
8
)
A+D2
1A+ i2D2A = 0 (31)
By differentiating Eqs. 27 and 28 twice with respect to T1 and recalling that
A = Ar + iAi, one obtains D2
1A = λ21A;
Mazzilli & Franzini (EPUSP) PEF5737 05/04/2022 38 / 40
Using the method of multiple scales (MMS)
Higher-order approximation
We define the auxiliary quantity γ = δ2 + 1
8
+ λ21 and, again, we separate A into its real
and imaginary part. Then, the solvability condition given by Eq. 31 reads:{
D2Ai
D2Ar
}
=
[
0 γ
2
− γ
2
0
]{
Ai
Ar
}
(32)
Similarly to whats has been already done in this lecture, the transition from bounded to
unbounded solutions are characterized by γ = 0;
Since λ21 = 1
4
(1− δ1)2 and δ1 = ±1, γ = 0 implies that:
δ2 = −
1
8
−
1
4
(1− δ21) = −
1
8
(33)
An approximation for the transition curves of interest, correct up to O(ε2), is given by
δ = 1± ε−
1
8
ε2 (34)
Mazzilli & Franzini (EPUSP) PEF5737 05/04/2022 39 / 40
Using the method of multiple scales (MMS)
Comparisons
0 0.5 1 1.5 2 2.5 3 3.5
0
1
2
3
4
5
6
(a) Solutions obtained with MMS. (b) Solution obtained with MMS superimposed
onto the stability map ρ∗(δ, ε).
Extracted from Franzini & Vernizzi (2022).
Mazzilli & Franzini (EPUSP) PEF5737 05/04/2022 40 / 40
	Objectives and references
	Autonomous systems
	Non-autonomous systems
	Bifurcation of periodic orbits
	Application to the Mathieu's equation
	Using the method of multiple scales (MMS)