the harmonic balance method: a theoretical basis and some ... the harmonic balance method: a...
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Outline of the talk Description of the HBM Non-autonomous equations Period function. Examples
The Harmonic Balance method: a theoretical basisand some practical applications
J. D. Garcıa-Saldana, A. G.
Departament de Matematiques, UAB
VI Workshop on Dynamical Systems - MAT 70
May 2014.
Harmonic balance method 1/40
Outline of the talk Description of the HBM Non-autonomous equations Period function. Examples
Outline of the talk
1 Description of the harmonic balance method
2 Actual periodic solutions near approximated solutions.
3 Approximations of the period function
Harmonic balance method 2/40
Outline of the talk Description of the HBM Non-autonomous equations Period function. Examples
Outline of the talk
1 Description of the harmonic balance method
2 Actual periodic solutions near approximated solutions.
3 Approximations of the period function
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Outline of the talk Description of the HBM Non-autonomous equations Period function. Examples
Preliminaries
Suppose that an ordinary differential equation
x = X (x),
has a T -periodic solution x(t) such that x(0) = A.
Then x(t) satisfies the functional equation
F(x(t), x(t)) = x(t)− X (x(t)) = 0.
We can represent x(t) by its Fourier series:
x(t) =a02
+∞∑k=1
(ak cos(kωt) + bk sin(kωt)
), ω :=
2π
T.
ak =2
T
∫ T
0
x(t) cos(kωt) dt and bk =2
T
∫ T
0
x(t) sin(kωt) dt for k ≥ 0.
ak := ak(A), bk := bk(A), and ω := ω(A).
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Outline of the talk Description of the HBM Non-autonomous equations Period function. Examples
Idea of the method
Given the equation x = X (x) the Harmonic Balance Method(HBM) is a method for obtaining analytic approximations of itsT -periodic solutions and of their corresponding periods T using
truncated Fourier series.
Notice that when the differential equation x = X (x , t) isnon-autonomous and T -periodic the problem is similar but then Tis no more an unknown.
The same method can also be applied to higher order differentialequations
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Outline of the talk Description of the HBM Non-autonomous equations Period function. Examples
Procedure of the HBM
1 Propose a trigonometric polynomial
xN
(t) =a02
+N∑
k=1
(ak cos(kωNt) + bk sin(kωNt)) .
2 Compute the T -periodic function FN := F(xN
(t), xN
(t)),
FN =A0
2+∞∑k=1
(Ak cos(kωNt) + Bk sin(kωNt)) ,
where Ak =Ak(a,b, ωN) and Bk =Bk(a,b, ωN), k ≥ 0, witha = (a0, a1, . . . , aN
) and b = (b1, . . . , bN).
3 Find the values of a, b, and ωN such that
Ak(a,b, ωN) = 0 and Bk(a,b, ωN) = 0 for 0 ≤ k ≤ N.
4 Every solution represents a candidate xN(t) to be an approxi-mation of a periodic solution of the differential equation.
5 TN = 2π/ωN is a candidate to approx. the corresp. period.
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Outline of the talk Description of the HBM Non-autonomous equations Period function. Examples
Example: a non-autonomous equation
Consider the system of differential equationsx = −y + x(−1 + 3x2 + 2xy + y2),y = x + y(−1 + 3x2 + 2xy + y2).
In polar coordinates
r ′ = −r + (cos(2t) + sin(2t) + 2)r3,
is a Bernoulli equation and it has a limit cycle
r∗(t) =1√
2 + cos(2t)
which is hyperbolic.
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Outline of the talk Description of the HBM Non-autonomous equations Period function. Examples
Example
The Fourier series of r∗(t) = 1√2+cos(2t)
is
a02
+∞∑k=1
a2k cos(2t)
with
a0 = 4K√3π≈ 1.491498374, a2 = 12E−8K√
3π≈ −0.2016837219,
a4 = −32E+20K√3π
≈ 0.04065713288, a6 = 476E−296K√3π
≈ −0.009092598292,
a8 = −10624E+6604K√3π
≈ 0.002133790322, a10 = 105548E−65608K√3π
≈ −0.0005148662408,
where K = K (√
6/3) and E = E (√
6/3) are elliptic integrals.
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Outline of the talk Description of the HBM Non-autonomous equations Period function. Examples
Example
Applying second order of the HBM, we look for a solution of theform
r(t) = r0 + r2 cos(2t),
of the functional equation
F(r(t)) = r ′(t) + r(t)− (cos(2t) + sin(2t) + 2)r3(t) = 0.
Imposing the vanishing of the coefficients 1 and cos(2t) in theFourier series of F(r(t)) we arrive to the system:
r0 − 2r30 − 32 r2r20 − 3r22 r0 − 3
8 r32 = 0,r2 − r30 − 6 r2r20 − 9
4 r22 r0 − 32 r32 = 0,
an approximated solution is r0 ≈ 0.7440, r2 ≈ −0.20139.(The exact Fourier series is a0/2 = 0.74574..., a2 = −0.201683...)
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Outline of the talk Description of the HBM Non-autonomous equations Period function. Examples
Example
Applying the HBM to order 8 we obtain
r(t) =4∑
k=0
r2k cos(2kt),
with
r0 = 0.7457489122, r2 = −0.2016836610, r4 = 0.04065712547,r6 = −0.0090925999, r8 = 0.002133823488.
Exact solutions:
a0/2 = 0.7457491.. a2 = −0.20168372.. a4 = 0.040657132..,a6 = −0.0090925982.. a8 = 0.0021337903..
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Outline of the talk Description of the HBM Non-autonomous equations Period function. Examples
Remarks about the HBM
The set of algebraic equations is a system of polynomial equa-tions which usually is very difficult to solve.
In many works only first and second orders are considered.
The coefficients of xN
(t) and xN+1
(t) do not coincide at all.
In general, although in many concrete applications the HBMseems to give quite accurate results, it is not proved that thefound Fourier polynomials are approximations of the actual pe-riodic solutions of differential equation.
In the autonomous case it seems that the values 2π/ωN giveapproximations of the periods of the corresponding periodicorbits.
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Outline of the talk Description of the HBM Non-autonomous equations Period function. Examples
Can we ensure that xN(t) approximates to x(t)?
In the autonomous case, can we ensure that TN = 2π/ωN
approximates the period T of a periodic orbit?
Harmonic balance method 12/40
Outline of the talk Description of the HBM Non-autonomous equations Period function. Examples
Outline of the talk
1 Description of the harmonic balance method
2 Actual periodic solutions near approximated solutions.
3 Approximations of the period function
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Outline of the talk Description of the HBM Non-autonomous equations Period function. Examples
The non-autonomous case: definitions
Consider the ODE:x ′(t) = X (x(t), t)
X : Ω × [0, 2π] → R, C 2 and 2π-periodic in t. Let x(t) be a2π-periodic C 1-function.We say that
x(t) is noncritical with respect to the ODE if∫ 2π
0
∂
∂xX (x(t), t) dt 6= 0. (1)
the accuracy S of x(t) is
S := ||s(t)||2 =
√1
2π
∫ 2π
0s2(t)dt,
where s(t) := F(x(t)) = x ′(t)− X (x(t), t).
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Outline of the talk Description of the HBM Non-autonomous equations Period function. Examples
Definitions
M ∈ R is a deformation constant associated to x(t) and X if
||yb(t)||∞ ≤ M||b(t)||2,
where yb(t) is the unique 2π-periodic solution of the linearperiodic system
y ′ =∂
∂xX (x(t), t) y + b(t),
b(t) is a 2π-periodic function and ||f ||∞ = maxx∈R |f (x)|.
Bound for the second derivative. Given
I := [ mint∈R
x(t)− 2MS , maxt∈R
x(t) + 2MS ] ⊂ Ω,
let K <∞ be a constant such that
max(x,t)∈I×[0,2π]
∣∣∣∣ ∂2
∂x2X (x , t)
∣∣∣∣ ≤ K . (2)
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Outline of the talk Description of the HBM Non-autonomous equations Period function. Examples
Theorem
Consider the 2π-periodic ODE
x ′(t) = X (x(t), t). (3)
Let x(t) be a 2π-periodic C 1–function, such that
is noncritical,
has accuracy S
M is the deformation constant
given I exists K satisfying (2).
If 2M2KS < 1, then there exists a 2π-periodic solution x∗(t) of (3)that satisfies ||x∗ − x ||∞ ≤ 2MS, and it is the unique periodicsolution of the equation entirely contained in this strip. If in addition,∣∣∣∣∫ 2π
0
∂
∂xX (x(t), t) dt
∣∣∣∣ > 2π
M,
then the periodic orbit x∗(t) is hyperbolic, and its stability is givenby the sign of this integral.
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Outline of the talk Description of the HBM Non-autonomous equations Period function. Examples
Stokes and Urabe
Our result is based on the classical papers:
• A. Stokes, On the approximation of Nonlinear Oscillations, J. Differ-ential Equations 12 (1972), 535–558.
• M. Urabe, Galerkin’s Procedure for Nonlinear Periodic Systems. Arch.Rational Mech. Anal. 20 (1965), 120–152.
It is published in:
• J. D. Garcıa-Saldana, A. Gasull, A theoretical basis for the harmonicbalance method J. Differential Equations 254 (2013), 67–80
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Outline of the talk Description of the HBM Non-autonomous equations Period function. Examples
Returning to the example
Consider the system of differential equationsx = −y + x(−1 + 3x2 + 2xy + y2),y = x + y(−1 + 3x2 + 2xy + y2).
In polar coordinates
r ′ = −r + (cos(2t) + sin(2t) + 2)r3,
is a Bernoulli equation and it has a limit cycle
r∗(t) =1√
2 + cos(2t)
which is hyperbolic.
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Outline of the talk Description of the HBM Non-autonomous equations Period function. Examples
Applying the 8 order of HBM we get
r(t) =4∑
k=0
r2k cos(2kt),
with
r0 = 0.7457489122, r2 = −0.2016836610, r4 = 0.04065712547,
r6 =− 0.009092599917, r8 = 0.002133823488.
Its accuracy is S = ||F(x(t))||2 ≤ 0.0039.
Using the theory of continued fractions we can find rational ap-proximations of ri such that the new approximation has an similaraccuracy.
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Outline of the talk Description of the HBM Non-autonomous equations Period function. Examples
For instance, the convergents of r0 ≈ 0.74574891 . . . are
1,2
3,
3
4,
41
55,
44
59, . . .
in particular 4459 ≈ 0.745762 . . .Consider
r(t) =44
59− 24
119cos(2t) +
2
49cos(4t)− 1
110cos(6t) +
1
468cos(8t).
The accuracy of r(t) is S ≤ 0.00394 (before was S ≤ 0.003935)
Computing S , M and K we have
2M2KS < 1
and then:
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Outline of the talk Description of the HBM Non-autonomous equations Period function. Examples
Proposition
There exists a periodic solution r∗(t) of the ODE such that
||r∗ − r ||∞ ≤ 0.0192,
and it is the unique in this region.
Remember that the real periodic solution is
r∗(t) =1√
2 + cos(2t).
and the approximated periodic solution is
r(t) =44
59− 24
119cos(2t) +
2
49cos(4t)− 1
110cos(6t) +
1
468cos(8t),
then, it can be seen
||r∗ − r ||∞ ≤ 0.0007.
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Outline of the talk Description of the HBM Non-autonomous equations Period function. Examples
Theorem
Consider the 2π-periodic ODE
x ′(t) = X (x(t), t). (4)
Let x(t) be a 2π-periodic C 1–function, such that
is noncritical,
has accuracy S
M is the deformation constant
given I exists K satisfying (2).
If 2M2KS < 1, then there exists a 2π-periodic solution x∗(t) of (4)that satisfies ||x∗ − x ||∞ ≤ 2MS, and it is the unique periodicsolution of the equation entirely contained in this strip. If in addition,∣∣∣∣∫ 2π
0
∂
∂xX (x(t), t) dt
∣∣∣∣ > 2π
M,
then the periodic orbit x∗(t) is hyperbolic, and its stability is givenby the sign of this integral.
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Outline of the talk Description of the HBM Non-autonomous equations Period function. Examples
Idea of the proof of theorem
• Change of coordinates to move the ODE to a neighborhood ofx(t).
• Formulation of the question as a fixed point problem.
• Prove the contraction property.
• Prove the function given by the contraction is in factdifferentiable.
• Prove the hyperbolicity of the periodic orbit.
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Outline of the talk Description of the HBM Non-autonomous equations Period function. Examples
Change of coordinates.
The function z(t) + x(t) is solution of
x ′ = X (x , t),
if and only if the function z(t) is solution of
z ′ = X (z(t) + x(t), t)− X (x(t), t)− s(t), (5)
Note that (5) can be written as
z ′ =∂
∂xX (x(t), t)z + R(z , t)− s(t)
where
R(z , t) = X (z + x(t), t)− X (x(t), t)− ∂
∂xX (x(t), t)z .
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Outline of the talk Description of the HBM Non-autonomous equations Period function. Examples
Fixed point formulation
Let P be the space of 2π-periodic C 0-functions.We look for a 2π-periodic solution of
z ′ =∂
∂xX (x(t), t)z + R(z , t)− s(t)
inN = z ∈ P : ||z ||∞ ≤ 2MS.
N is a complete metric space with ||z ||∞ .Since x(t) is noncritical, if we consider the linear ODE
y ′ =∂
∂xX (x(t), t)y + R(z , t)− s(t),
then, with each z ∈ N we can associate the unique 2π-periodicsolution of the linear ODE, called T (z).
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Outline of the talk Description of the HBM Non-autonomous equations Period function. Examples
Fixed point formulation
For instance,
||T (z)||∞ ≤ M||R(z(·), ·)− s(·)||2 ≤ M (||R(z(·), ·)||2 + S)
≤ M(||R(z(·), ·)||∞ + S) ≤ M(K
2||z ||2∞ + S)
≤ M(2KM2S2 + S) < 2MS ,
where in the last inequality we have used 2M2KS < 1.Therefore T : N −→ N.
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Outline of the talk Description of the HBM Non-autonomous equations Period function. Examples
• It is no difficult to see that T is a contraction in N if
2M2KS < 1.
• In principle, we can only ensure that the fixed-point function isonly continuous, but because it satisfies
x∗(t) = x∗(0)+
∫ t
0
(∂
∂xX (x(w),w)x∗(w) + R(x∗(w),w)− s(w)
)dw ,
it is in fact differentiable.
• Finally, imposing that∣∣∣∣∫ 2π
0
∂
∂xX (x(t), t) dt
∣∣∣∣ > 2π
M,
we can prove that the periodic orbit x∗(t) is hyperbolic.
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Outline of the talk Description of the HBM Non-autonomous equations Period function. Examples
Outline of the talk
1 Description of the harmonic balance method
2 Actual periodic solutions near approximated solutions.
3 Approximations of the period function
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Outline of the talk Description of the HBM Non-autonomous equations Period function. Examples
Approximations of the period function
Consider a planar differential system having a continuum of periodicorbits.
Let T (A) be its period function: the function that associates toeach periodic orbit its period.
Let TN(A) be the approximation of T (A) obtained with the N-thorder of the HBM.
Harmonic Balance Method vs Analytical study
monotonicity
local behavior near the critical points
behavior near the infinity
critical periods (oscillations).
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Outline of the talk Description of the HBM Non-autonomous equations Period function. Examples
Examples of application
Example 1: Potential systems
Example 2: A singular second order equation
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Outline of the talk Description of the HBM Non-autonomous equations Period function. Examples
Potential systemsTheorem
The systemx = y ,y = − x
(x2+k2)m, k ∈ R \ 0, m ∈ [1,∞).
(6)
has a center at the origin and its period function T (A) isincreasing. Moreover, the center is global for m = 1 andnon-global otherwise.
Proposition
By applying the first-order HBM to system (6) we obtain theincreasing function
T1(A) = 2π
√√√√ m∑j=0
(1
2
)2j (m
j
)(2j + 1
j
)k2(m−j)A2j .
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Outline of the talk Description of the HBM Non-autonomous equations Period function. Examplesx = −y ,y = x + x2m−1, m ∈ N and m ≥ 2,
Theorem
• T (A) is decreasing.
• In A = 0,
T (A) = 2π(1− S1 A2m−2 + S2 A4m−4 + O(A6m−6)
)
S1 = (2m−1)!!(2m)!!
,
S2 = (2m−1)(4m−1)!!m(4m)!!
− (m−1)(2m−1)!!m(2m)!!
• When A→∞,
T (A) ∼ B
(1
2m,
1
2
)2√
mAm−1.
Proposition
T1(A) =2mπ√
(2m−1)!(m−1)!m!
A2m−2 + 22m−2.
• In A = 0,
T1(A) = 2π(1− S1 A2m−2 + S2 A4m−4 + O(A6m−6)
)
S1 = (2m−1)!!(2m)!!
S2 = 32
((2m−1)!!(2m)!!
)2• When A→∞,
T1(A) ∼ 2mπ√(2m−1)!(m−1)!m!
Am−1.
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Outline of the talk Description of the HBM Non-autonomous equations Period function. Examples
x = −y ,y = x + k x3 + x5, k ∈ (−2,∞).
Theorem
• T (A) decreasing if k ≥ 0.
• T (A) has a critical period ifk ∈ (−2, 0) (Manosas, Villadelprat).
• At the origin
T (A) = 2π−3
4kπA2+
57k2 − 80
128πA4+O(A6)
• At the infinity
T (A) ∼2B( 1
6, 12)
√3
1
A2≈ 8.4131
A2.
Proposition
T1(A) =8π
√16 + 12kA2 + 10A4
.
• T1(A) has a critical period ifk ∈ (−2, 0).
• At the origin
T1(A) = 2π−3
4kπA2+
54k2 − 80
128πA4+O(A6)
• At the infinity
T1(A) ∼ 4π√
10
5
1
A2≈ 7.9477
A2.
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Outline of the talk Description of the HBM Non-autonomous equations Period function. Examples
Example 2: The HBM and a weak periodic solution
The nonlinear differential equation
xx + 1 = 0, (7)
appears in the modeling of certain physical phenomena.
Mickens calculates the period of its periodic orbits and alsouses the first and second order HBM to obtain approximationsof these periodic solutions and of their corresponding periods.
Strictly speaking, neither (7),nor its associated system
x = y ,y = − 1
x ,
have periodic solutions.
R. E. Mickens, Harmonic balance and iteration calculations of periodic solutions to
y + y−1 = 0, J. Sound Vibration 306 (2007), 968–972.
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Outline of the talk Description of the HBM Non-autonomous equations Period function. Examples
The equation xx + 1 = 0
Two different interpretations of Mickens’ computation of the period.
We prove that the equation has weak periodic solutions andcompute its period.As the limit, when k → 0 tends to zero, of the period of actualperiodic solutions of the extended planar differential system
x = y ,y = − x
x2+k2 ,(8)
which, for k 6= 0, has a global center at the origin.
k = 1 k = 150 k = 1
1000
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Outline of the talk Description of the HBM Non-autonomous equations Period function. Examples
Theorem
For the initial cond. x(0) = A, x(0) = 0, the differential equa-tion xx + 1 = 0 has a weak C 0-periodic solution with periodT (A) = 2
√2πA.
Let T (A; k) be the period of the periodic orbit of system x = y,y = −x/(x2 + k2) with initial cond. x(0) = A, y(0) = 0. Then
T (A; k) = 4 A
∫ 1
0
ds√ln(
A2+k2
A2s2+k2
)and
limk→0
T (A; k) = 4A
∫ 1
0
1√−2 ln s
ds = 2√
2πA.
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Outline of the talk Description of the HBM Non-autonomous equations Period function. Examples
Proof the theorem
xx + 1 = 0 for x(0) = A, x(0) = 0 and t ∈(−√2π2 A,
√2π2 A
).
The solution is
x(t) = φ0(t) := Ae−(erf−1
(2 t√2πA
))2
, where erf(z) =2√π
∫ z
0
e−s2
ds.
We define the weak solution
φ(t) =
(−1)nφ0(t − n
√2π), for t ∈
(2n−12
√2π, 2n+1
2
√2π), n ∈ Z,
0 for t = 2n+12
√2π, n ∈ Z,
φ(t) is a C 0-function of period T (A) = 2√
2πA and satisfies theequation for any t ∈ R \ ∪n∈Z2n+1
2
√2π.
Two solutions A weak solutionHarmonic balance method 37/40
Outline of the talk Description of the HBM Non-autonomous equations Period function. Examples
Equation xx + 1 = 0 is equivalent to the differential equations
xm+1x + xm = 0, m ∈ N ∪ 0. (9)
It is natural to approach T (A) = 2√
2πA ≈ 5.0132A, by the periodsof the trigonometric polynomials obtained applying the N-th orderHBM to (9).
Theorem
Let TN(A; m) be the period of the truncated Fourier seriesobtained applying the N-th order HBM to equation (9). For allm ∈ N ∪ 0,
T1(A; m) = 2π
√2[m+1
2 ] + 1
2[m+12 ] + 2
A.
T1(A; 0) =√
2πA ≈ 4.4428A,T1(A; 1) =
√3πA ≈ 5.4414A.
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Outline of the talk Description of the HBM Non-autonomous equations Period function. Examples
Comparison of the percentage of the relative errors.
eN (m) = 100
∣∣∣∣∣TN (A;m)− T (A)
T (A)
∣∣∣∣∣ .
eN(m) m = 0 m = 1 m = 2
N = 1 11.38% 8.54% 8.54%
N = 2 2.80% 5.19% 5.17%
N = 3 1.55% 2.68% 2.56%
N = 4 0.64% 2.10% −N = 5 0.58% − −N = 6 0.25% − −
Consider x4(t) = a1 cos(ω4t)+a3 cos(3ω4t)+a5 cos(5ω4t)+a7 cos(7ω4t),
2−(a21 + 9a23 + 25a25 + 49a27
)ω24 = 0,
a21 + 10a1a3 + 34a3a5 + 74a5a7 = 0,
5a1a3 + 13a1a5 + 29a3a7 = 0,
9a23 + 50a1a7 + 26a1a5 = 0,
a1 + a3 + a5 + a7 − A = 0.
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Outline of the talk Description of the HBM Non-autonomous equations Period function. Examples
Despite the functions Tn(A), obtained by applying n-th orderHBM, capture and reproduce quite well the actual behavior ofT (A), there are no results that guarantee this fact in general.
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