The Harmonic Balance method: a theoretical basis and some ... The Harmonic Balance method: a...

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

1 Description of the harmonic balance method

2 Actual periodic solutions near approximated solutions.

3 Approximations of the period function



Outline of the talk






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).



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



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.



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.



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.



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...)



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..



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.



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?



Outline of the talk






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).



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)



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.



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



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.



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.



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:



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.



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.



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.



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 .



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).



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.



• 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.



Outline of the talk






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).



Examples of application

Example 1: Potential systems

Example 2: A singular second order equation



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 .


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.



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.



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.



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



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.



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


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.



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.



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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