Asymmetric Mathieu equations

doi: 10.1098/rspa.2005.1632, 1643-1659462 2006 Proc. R. Soc. A

Amol Marathe and Anindya Chatterjee Asymmetric Mathieu equations

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Asymmetric Mathieu equations

BY AMOL MARATHE* AND ANINDYA CHATTERJEE

Mechanical Engineering, Indian Institute of Science, Bangalore 560012, India

An inverted pendulum with asymmetric elastic restraints (e.g. a one-sided spring), whensubjected to harmonic vertical base excitation, on linearizing trigonometric terms, isgoverned by an asymmetric Mathieu equation. This system is parametrically forced andstrongly nonlinear (linearization for small motions is not possible). However, solutionsare scaleable: if x(t) is a solution, then so is ax(t) for any real aO0. We numerically studythe stability regions in the parameter plane of this system for a fixed degree ofasymmetry in the elastic restraints. A Lyapunov-like exponent is defined andnumerically evaluated to find these regions of stable and unstable behaviour. Thesenumerics indicate that there are infinitely many possibilities of instabilities in this systemthat are missing in the usual or symmetric Mathieu equation. We find numerically thatthere are periodic solutions at the boundaries of stable regions in the parameter plane,analogous to the symmetric Mathieu equation. We compute and plot several of thesesolution branches, which provide a relatively simpler means of computing the stabilitytransition curves of this system. We prove theoretically that such periodic solutions mustexist on all stability boundaries. Our theoretical results apply to the asymmetric Hill’sequation, of which the pendulum system is a special case. We demonstrate this withnumerical studies of a more general asymmetric Mathieu equation.

Keywords: asymmetric; Mathieu; Lyapunov exponent; stability; transition curves

*A

RecAcc

1. Introduction

Consider an inverted pendulum with asymmetric elastic restraints in which thepivot is given a vertical periodic oscillation, as shown schematically in figure 1.Spring stiffnesses on the left and right are taken as dð1K�aÞ and dð1C �aÞ,respectively, with 0% �a!1. The springs may be taken as straight, as shown,with a long free length so that they remain effectively horizontal even whentheir endpoints move up and down; they may alternatively, for simplicity, bethought of as torsion springs at the pivot point. Gravity may be incorporated inthe spring stiffnesses and is not explicitly accounted for. There is no damping.

Let the angular displacement of the pendulum be q(t). Taking the length ofthe pendulum to be l and the origin of the coordinate system at some point on theqZ0 line, the coordinates of the centre of mass of the pendulum are

x Z l sin q; y Z uðtÞC l cos q;

Proc. R. Soc. A (2006) 462, 1643–1659

doi:10.1098/rspa.2005.1632

Published online 14 February 2006

uthor for correspondence ([email protected]).

eived 20 July 2005epted 29 November 2005 1643 q 2006 The Royal Society


x

y

d (1–a) d (1+a)

e cos(w t)

spring is unloaded spring is compressed

q

Figure 1. An asymmetrically supported inverted pendulum with base excitation.

A. Marathe and A. Chatterjee1644


where u(t) is the imposed displacement of the pivot. Differentiating,

_x Z l cos q _q; _y Z _uKl sin q _q:

The system kinetic energy is

T Z1

2mð _x2 C _y2Þ;

where m is the mass. We take the potential energy to be of the form (recall thatgravity is not explicitly included)

V Z1

2dð1CsgnðqÞ�aÞl2q2;

which is correct for torsion springs, and approximately correct for straightsprings with long free lengths as mentioned earlier.

The equation of motion can now be routinely obtained using Lagrange’smethod.On linearizing trigonometric terms by substituting cos qZ1 and sin qZq, andletting lZ1, mZ1, we obtain

€qK€uqCdð1CsgnðqÞ�aÞqZ 0:

Assuming uZe cos(t), writing x in place of q, and noting that

sgnðxÞx Z jxj;we obtain

€x CðdCe cosðtÞÞxCd�ajxjZ 0: ð1:1ÞWe call equation (1.1) an asymmetric Mathieu equation; setting �aZ0 gives theusual Mathieu equation (Stoker 1950; Winkler 1966; Rand 2003). We mention inpassing that the dynamics of a gear-pair system involving backlash can be

Proc. R. Soc. A (2006)


1645Asymmetric Mathieu equations


modelled by a similar equation (Theodossiades & Natsiavas 2000). In this paper,we investigate some of the similarities and differences between the stabilitydiagrams (in the d–e parameter plane) for the usual and asymmetric Mathieuequations. Our theoretical treatment allows parametric forcing of the jxj term aswell, which introduces two more free parameters as discussed later.

As is well known, the usual Mathieu equation has alternating stable andunstable regions in the parameter plane. On the transition curves (boundaries ofthe stability regions), there are periodic solutions with period either 2p or 4p (2:1subharmonic motion) (Stoker 1950; Winkler 1966; Rand 2003). How are theserelated to the stability regions, transition curves, and periodic solutions of theasymmetric Mathieu equation?

Equation (1.1) is a homogeneous second-order differential equation with realperiodic coefficients. It is also essentially nonlinear: linearization near xZ0 is notpossible. If x1(t) and x2(t) are two solutions, then ax1(t)Cbx2(t) is not a solutionin general. However, solutions are scaleable, i.e. if x(t) is a solution, then ax(t) isalso a solution for any real aO0. This scaleability will be important in thesubsequent analysis.

2. Unforced system

The unforced system (eZ0) is

€x CdxCd�ajxjZ 0: ð2:1ÞThe asymmetric potential energy of this unforced system and the correspondingasymmetric spring force are shown in figure 2, left and right, respectively.

All solutions of the unforced system are periodic, and have the same period.Each such periodic solution consists of two half-sinusoids, one for xO0 and onefor x!0, with different amplitudes and time durations. The time period T andcorresponding fundamental frequency u are

T Zpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

dð1C �aÞp C

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffidð1K�aÞ

p and uZ2p

T: ð2:2Þ

3. A Lyapunov-like exponent for scaleable systems

Since the fully nonlinear forced system seems analytically intractable, we resortto numerics. A few numerically obtained solutions for equation (1.1) are shown infigure 3. It is seen that solutions grow without bound (unstably) for someparameter values, but remain bounded for others. The unstable solutions growexponentially, and we begin by characterizing their exponential growth rate.

A Lyapunov-like exponent is defined as

sðd; eÞZ limN/N

1

N

XNkZ0

lnðjjX2pk jjÞ; ð3:1Þ

where the state vector XZfx; _xgT; k$k denotes the Euclidean norm; thesubscript k in X2p

k denotes the time in number of forcing periods; and asuperscript, either 0 or 2p, denotes the start or end of a forcing period,respectively.



2

4

6

8

10

–10 –8 –6 –4 –2 2 4

V (x )

x

–1

1

2

3

4

–10 –8 –6 –4 –2 2 4

Fs

x

(b)(a)

Figure 2. dZ0.5, �aZ0:7. (a) Potential energy. (b) Restoring spring force.



The above Lyapunov-like exponent is calculated as follows. We numericallyintegrate equation (1.1) over an interval of 2p with random initial conditions X0

0

that satisfy jjX00 jjZ1. At the end of this integration, we have the state X2p

0 .Initial conditions for the next cycle are then obtained by rescaling to unit norm:

X01 Z

X2p0

jjX2p0 jj :

Integration over another period of forcing then gives X2p1 . The above steps are

repeated N times, for some large N. We discard the first several states in thecalculation to get rid of transients, and take the first retained state as X0

0 . N istaken large enough to obtain convergence, which in this case is relatively rapidpresumably because solutions are not chaotic.

An exponent sZ0 implies the corresponding d–e point is stable, while sO0implies it is unstable. In numerics, sZ0 is not exactly observed, and a non-zeronumerical tolerance or threshold is set by the analyst.

4. Numerical stability diagram

We first present the results of a numerical stability analysis of the asymmetricMathieu equation on the d–e parameter plane for the arbitrarily chosen value of�aZ0:7 (we will show some more results for �aZ0:8 later in this same section). Inthese calculations, numerical integration of ODEs was done using MATLAB’sbuilt-in ODE solver ‘ODE45’ with ‘event detection’ so as to ensure that everyzero-crossing of x coincides with a mesh point (or sampling instant) in the time-discretized solution.



0

2

4

6x 108

x(t)

x(t)

x(t)

x(t)

x(t)

x(t)

d=0.0233

–10

–5

0

5 d=0.04193

–400

–200

0

200 d=0.1248

–10

–5

0

5 d=0.2006

0 50 100 150–10000

–5000

0

5000

t

d=0.3699

0 50 100 150–6

–4

–2

0

2

4

t

d=0.5392

(a)

(c)

(e)

(b)

(d)

( f )

Figure 3. Time histories of solutions for eZ0.2 fixed and d values as indicated. Cases (a, c, e) areunstable and (b, d, f ) are stable.



Figure 4 shows the Lyapunov-like exponent on a 500!500 grid coveringd2[K0.1,1.3] and e2[0,1], for NZ600. The calculation took several days,distributed among a few PCs. Figure 5 shows a closer and refined view of theboxed, bottom left, region of figure 4. Now d2[0,0.45] and e2[0,0.4], the grid is450!400, and NZ2400.

The figures show several instability regions. Limitations in numerics andgraphics limit the number of visible regions; however, it seems that there areinfinitely many such regions, successively narrower and more weakly unstable,nested between a few large and strongly unstable regions. These large andstrongly unstable regions have a one-to-one correspondence with regions ofinstability for the usual Mathieu equation, as may be seen from the following.Narrower and finer resonance regions are associated with smaller numericalvalues of Lyapunov-like exponents: decreasing N or increasing the numericalvalue of the cutoff tolerance in the stability diagram will prevent us fromdetecting these regions.



d0 0.2 0.4 0.6 0.8 1.0 1.2

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

Figure 4. Stability diagram for �aZ0:7, NZ600. Grey, stable; black, unstable.

d0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45

0.40

0.35

0.30

0.25

0.20

0.15

0.10

0.05

0

Figure 5. Zoomed portion of figure 4: �aZ0:7, NZ2400. Grey, stable; black, unstable.



For the usual or symmetric Mathieu equation (�aZ0), there are regions ofinstability emanating from the d-axis at dsymZ0, 0.25, 1,/. These correspond tosimple rational relationships between the unforced time period, 2p=

ffiffiffiffiffiffiffiffiffidsym

p, and

the forcing period, 2p.For the asymmetric Mathieu equation with �aZ0:7, the time period of the

unforced system is, from equation (2.2),

1:29642pffiffiffiffiffiffiffiffiffiffiffidasym

p :

For the same rational relationships to hold so that the same resonances mayoccur, we require this time period to bear the same ratio to 2p as in thesymmetric case.





This gives

1:29642pffiffiffiffiffiffiffiffiffiffiffidasym

p Z2pffiffiffiffiffiffiffiffiffidsym

p ;

or

dasym Z 1:29642dsym Z 1:6805dsym:

Thus, we expect corresponding instability regions for the �aZ0:7 case to emanatefrom the d-axis at dZ0, 0.4201, 1.6805,/. This is supported by the numericalresults.

The asymmetric and symmetric Mathieu equations are identical on the e-axis(with dZ0),

€x Ce cosðtÞx Z 0:

Therefore, stability intervals on the e-axis for both equations (for any �a) arethe same. So, all the new instability regions of the asymmetric equation musthave zero widths on the e-axis. This is observed in the numerics. Similarly, thewidths of the corresponding strong instability regions that occur for both theasymmetric as well as symmetric equations coincide on the e-axis; this is outsidethe region covered by the numerical results, and cannot be seen in the figure.

We now consider other possible resonances. Let r be a simple rational number,such as (say) 2.5, 3 or 4. Consider a situation where the ratio between theunforced time period and the forcing period is r, i.e.

1:2964!2pffiffiffid

p Z 2pr ð4:1Þ

or

dZ1:29642

r2Z

1:6805

r2:

The d values corresponding to rZ2.5, 3 and 4 are then 0.2689, 0.1867 and 0.1050,respectively. There are, in fact, instability regions corresponding to these valueson the d-axis, as may be seen with a little faith from figure 5, or moreconvincingly from figure 7 in §5.

The qualitative results obtained above are not special for �aZ0:7. Figure 6shows the Lyapunov-like exponent for �aZ0:8 on a 500!500 grid coveringd2[K0.1,1.5] and e2[0,1], for NZ600. There is agreement with the results for�aZ0:7.

From the numerical results of this section, it seems likely that all the narrowinstability regions seen in figure 5 do in fact continue all the way to the d-axis.This is verified numerically in §5, and supported further with theory later inthe paper.

5. Periodic solutions on stability boundaries

As is well known, for the usual or symmetric Mathieu equation, there are periodicsolutions of period either 2p or 4p on each stability boundary on the parameterplane. Are there periodic solutions (of possibly other periods) for each stability



d0.06 0.22 0.38 0.54 0.70 0.86 1.02 1.18 1.34 1.50

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

Figure 6. Stability diagram for �aZ0:8, NZ600. Grey, stable; black, unstable.



boundary for the asymmetric Mathieu equation? In this section, we investigatethis question numerically. In particular, we seek curves on the d–e parameterplane where periodic solutions exist; and anticipate that these curves will startfrom the d-axis at the points from which instability regions emanate (as discussedearlier).

We present here the results obtained for dZ0.4201, 0.2689, 0.1867 and 0.1050.The corresponding time periods expected, and obtained, are 4p, 10p, 6p and 8p,respectively (note: 10p corresponds to rZ2.5 and not 5 in equation (4.1)). Fromeach of these points on the d-axis, two curves are found to emanate. These curveswere obtained using a numerical arc-length based continuation method which isdescribed in appendix A. Results, shown using white lines on the stabilitydiagram, are given in figure 7. The picture, presented here for eR0, is symmetricabout the d-axis.

It is clear that, at least for the instability regions considered, stabilityboundaries correspond to the existence of periodic solutions, even for theunstable regions that are absent for the symmetric Mathieu equation. It seemslikely that there are also periodic solutions at all other stability boundaries. In§7, we show theoretically that this is in fact the case.

A few points regarding the numerical search for periodic solutions are nowpresented.

Since solutions are scaleable, we may assume that the initial conditions at tZ0satisfy either x0ZG1, or _x0ZG1, or even x20C _x20Z1. It turns out that for theperiodic solutions plotted in figure 7, the solution branch to the left of eachinstability region has _x0Z0 and x0Z1; while for the solution branch on the rightside of each region, x0Z0 and _x0Z1. The numerical strategy, however, assumesone of these (x0 or _x0) to be non-zero (equal to G1), and lets the numericalroutine discover that the other is zero (if in fact it is).

Numerically, we proceed as follows. Starting at tZ0 with some values of d ande (along with, say, �aZ0:7 fixed and x0Z1), we numerically integrate forward intime to time T (for suitable T, which is an even multiple of p that we know in



d0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45

0.40

0.35

0.30

0.25

0.20

0.15

0.10

0.05

0

Figure 7. 4p, 6p, 8p and 10p period solutions on the stability boundaries in the d–e parameterplane for the asymmetric Mathieu equation with �aZ0:7.



advance as indicated earlier), and check that the final state is identical to theinitial state. We then iteratively look for a nearby point on the d–e plane wherethese conditions are satisfied again. This well-known procedure, called arc-lengthbased continuation, is described for completeness in appendix A.

Note that these periodic solution branches can (at least potentially) becomputed in a small fraction of the time needed to generate, say, figure 5. Thus,as for the symmetric Mathieu equation, finding periodic solutions is an efficientway of computing the stability transition curves for the asymmetric Mathieuequation.

6. Theoretical considerations

The asymmetric Mathieu equation (1.1) can be rewritten as

_x Z f ðx; tÞ; ð6:1Þwith

x Zx

y

� �and

f Zy

KðdCe cosðtÞÞxKd�ajxj

( ):

Formally, we observe that equation (6.1) is divergence free, i.e. V$fZ0. Thismeans areas are preserved in the ðx; _xÞ plane.

Let xZR cos(f) and yZR sin(f). Consider a Poincare map that takes x fromthe start of a forcing period to the start of the next (i.e. through tZ2p). Let



x

y(R0,f0)

(R0,f0+∆f0)∆f0

x

y

(R0 f (f0+∆f0), g(f0+∆f0))

(R0 f (f0), g(f0))

∆g(f0)

Figure 8. Area preservation of the flow.



(R0, f0) be an initial point. The Poincare map sends ðR0;f0Þ1ðR1;f1Þ withR1 Z �f ðR0;f0Þ;

f1 Z �gðR0;f0Þ;for some as yet unknown continuous functions �f and �g. Since solutions toequation (1.1) are scaleable, we can write

�f ðR0;f0ÞZR0�f ð1;f0ÞZR0f ðf0Þ:

Similarly, scaleability requires

�gðR0;f0ÞZ gðf0Þ:Thus, the point (R0, f0) gets mapped to (R0f(f0), g(f0)).

The system is reversible in time. Zero initial conditions lead to zero solutionsfor all time. These two together imply that non-zero solutions do not become zerowithin finite time. In turn, this means

f ðf0Þs0:

We may, without loss of generality, assume fO0.Consider two nearby initial conditions (R0, f0) and (R0, f0CDf0), as sketched

in figure 8. The Poincare map sends these initial conditions to (R0f(f0), g(f0))and (R0f(f0CDf0), g(f0CDf0)) as sketched in figure 8.

If we think of the entire triangle as composed of initial conditions, then thetriangle gets mapped to a triangle because solutions are scaleable (the edgesremain straight lines).

Since f and g are continuous, we write

f ðf0 CDf0ÞZ f ðf0ÞCDf ; and gðf0 CDf0ÞZ gðf0ÞCDg;

where the D symbol is now taken to denote ‘small’.Area preservation now gives

jDgðf0ÞjDf0

Z1

f ðf0Þ2;

where we have ignored small quantities of second-order.We now come to an important point. No matter what finite values we assign to

the parameters d and e, the functions f and g depend continuously on them as





well as on f0. So, if we now change any combination of d, e and/or f0 in any waythat we like, f(f0) always remains finite and non-zero. This means Dg(f0) isalways non-zero as well, because Df0 is non-zero and positive by choice; it neverchanges sign. It is possible to conclude from a consideration of the case dZeZ0and f0Zp/2, i.e.

€x Z 0;

that DgO0. From here, by continuous changes in parameters and initialconditions, we can arrive at the point of interest to conclude that the absolutevalue sign may be removed, and so

Dgðf0ÞDf0

Z1

f ðf0Þ2;

which in the limit shows that g is differentiable and satisfies

g 0ðf0ÞZ1

f 2ðf0Þ: ð6:2Þ

It follows that g0(f0)O0 for all f0.We observe from equation (6.2) that if any solution settles down to some

stable point f�, then g(f�)Zf� and g0(f�)!1 (the condition for stability of fixedpoints of iterated scalar maps). This in turn implies that f(f�)O1. Thus, anysolution that settles exponentially to some f� must grow exponentially inmagnitude.

What happens if, instead of a fixed point, g has a k-cycle, i.e. the k th iterate ofsome f� equals itself, or gk(f�)Zf�? Let g(f�)Zf1, g(f1)Zf2, and so on. Then,fkZf� for a k-cycle.

If, in addition, the k-cycle is exponentially stable, i.e.

g 0ðf�Þ$g 0ðf1Þ$g 0ðf2Þ/g 0ðfkK1Þ!1; ð6:3Þthen it follows that

f ðf�Þ$f ðf1Þ$f ðf2Þ/f ðfkK1ÞO1;

and the solution grows exponentially in magnitude (the Lyapunov-like exponentis positive).

Consider, now, a gradual change in parameters that causes this unstable pointin parameter space to approach a stability boundary. The k-cycle (correspondingto f�) depends on parameters, and changes gradually as well; it is structurallystable as long as inequality (6.3) holds. Thus, loss of stability can only occurwhen

g 0ðf�Þ$g 0ðf1Þ$g 0ðf2Þ/g 0ðfkK1ÞZ 1;

at which point we also have

f ðf�Þ$f ðf1Þ$f ðf2Þ/f ðfkK1ÞZ 1:

In other words, an unstable solution (i.e. a growing solution with a positive valueof the Lyapunov-like exponent used in this paper; but also a solution where f�

corresponds to a stable k-cycle of the iterated function g) can only lose instability(or gain stability; or reach a stability margin) by deforming continuously intoa periodic solution as parameters are slowly changed so as to reach a point on astability boundary.





Conclusion 6.1. A stable k-cycle in the iterated function g implies instability inthe system solution. From the corresponding (unstable) point on the parameterplane, moving towards a stable point requires appearance of a periodic solutionon the stability boundary.

What we wish to prove, however, is more general. We wish to prove that everystability boundary corresponds to the existence of a periodic solution. We will dothis by showing that every unstable point in the parameter plane corresponds tothe existence of a stable k-cycle in the iterated function g. Conclusion 6.1 willthen be applicable, and the desired result will be established.

Accordingly, we now assume that the solution is unstable, i.e. for someparameter values d and e, and referring to equation (3.1),

sZ limN/N

1

N

XNkZ0

lnðjjX2pk jjÞO0: ð6:4Þ

This is equivalent toEðlnðf ðgnðf0ÞÞÞÞO0; ð6:5Þ

where E represents expected value, and n is sufficiently large that initialtransients are not important and the final steady-state behaviour of the iteratesof g, whether periodic or not, is obtained. From equation (6.2), we have

Eðlnðg 0ÞÞZK2Eðlnðf ÞÞ;i.e.

Eðlnðg 0ðgnðfÞÞÞÞZK2Eðlnðf ðgnðfÞÞÞÞ:Therefore

Eðlnðg 0ðgnðfÞÞÞÞ!0:

LetEðlnðg 0ðgnðfÞÞÞÞZKa; ð6:6Þ

for some strictly positive number a. Moreover, since we assume the system hasreached steady-state behaviour, we can also define the variance of ln g0 as

varðlnðg 0ðgnðfÞÞÞÞZ b2; ð6:7Þfor some bO0.

Let fpZgp(f0) for pO1. Since f is an angle, we can look at its values modulo 2p.The dynamics of the system generates a sequence f0, f1, f2, f3,/, which is now

bounded (because we look at the values modulo 2p). Every bounded sequence has aconvergent subsequence (Rudin 1978). Let the subsequence be fc1

, fc2, fc3

, /,where ciOcj if iOj. Let the subsequence converge to f�. Then, for any given eO0,there exists a finite M such that for all nOM, fcnKf�!3. We choose two pointsfrom the subsequence, not necessarily consecutive, say fcn1

, fcn2with n2On1OM.

Now consider the function

hðfÞZ gcn2Kcn1 ðfÞKf: ð6:8ÞThen, applying the chain rule of differentiation,

h 0ðfÞZ g 0ðgcn2Kcn1K1ðfÞÞg 0ðgcn2Kcn1K2ðfÞÞ/g 0ðgðfÞÞg 0ðfÞK1: ð6:9Þ





Considering h0(fm) for some mOM in the subsequence c1, c2, /, we have

h 0ðfmÞZ g 0ðfcn2Kcn1CmK1Þg 0ðfcn2Kcn1CmK2Þ/g 0ðfmC1Þg 0ðfmÞK1:

Considering the logarithm of the first-term on the right-hand side, we have(calling it, say, Z)

Z ZXcn2Kcn1K1

iZ0

ln g 0ðfmCiÞ:

For cn2 sufficiently larger than cn1 (and we are free to choose it so), the centrallimit theorem applies; in particular, the expected value of Z isKðcn2 Kcn1Þa (seeequation (6.6)), and its standard deviation is b

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifficn2Kcn1

p(see equation (6.7)),

which is much smaller. Thus, Z is strictly negative, and can be as large as wewish to make it: its exponential is a positive number which can be as small as welike, independent of e. It follows that

h 0ðfmÞZ g 0ðfcn2Kcn1CmK1Þg 0ðfcn2Kcn1CmK2Þ/g 0ðfmC1Þg 0ðfmÞK1;

lies between K1 and 0. In particular, it can be bounded away from 0 by a non-zero amount, such as 1/2, independent of e.

Now, consider equation (6.8)

jhðfmÞjZ jfmCcn2Kcn1Kfmj!2e:

Thus, h(fm) is small; and h 0(fm) is non-zero. By the implicit function theorem, hhas a zero close to fm, say at ~f. This zero corresponds to a stable k-cycle of theiterated function g.

Conclusion 6.2. An unstable solution of the system implies the existence of astable k-cycle of the iterated function g.

By Conclusion 6.1, every point on a stability boundary in the parameter plane(or parameter space, if we introducemore parameters) corresponds to the existenceof a periodic solution. However, unlike the usual or linear Mathieu (or Hill)equation, the periodneednot be solely 2p or 4p, but could be a highermultiple of 2p.

7. General asymmetric Mathieu equations

More generally, our foregoing results apply to

€x CðdCe0 cosðtÞÞxC �aðdCe1 cosðtCjÞÞjxjZ 0; ð7:1Þwhere e1 and j are additional free parameters (compare with equation (1.1)).

The theoretical results presented above actually hold for the generalasymmetric Hill’s equation

€x Cp1ðtÞxCp2ðtÞjxjZ 0; ð7:2Þwith p1(tCT)Zp1(t) and p2(tCT)Zp2(t)ct and for some TO0 (T can be takenas 2p upon scaling time suitably). We are not presently aware of actual physicalsystems governed by such equations, except for the restricted case we began thispaper with.

A few numerically generated stability diagrams for equation (7.1) are nowpresented. We restrict ourselves to �aZ0:7 as before along with jZp/4.



0

d

0= 0 plane

1= 0.2plane

1= 0 plane

1

Figure 9. Parameter planes in the (d, e0, e1) space where numerical stability results are presented.

d

1

0 0.2 0.4 0.6 0.8 1.0 1.2

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

Figure 10. Stability diagram for equation (7.1) with e0Z0, jZp/4, �aZ0:7 and NZ600. Grey,stable; black, unstable; white, periodic solution.



A sketch of a subset of the full parameter space is shown in figure 9. Figure 10shows the stability diagram in the parameter plane e0Z0, and figure 11 shows thestability diagrams on portions of the two orthogonal planes e1Z0 and e0Z0,respectively.

Figure 12 shows the stability diagram on e1Z0.2 plane and figure 13 shows thestability diagrams on planes e1Z0.2 and e0Z0, respectively. Numerical



0

1d

Figure 11. Stability diagram for equation (7.1) with jZp/4, �aZ0:7 and NZ600, on two differentparameter planes. Figure 10 is included but now is horizontal.

d

0

0 0.2 0.4 0.6 0.8 1.0 1.2

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

Figure 12. Stability diagram for equation (7.1) with e1Z0.2, jZp/4, �aZ0:7 and NZ600. Grey,stable; black, unstable; white, periodic solution.

0

1 d

Figure 13. Stability diagram on the planes e1Z0.2 and e0Z0; again, �aZ0:7 and jZp/4 andNZ600.







resolution limits the degree of detail that can be trusted in the figures. With morecomputation time, any of these figures could be regenerated with higher precisionand resolution. However, our key point in producing these figures is to emphasizethat the white lines, representing numerically obtained periodic solutions, werein fact computed accurate to nine decimal places. Moreover, we did find periodicsolutions on every stability boundary that we examined, verifying the theoreticalresults obtained above.

8. Conclusions and further work

Wehave numerically and theoretically studied the asymmetricMathieu equations,which are strongly nonlinear but conservative, and have scaleable solutions (if x(t)is a solution, then so is ax(t) with aO 0). We have found that there are infinitelymanymore instabilities for this system than for the usual Mathieu equation. Thereare periodic solutions on every stability boundary in the parameter space. Theperiods of these solutions are not confined to either 2p or 4p; higher multiples of 2poccur. Our theoretical results are also applicable to asymmetric Hill’s equations.

Several questions seem interesting and relevant which we have been unable toanswer here. For equation (1.1), are there in fact infinitely many instability regionsof strictly non-zero width emanating from any finite interval on the d-axis? How dothe widths of these regions, say for small e and for anm:n resonance, depend onm, nand e? Given an arbitrary point (d, e) with eO0, does every open set in theparameter plane that contains this point also contain an unstable point? Givensome small damping, how much of which instability regions will survive? We hopethat future work may shed some light on these issues.

Appendix A. An arc-length based continuation method

We numerically seek periodic solutions to equation (1.1). Consider a solutionstarting at tZ0. If x0 and _x0 are the initial conditions and we numericallyintegrate equation (1.1) to tZT for suitable T, then x(T) and _xðTÞ should satisfy

xðTÞKx0 Z 0; ðA 1Þ

_xðTÞK _x0 Z 0: ðA 2Þ

The routine continuation method described below was also used inNandakumar & Chatterjee (2005). Assume that we are on a branch where wecan take x0Z1 (recall that solutions are scaleable; the corresponding procedurefor x0ZK1 or _x0ZG1 will be obvious below). We define

y Z

_x0

e

d

8><>:

9>=>;: ðA 3Þ

Assume that for some known y, i.e. for some known choice of d, e aswell as _x0, we havea periodic solution of periodT. (Such a y can be found by an initial numerical searchthat is simpler than the continuation method and is not described here. Note, inparticular, that resonance conditions provide the d values corresponding to eZ0,





so only _x0 need to be found for these points.) Let us refer to this value of y by the nameyold. The continuation method is based on defining a function g(ynew) as follows.

Given any vector ynew, we have dnew, enew and _x0;new. Inserting dnew and enewinto equation (1.1), and using initial conditions x0Z1 and _x0Z _x0;new, wenumerically integrate forward to time T to obtain x(T) and _xðTÞ. We also choosea small positive number s, which represents arc-length. Then, we define

gðynewÞZxðTÞK1

_xðTÞK _x0;new

syoldKynewsKs

8><>:

9>=>;: ðA 4Þ

We now numerically evaluate the function g as accurately as we like; and use anystandard iterative technique (like Newton–Raphson) to find a value ynew thatsatisfies g(ynew)Z0. Finally, we redefine yold to be the freshly obtained ynew, andrepeat the above procedure. This completes the arc-length based continuationmethod.

In numerical implementation, in the Newton–Raphson iterative stage, it ishelpful to have a good initial guess. Using the two previously obtained y vectors,say y1 and y2, to extrapolate linearly to

yguess Z 2y2Ky1;

gives a good guess for small s in most cases.

References

Nandakumar, K. & Chatterjee, A. 2005 Resonance, parameter estimation, and modal interactionsin a strongly nonlinear benchtop oscillator. Nonlin. Dyn. 40, 149–167. (doi:10.1007/s11071-005-4228-3)

Rand, R. H. 2003 Lecture notes on nonlinear vibrations—version 45. Available at http://www.tam.cornell.edu/randdocs/.

Rudin, W. 1976 Principles of mathematical analysis. Singapore: McGraw-Hill.Stoker, J. J. 1950 Nonlinear vibrations in mechanical and electrical systems. New York: Wiley.Theodossiades, S. & Natsiavas, S. 2000 Nonlinear dynamics of gear pair systems with periodic

stiffness and backlash. J. Sound Vib. 229, 287–310. (doi:10.1006/jsvi.1999.2490)Winkler, S. & Magnus, W. 1966 Hill’s equation. New York: Dover.


http://dx.doi.org/doi:10.1007/s11071-005-4228-3

http://dx.doi.org/doi:10.1007/s11071-005-4228-3

http://www.tam.cornell.edu/randdocs/

http://www.tam.cornell.edu/randdocs/

http://dx.doi.org/doi:10.1006/jsvi.1999.2490


Asymmetric Mathieu equations

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