Bounding periodic orbits in second order systems

ANDRÉS GABRIEL GARCÍA

Departamento de Ingeniería Eléctrica

Universidad Tecnológica Nacional-FRBB

11 de Abril 461, Bahía Blanca, Buenos Aires

ARGENTINA

Abstract: - This paper provides an upper bound for the number of periodic orbits in planar systems. The research

results in, [7], and, [8], allows one to produce a bound on the number of periodic orbits/limit cycles.

Introducing the concept of Maximal Grade and Maximal Number of Periodic Orbits, a simple algebraic calculation

leads to an upper bound on the number of periodic trajectories for general second order systems. In particular, it

also applies to polynomial ODE’s.

As far as the author is aware, such a powerful result is not available in the literature. Instead, the methods in this

paper provide a tool to determine an upper bound on the periodic orbits/limit cycles for a wide range of dynamical

systems.

Key-Words: - Periodic orbit, Limit cycles, Nonlinear ODE.

Received: October 29, 2021. Revised: October 25, 2022. Accepted: November 27, 2022. Published: December 9, 2022.

1 Introduction

In recent papers, [7] and [8], a necessary and suf-

ficient condition for the existence of periodic orbits

in general second order oscillators (non-conservative)

were presented. Moreover, the periodic orbit’s phase

portrait can be constructed (at least for a trajectory

portion) by solving a first order singular nonlinear

ODE.

However, sometimes the exact determination of a

periodic orbit is not required and a bound on the num-

ber of them is enough, [1], [9], [10]. In this direc-

tion, even simple polynomial systems can oppose re-

sistance to the possibility of finding bounds for their

number of limit cycles, [6], [11].

This paper aims to provide an upper bound for the

number of periodic orbits in a second order oscillator

to the form:

¨x(t) = f(x(t),˙x(t), a1, a2, . . . , am)(1)

x(t)∈ <, f ∈Cn+1(<)

Where {a1, a2, . . . , am}is a real set of numeric

parameters and nis the minimal derivative order such

that:

dnf(x(t),˙x(t),a1,a2,...,am)

˙x(t)|˙x=φ(x)

dxn=g(x(t))

This means that, after nderivatives, a function g(.)

is obtained that does not depend on the parameters

{a1, a2, . . . , am}.

Despite simplified versions of the nonlinear os-

cillators considered in this paper, up to the author’s

knowledge, no other available method is able to find

an explicit upper bound for the number of periodic

orbits.

The paper is organized as follows: Section 2

presents all the machinery necessary and the main re-

sults, Section 3 presents some examples of application

and finally Section 4 some conclusions.

2 A bound for the number of periodic

orbits

This section introduces some machinery needed to

prove the main theorem:

Lemma 1 Given a function f∈Cn+1(<), and a

function φ(x)∈C0(<), that is a solution of:

(dφ(x)

dx =f(x,φ(x),a1,a2,...,am)

φ(x), φ(Ai) = 0

dφ(x)

dx |x=Ai→ ∞, i = 1, . . . L

Then, φ(x)∈Cn+1(<\Ai).

proof By induction, taking a derivative:

∂ f (x,φ(x),a1,a2,...,am)

∂x

φ(x)+

−f(x,φ(x),a1,a2,...,am)·dφ(x)

φ(x)2∈C0(<\Ai)

In other words: d2φ(x)

dx2∈C0(<\Ai). Then, as-

suming that φ(x)∈Cn(<\Ai)and taking an extra

derivative:

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dx 





dn−1

dxn−1f(x, φ(x), a1, a2, . . . , am)

φ(x)

| {z }

Cn−1(<\Ai)







Then, according to the derivative process above,

the n−1derivatives will lead a quotient:

dn−1

dxn−1f(x, φ(x), a1, a2, . . . , am)

φ(x)=

=P(x)

Q(x),{P(x), Q(x)} ∈ Cn−1(<\Ai)

Finally, an extra derivative in this quotient will

lead: dn+1φ(x)

dxn+1 ∈Cn+1(<\Ai). This completes the

proof.

Lemma 2 If a function φ(x)∈Cn+1(<)is solution

of:

(dφ(x)

dx =f(x,φ(x),a1,a2,...,am)

φ(x), φ(Ai) = 0, i = 1, . . . L

d(i)φ(x)

dxi|x=Ai→ ∞, i = 1, . . . L

Allowing L→ ∞, then is solution of:







d(n+1)φ(x)

dxn+1 =dnf(x(t),φ(x),a1,a2,...,am)

φ(x)

dxn=g(x(t))

φ(Ai) = 0,d(i)φ(x)

dxi|x=Ai→ ∞, i = 1, . . . L

proof The proof is rather straightforward taking n

derivatives from the given ODE, in the view of Lemma

1. This completes the proof.

With this lemma, it is clear that any periodic orbit

of (1), it is a solution contained in, [8]:







d(n+1)φ(x)

dxn+1 =dnf(x(t),φ(x),a1,a2,...,am)

φ(x)

dxn=g(x(t))

φ(Ai) = 0,d(i)φ(x)

dxi|x=Ai→ ∞, i = 1, . . . L

(2)

However, this ODE does not depend on the param-

eters {a1, a2, . . . , am}, or in other words, equation (2)

serves as a universal ODE containing all the possible

periodic orbits corresponding to (1).

The final preliminary result is about uniqueness of

solutions:

Lemma 3 (Uniqueness of solutions) Given an

ODE:

dφ(x)

dx =f(x, φ(x), a1, a2, . . . , am)

φ(x)

φ(Ai) = 0, i = 1, . . . L

Every solution satisfying the initial condition is

unique.

proof Let’s suppose that two different solutions

{φ1(x), φ2(x)}, φ1(x)6=φ2(x)∀x6=Aiexists:

(dφ1(x)

dx =f(x,φ1(x),a1,a2,...,am)

φ1(x)

dφ2(x)

dx =f(x,φ2(x),a1,a2,...,am)

φ2(x)

Integrating between xand Ai:

(−φ1(x)2

2=RAi

xf(σ, φ1(σ), a1, a2, . . . , am)·dσ

−φ2(x)2

2=RAi

xf(σ, φ2(σ), a1, a2, . . . , am)·dσ

Subtracting both equations and taking absolute

value:

2·φ1(x)2−φ2(x)2=

=ZAi

(f(σ, φ1(σ), a1, a2, . . . , am)+

−f(σ, φ2(σ), a1, a2, . . . , am)) ·dσ|

Moreover:

2·φ1(x)2−φ2(x)22=

2·φ1(x)2−φ2(x)2·

ZAi

(f(σ, φ1(σ), a1, aa, . . . , am)+

−f(σ, φ2(σ), a1, a2, . . . , am)) ·dσ|

Then:

2·φ1(x)2−φ2(x)2≤ZAi

ν(σ)·φ1(x)2−φ2(x)2·dσ

where ν(x) = |f(σ, φ1(σ), a1, aa, . . . , am)+

−f(σ, φ2(σ), a1, aa, . . . , am)|.

Considering the Gronwall-Bellman inequality,

[2], pp. 45, with zero independent term, then:

φ1(x)2−φ2(x)2≤0·e2·RAi

xν(σ)·dσ = 0 →φ1(x) = φ2

This completes the proof.

Finally, noticing that successive derivatives in (1)

up to the order n+ 1 will lead:

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









d(n+1)φ(x)

dxn+1 =dnf(x(t),˙x(t),a1,a2,...,am)

˙x(t)|˙x=φ(x)

dxn=

=gx(t),˙x(t),¨x(t), . . . , dnx(t)

dxn

φ(Ai) = 0,d(i)φ(x)

dxi|x=Ai→ ∞, i = 1, . . . L

This motivates the following definition:

Definition 1 (Maximal Grade (MG)) Given a sec-

ond order ODE (1) with its complementary ODE:











dφ(x)

dx =f(x,φ(x),a1,a2,...,am)

φ(x)

φ(Ai) = 0, i = 1, . . . L

d(i)φ(x)

dxi|x=Ai→ ∞, i = 1, . . . L

The maximal grade (MG) is defined to be the min-

imal derivative order, such that:











d(n+1)φ(x)

dxn+1 =dnf(x(t),˙x(t),a1,a2,...,am)

˙x(t)

dxn=

=gx(t),˙x(t),¨x(t), . . . , dnx(t)

dxn

φ(Ai) = 0,d(i)φ(x)

dxi|x=Ai→ ∞, i = 1, . . . L

(3)

In the same manner, the maximal number of peri-

odic orbits is defined to be:

Definition 2 (MNPO) Given a second order ODE

(1) with its maximal complementary ODE (3) and

defining its solution by φ(x) = ζ(x)∈Cn+1(<),

the maximal number of periodic orbits is defined as

the number of real solutions of ζ(x) = 0.

Then, the following theorem provides an upper

bound for the number of periodic orbits:

Theorem 1 A second order ODE : ¨x(t) =

f(x(t),˙x(t), a1, a2, . . . , an), x(t)∈ <, f ∈

Cn+1(<)possessing L limit cycles: {x(0) = Ai∈

<+, x(T) = Ai,˙x(0) = 0, i = 1, . . . , L}, with a

MG =M, M 6= 0 and MNP O =R(possibly

with R→ ∞) satisfies:

L≤min{R, M }

proof Recalling that the solution to (3) is defined to

be: φ(x) = ζ(x)∈Cn+1(<). It turns out that

this solution does not depend on the set of parame-

ters {a1, a2, . . . , am}, moreover: ζ(Ai)=0,∀i=

1, . . . , L.

On the other hand, let’s recall that according to,

[8], given a second order oscillator, the periodic or-

bits can be computed by solving the following auxil-

iary ODE:

(dφ(x)

dx =f(x,φ(x),a1,a2,...,am)

φ(x)

φ(Ai) = 0, i = 1, . . . L

Let’s denote the solution to this ODE by

φ(x, a1, a2, . . . , am) = ζ(x, a1, a2, . . . , am)∈

Cn+1(<). Notice that in this case, the solution does

depends on the set of parameters {a1, a2, . . . , am}.

Performing an asymptotic approximation for the

solution φ(x, a1, a2, . . . , am)with x→Ai,∀i=

1, . . . , L to the form, [5]:

ζ(x, a1, a2, . . . , am)∼ζ(x) + αL(x, a1, a2, . . . , am)(4)

(x→Ai)

where αLis a polynomial of degree Lsuch that:

αL(Ai, a1, a2, . . . , am) = 0. Moreover, let’s prove

that this asymptotic approximation satisfies (solution)

the ODE:

(dφ(x)

dx =f(x,φ(x),a1,a2,...,am)

φ(x)

φ(Ai) = 0,φ(x)

dx |x=Ai→ ∞, i = 1, . . . L

Asymptotically:

limx→Ai

d(ζ(x) + αL(x, a1, a2, . . . , am))

dx =

=limx→Ai

f(x, ζ(x) + αL(x, a1, . . . , am), a1, . . . , am)

ζ(x) + αL(x, a1, a2, . . . , am)

Rewriting this limit as:

limx→Ai

dζ(x)·1 + αL(x,a1,...,am)

ζ(x)

dx =

=limx→Ai

f(x, ζ(x)·1 + αL(x,a1,...,am)

ζ(x), a1, . . . , am)

ζ(x)·1 + αL(x,a1,...,am)

ζ(x)

Since αL(Ai, a1, a2, . . . , am) = 0 and ζ(Ai) = 0,

applying L’Hospital rule:

limx→Ai

dζ(x)

dx =

limx→Ai

f(x, ζ(x))

ζ(x)

Showing that ζ(x)is actually a solution to the

given ODE confirming the universality of ζ(x).

Note: The polynomial derivative: α0

dαL(x,a1,...,am)

dx is another polynomial with one de-

gree less, so this function is bounded. More-

over, the derivative: dζ(x)

dx =f(x,ζ)

ζ(x), in the view

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of the solution’s ζ(x)universality. In this case,

limx→Ai

α0

L·ζ(x)

f(x,ζ(x)= 0.

On the other hand, according to Lemma 3 a so-

lution ζ(x, a1, a2, . . . , am)is unique, so let‘s prove

that the polynomial αL(x, a1, a2, . . . , am)) is, in fact,

asymptotically unique.

Moreover, it is not difficult to conclude this

assumption: two polynomials with identical roots

(Ai, i = 1, . . . , L) are identical up to a constant.

Concluding that, asymptotically, the polynomial

αLis unique. Next, by virtue of Lemma 2, the solu-

tion ζ(x) + αL(x, a1, a2, . . . , am)should provide an

asymptotic solution to:

d(M+1) (ζ(x) + αL(x, a1, . . . , am))

dxM+1 =

dMf(x(t),ζ(x)+αL(x,a1,a2,...,am),a1,...,am)

ζ(x)+αL(x,a1,a2,...,am)

dxM,(x→Ai)

That is:

d(M+1)ζ(x)

dxM+1 +dM+1αL(x, a1, . . . , am)

dxM+1 =

dMf(x(t),ζ(x)+αL(x,a1,...,am),a1,...,am)

ζ(x)+αL(x,a1,a2,...,am)

dxM

Moreover, by virtue of the universality of ζ(x):

d(M+1)ζ(x)

dxM+1 =

dMf(x(t),ζ(x),a1,aa,...,am)

ζ(x)

dxM

Then:

dMf(x(t),ζ(x),a1,...,am)

ζ(x)

dxM+

+dM+1αL(x, a1, . . . , am)

dxM+1 =

dMf(x(t),ζ(x)+αL(x,a1,a2,...,am),a1,...,am)

ζ(x)+αL(x,a1,...,am)

dxM

Asymptotically:

dMf(x(t),ζ(x),a1,...,am)

ζ(x)

dxM+dM+1αL(x, a1, . . . , am)

dxM+1 =

dMf(x(t),ζ(x)+αL(x,a1,...,am),a1,aa,...,am)

ζ(x)+αL(x,a1,a2,...,am)

dxM∼

dM+1αL(x, a1, a2, . . . , am)

dxM+1 = 0 (x→Ai)

Having considered that: ζ(x) +

αL(x, a1, a2, . . . , am)∼ζ(x→Ai). This equiv-

alence shows that all the derivatives above the order

Mwill lead: dpαL(x,a1,a2,...,am)

dxp= 0, p ≥M+ 1,

then at most Mdistinct roots are possible for the

polynomial αL(x, a1, a2, . . . , am).

The key conclusion is about Land M:L≤M.

So the expression for the asymptotic equivalence can

be written as follows:

ζ(x, a1, a2, . . . , am)∼ζ(x) + αM(x, a1, a2, . . . , am)

(x→Ai)

On the other hand, by virtue of (4), it is clear that,

asymptotically:

ζ(x, a1, a2, . . . , am) = 0 ∼αM(x, a1, a2, . . . , am) = 0

(x→Ai)

Recalling that, by definition, ζ(Ai)=0,∀i=

1, . . . , L. In other words, Aiis the amplitude of a pe-

riodic orbit if and only if: ζ(Ai, a1, a2, . . . , am) = 0

or if and only if: αM= 0, that is, at most Mperi-

odic orbits for a second order oscillator with MG =

M, M 6= 0.

The number of actual periodic orbits is, in fact,

L≤M. Therefore, not all the zeroes of αMindicate

the actual orbits, but this serves as an upper bound

as requested. Furthermore, if M= 0, then αM=

0provides no insight into this bound on the periodic

orbits’ number.

Finally, it should be noticed that the general so-

lution to (3), that does not depend on the parameters

{a1, a2, . . . , am}, it contains all the possible periodic

orbits for any possible combination of these parame-

ters. Similarly, for a given oscillator, certain param-

eter selections will raise a center (R→ ∞) or, with

another choice, limit cycles (R < ∞).

Looking for an upper bound on the limit cycles, if

R→ ∞ the conclusion is clear: L≤M, however, if

R < ∞, two possibilities arise by virtue of the asymp-

totic equivalence (4):

•R > M , then only the Mzeroes of αMcould

satisfy the asymptotic equivalence, so: L≤M

•R < M , then only the zeroes of ζ(x)could satisfy

the asymptotic equivalence, so: L≤R

In summary: L≤min{R, M }. This completes

the proof.

Finally, specializing this result to oscillators:

¨x(t) = −x(t) + dF (x(t))

dx(t)·˙x(t), with F(x)a poly-

nomial of degree N:

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Corollary 1 A second order ODE : ¨x(t) = ¨x(t) =

−x(t) + dF (x(t))

dx(t)·˙x(t), with F(x)a polynomial of

degree Nand the number of isolated periodic orbits

L, a bounded from above is given by: L≤N+ 1.

proof Applying Theorem 1: M=N+ 1, in addition

to setting all the coefficients to zero in F(x), a center

is obtained, then R=∞. This completes the proof.

3 Examples

3.1 Classical Van der Pol’s Oscillator

The classical Van der Pol’s equation can be written to

be, [3], pp. 6:

¨x(t) = −x(t) + ·1−x(t)2·˙x(t)

Then, it is not difficult to obtain:

dφ(x)

dx =−x

φ(x)+·1−x2

Setting = 0 clearly results in a center, so: R=

∞. On the other hand, taking derivatives:

d4φ(x)

dx4=d3

dx3−x

φ(x)

In this case: M G = 3, then: L≤3. It is known

that for a Van der Pol system L= 1.

3.2 Polynomial systems

Considering the important case analysed by Du-

mortier, et. al, [4]:

¨x(t) = −·(x−λ)−∂H(c, e, a, x)

∂x ·˙x

Where H(c, e, a, x) = x2−12·(c·e·x+ 1) ·

x2+e·x+1

8−a·x. Considering polynomial’s

degree of H(c, e, a, x)equal to 7 and according to

Corollary 1: M= 8.

On the other hand, let’s consider a generalization

to the given polynomial H(c, e, a, x)by:

H∗(c, e, a, x, d) = x2−12·(c·e·x+d)·

·x2+e·x+1

8−a·x

It is clear that the conclusion about the number

of limit cycles includes the case d= 0, or, in other

words, the given system.

In this way, setting all the coefficients to zero:

{a= 0, c = 0, d = 0, e = 0}:

¨x(t) = −·(x−λ)

Which is not more than a center: R=∞. Then,

the conclusion, according to Theorem 1 is: L≤8.

This is in complete agreement with, [4], where the

bound was: L≥4.

4 Conclusions

In this paper, in light of the recent necessary and suffi-

cient conditions for periodic orbits in planar systems,

[7], and, [8], a bound from above is presented for the

number of periodic orbits, based on the concept of

maximal grade (MG) and maximum number of pe-

riodic orbits (MNPO).

This result allowed, for instance, to establish an

upper bound for polynomial systems based on a poly-

nomial’s degree, in particular, the influential case

studied by Dumortier, et. al, was analysed and an up-

per bound obtained,[4].

In a future paper, oscillators without parameters or

even bifurcation will be studied utilizing the method

in this paper.

Acknowledgments

This work is supported by Universidad Tecnológica

Nacional.

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