Perturbed damped pendulum: finding periodic solutions via averaging method

Perturbações do pêndulo amortecido: encontrando soluções periódicas via método averaging

ARTIGOS GERAIS

Perturbed damped pendulum: finding periodic solutions via averaging method

Perturbações do pêndulo amortecido: encontrando soluções periódicas via método averaging

Douglas D. Novaes¹

Departamento de Matematica, Universidade Estadual de Campinas, Campinas, BP, Brasil

ABSTRACT

Using the damped pendulum model we introduce the averaging method to study the periodic solutions of dynamical systems with small non-autonomous perturbation. We provide sufficient conditions for the existence of periodic solutions with small amplitude of the non-linear perturbed damped pendulum. The averaging method provides a useful means to study dynamical systems, accessible to Master and PhD students.

Keywords: averaging method, periodic solutions, nonlinear systems, damped pendulum.

RESUMO

Utilizando o modelo do pêndulo amortecido, introduzimos o método "averaging" no estudo de soluções periódicas de sistemas dinâmicos com pequenas perturbações não autónomas. Considerando perturbações do sistema do pêndulo amortecido, fornecemos condições suficientes para a existência de soluções periódicas de pequena amplitude. O método "averaging" fornece uma ferramenta útil no estudo de sistemas dinâmicos e é acessível a estudantes de pós-graduação.

Palavras-chave: método "averaging", soluções periódicas, sistemas não lineares, pêndulo amortecido.

1. Introduction

Systems derived from the pendulum give to students important and practical examples of dynamical sy&tems. For instance, we can see the weight-driven pendulum clocks which had its historical and dynamical aspects studied by Denny in a recent paper [I]. This system has been revisited by Llibre and Teixeira in [2], and using some simple techniques, from averaging theory, they got the sarne results. Usually, the systems involving pendulums have also been used to introduce mathematical concepts of classical mechanics, as we can see in Ref. [3].

ln this paper we attempt to use a simpIe physical system, as the damped pendulum, to introduce some concepts and techniques of the important and useful averaging theory, which can be used to study the periodic solutions of dynamical systems. For instance, in Ref. [4], Llibre, Novaes and Teixeira have used the averaging theory to provide sufficient conditions for the existence of periodic solutions of the planar double pendulum with small oscillations perturbed by non-linear functions.

2. The damped pendulum

We consider a system composed of a point mass m moving in the plane, under gravity force, such that the distance between the point mass m and a given point P is fixed and equal to l. We also consider that the motion of the particle suffers a resistance proportional to its velocity. This system is called Damped Pendulum.

The position of the pendulum is determined by the angle θ shown in Fig. 1. The equation of motion of this system is given by

where a > 0 and b > 0 are real pararneters, with a =g/l, g the acceleration of the gravity, l the length of the rod and b the damping coefficient. We shall also assume that the damping coefficient b is a small parameter.

There are many other kinds of resistance that the particle motion can suffer, providing many different dynamical behaviors (see Remark 1). For instance, the Coulomb Friction introduces a discontinuous term in the equation of motion (1). For more details about this last issue, see the book of Andronov et al. [5].

ln the qualitative theory of dynamical systems, a singularity x* of an autonomous differential system (t) = F(x), i.e. F(x* ) = 0, is called Hyperbolic if the eigenvalues of the linear transformation DF(x*) (derivative of F in x*) has non-zero real components. ln this case, applying the classical Hariman-Grobman Theorem (see Theorem 2.2.3 from Ref. [6]) we can study the local behavior of the system looking to the linearized system (t) = D(x*)y.

Remark 1. Here, the behavior of a dynamical system can be informally understood as how the phase portrait looks like. For a general introduction to qualitative theory of dynamical systems see for instance the book of Arrowsmith and Place [6].

The linearized equation of motion of the damped pendulum is given by

Considering the coordinates (

), the system (2) has the following eigenvalues

which is the eigenvalues of the matrix

Note that for b²> 4a the eigenvalues λ₁ end λ₂ are both negative, then the singularity () = (0, 0) is a attractor node, represented in the Fig. 2. Now, for b²< 4a both eigenvalues λ₁ end λ₂ have the imaginary part different of zero and negative real part, then the singularity () = (0,0) is a attractor focus, represented in the Fig. 3. Both cases are topologically equivalent.

Observe that without the damping effect, i.e. b = 0, both eigenvalues λ₁ end λ₂ would be purely imaginary, then the singularity () = (0,0) would be a center (which is a non-hyperbolic singularity), represented in the Fig. 4. This last case is not topologically equivalent to the two cases above.

We emphasize here that, for b #-0, the linearized Eq. (2) can only be used to study the local behavior, at () = (0,0) , ofthe original (non-linear) system (1). However, a periodic solution of a differential system is a global element of the phase portrait. Thus to study these elements we have to consider the non-linear equation, which has the global information of the system.

As we have seen, when b ≠-0, the origin is an attractor singularity, thus the orbits of the system starting sufficiently doser to the origin tends to it. ln other words, the damped pendulum always tends to stop. The following study provide conditions to drop this regime obtaining thus a periodic solution of the system which never reaches the origin.

3. Perturbed system

Let (f, g) be an ordered pair of functions, such that

We consider the non-autonomous perturbation of the system (1)

By a non-autonomous perturbation we understand thal the function (f, g) (t,) is dependent of the variable t. We also assume that the ordered pair of functions (1,g) satisfies the following conditions:

(C1) f(t,) and g (t,) are C² functions;

(C2) f(t,) and g (t,) are locally Lipschitz with respect lo ();

(C3) f(t,) and g(t,) are respectively T_ƒ and T_g periodic in the variable t with

being ρ_i and q_i relatively prime positive integers for i = f, g;

(C4) and f(t, 0, 0) = 0.

We say that the Basic Conditions are satisfied for an ordered pair of functions (f,g) (which define the non-autonomous perturbation on the system (3)) when (f,g) satisfies the conditions C_i, for i = 1,2,3,4.

Remark 2. For simplicity, instead condition C3 we can assume, without loss of generality, that the functions f(t,) and g (t,) are T-periodic with T = 2pπ/√a for some integer p. lndeed, taking p the least common multiple between P_ƒ and P_g, we have that there exists integers n_ƒ and n_g such that p = n_ƒ p_ƒ = n_g P_g. Hence

4. Change of coordinates

As we shall see in the Section 6. the main result from the Averaging Theory, used in this present work, assumes that the perturbed system is given in an Standard Form (8). To get it, we have to introduce two changes of coordinates. The first one is done in the following.

If we lake θ = εΦ with |ε| > 0, then the system (3) becomes

Note that, since b is a small parameter, we can assume that b = ε, with > O if we consider perturbations for ε > 0; and with < 0 if we consider perturbations for ε < 0. Henceforward we assume that ε >0.

As a consequence of this coordinates change we have the following lemma.

Lemma 1. There exists a continuous function r(t, Φ, Φ, ε), T–periodic in the variable t with T = 2pπ/√a, such that the system (4) is written as

where

Demonstração. Sei the variables t, Φ and Φ as the parameters of the C² function s → F(s; t, Φ, Φ) defined as

Applying the Taylor's Formula with Lagrange Remainder for the function F(s; t,Φ, Φ) at as s = 0 we conclude thal there exisls 0 < h < 1 such that

Since

and

It follows that the function

is continuous in the variable ε. By the other hand

Thus we conclude that the function r(t,Φ, Φ) is continuous in the variables t ,Φ and Φ, and T-periodic in the variable t.

The Taylor's Formula with Lagrange Remainder has its statement and proof done in the book of Lima [7].

To study the periodic solutions of the system (3) we shall study the periodic solutions of the system (5). Indeed, if (t, ε) is a solution of system (5), then φ(t, ε) = ε(t, ε) is a solution of the system (3). However, the change of coordinate, introduced above, restricts our study only for periodic solutions close to the origin, since |φ(t, ε)| → 0 as ε → 0 for t ∈ I for every I compact interval of . In this case, we say that this periodic solution is bifurcating of the origin.

Now, denoting (x, y) = (

), the second order differential Eq. (5) can be written as the first order differential system

Observe that the unperturbed system, i.e. ε = 0,

We define the matrix M = (M_ij)_2x2 as has the following eigenvalues

thus it is a center at the origin, see Fig. 4.

There are many works which deal with perturbation of centers, even in higher dimensions. For instance we can see the paper of Llibre and Teixeira [8].

The second change of coordinates (13) will be done in the Section 7..

5. Statements of the main results

Our main goal, in this present work, is to find sufficient conditions on the ordered pair of functions (f, g) to assure the existence of periodic solutions of the system (3). For this, we shall provide a matrix M such that its non-singularity, i.e. det(M) ≠ 0, implies the existence of at least one periodic solution, for ε > 0 sufficiently small, of the system (3).

Our main result on the periodic solutions of the damped pendulum with small non-autonomous perturbation (3) is the following.

Theorem 2.Assume that the Basic Conditions are satisfied for the pair of functions (f, g), which define the non-autonomous perturbation on the system (3). If det(M) ≠ 0, then for ε> 0 sufficiently small the perturbed damped pendulum (3) has a T-periodic solution θ(t, ε), such that

when ε → 0.

Theorem 2 is proved in section 7.. Its proof is based in the averaging theory for computing periodic solutions, which will be introduce in section 6..

We provide an application of Theorem 2 in the following corollary.

Corollary 3. Assume that the Basic Conditions are satisfied for the pair of functions (f, g), which define the non-autonomous perturbation on the system (3). Moreover, suppose that

If (C₁,C₂) ̸= (0, ), then there exists a T-periodic solution θ(t, ε) of the perturbed damped pendulum (3), such that

when ε → 0.

The Corollary 3 will be proved in section 7..

6. Averaging theory

We present in this section a basic result known as First Order Averaging Theorem. For a general introduction to averaging theory see for instance the book of Sanders and Verhulst [9] and the book of Verhulst [10]. We consider the differential equation

where F₁ : x U → ⁿ is a smooth function and R : x U x (−ε_ƒ , ε_ƒ ) → ⁿ is a continuous function. These functions are both T-periodic in the first variable t and U is an open subset of ⁿ.

Remark 3. The Eq. (8) is the Standard Form, of a differential system, to apply the first order averaging theorem.

We define the averaged system associated to system (8) as

where ƒ₁ : U → ⁿ is given by

ln resume, the averaging theory gives a quantitative relation between the solutions of some non-autonomous differential system and the solutions of the averaged differential system, which is an autonomous one. ln our case, as we are working with periodic systems, the averaging method also leads to the existence of periodic solutions.

The next theorem associates the singularities of the system (9) with the periodic solulions of the differential system (8).

Theorem 4. Assume that

(i) F, and R are 10caUy Lipschitz with respect to X;

(ii) for a ∈ U with ƒ₁(a) = 0, there exist a neighborhood V of a such that ƒ₁(Z) ≠ 0 for al z ∈ \{a} and det(dƒ₁(a)) ≠ 0.

Then, for |ε|> 0 sufticiently smal, there exist a T-periodic solution X(t,ε) of the system (8) such that X(0, ε) → a as ε → 0.

Using Brower degree theory, the hypotheses of Theorem 4 becomes weaker. For a proof of Theorem 4 see Buica and Llibre [11].

Remark 4. Using the Averaging Theorem 4 we can study some solutions of Eq. (8) only studying the algebraic equation ƒ₁(Z) = 0, instead solving the differential equation. This is one of the main characieristic of Averaging Theory.

7. Proofs of Theorem 2 and Corollary 3

ln order to use the Theorem 4 in the proof of Theorem 2, we have lo modify the Eq. (6). If we denote

and

then the Eq. (6) can be written in the matrix form

Now, we introduce the main change of coordinates which makes us able lo wrile the differenlial Eq. (12) in the Standard Form (8).

Take y(t) ∈ ² as

where

is the matrix of the fundamental solution of the unperturbed differential system (12), i.e., ε = 0. This change of coordinates is done in the book of Sanders and Verhulst [12].

Observe that the application t → e^-At is T-periodic function with T = 2π/ √a, since the eigenvalues of A are purely imaginary. Moreover y(0) = x(0).

Lemma 5. The system (12) is written in the new variable y as

Demonstração.

Note that the system (14) is wrillen in the slandard form (8). Thus we are ready lo prove the Theorem 2.

Proof of Theorem 2. The smooth funclions ƒ(t, x) and g(t,x) are T-periodic in the variable t, with T = 2ρπ/ √, which implies thal the smooth functions F(t, y) and (t, y, ε) are also T-periodic in t.

We shall apply the Theorem 4 lo the differenlial Eq. (14). Nole thal the Eq. (14) can be wrillen as the Eq. (8) taking

Observe that, by Lemma 1, the function F₁ and R satisfy the assumptions of Theorem 4.

Given z ∈ ² , we can compute the averaged function ƒ₁ : ² → ², defined in (10), as

where M is defined in (7) and

Assuming that det(M) ≠ 0, we conclude thal there exists a solution z₀ = (x₀, y₀) of the linear system ƒ₁(z) = 0 given by z_o = M^-1v which satisfies the hypotheses of Theorem 4. Indeed

Moreover, del(M) ≠ 0 also implies the uniqueness of the solulion z₀ of the system Mz = v, thus ƒ₁(z) ≠ 0 for ali z ∈ ²\{z₀}.

Hence, applying Theorem 4, follows that there exists a T-periodic solulion y(t,ε) of the system (14) such that

which implies the existence of a periodic solution (x(t,ε),y(t,ε)) of the system (6) such that

Since x(0) = y(0) it follows that

when ε → 0. Hence

is a T–periodic solution of the system (3) such that implies that

when ε → 0.

Proof of Corollary 3. The hypotheses of Corollary 3

Thus

Hence det(M) = 0 if and only if (C₁,C₂) = (0, ). Since (C₁,C₂) ≠ (0, ), the result follows by applying Theorem 2.

8. Simulation

Consider the following differential equation

Observe that the Eq. (16) is a small perturbation of the damped pendulum system. lndeed, Eq. (3) becomes Eq. (16) by taking the parameters a = 1, and b = ε; and the functions ƒ(t, θ, θ) = 0, and g(t, θ, θ) = sin(t). Therefore, considering C₁ = C₂ = 0 we can apply the Corollary 3 to assure the existence of a periodic solution of the system (16).

lndeed, proceeding with the numerical simulation we find this periodic solution, represented by the blue line in Fig. 5.

This simulation has been done using the Wolfram Mathematica^® 8 software.

9. Conclusions and future directions

The averaging theory is a collection of techniques to study, via approximations, the behavior of the solutions of a dynamical system under small perturbations. As we have seen, it can also be used to find periodic solutions.

ln this paper, we have presented one of these techniques and used it to find conditions that assure the existence of a periodic solution of the non-autonomous perturbed damped pendulum system. ln resume, we have got an algebraic non-homogeneous linear system M z = v such that its solution, when det(M) ≠ 0, is associated with a periodic solution of such perturbed system.

Recently, Llibre et al. [13] have extended the averaging method for studying the periodic solutions of a class of differential equations with discontinuous second member. Therefore we are able to consider for instance equations of kind

Here, the term bsign() represents the Coulomb Friction, where the function sgn(z) denotes the sign function, i.e.

For instance, in Ref. [14], Llibre et al have used the averaging theory to provide sufficient conditions for the existence of periodic solutions with small amplitude of the non-linear planar double pendulum perturbed by smooth or non-smooth functions.

Acknowledgements

The author is supported by a FAPESP-BRAZlL grant 2012/10231-7.

Recebido em 3/2/2012

Aceito em 15/12/2012

Publicado em 18/3/2013

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• [13] J. Llibre, D.D. Novaes and M.A. Teixeira, arXiv:1205.4211 [math.DS], http://arxiv.org/abs/1205.4211
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Publication Dates

• Publication in this collection
  07 May 2013
• Date of issue
  Mar 2013

History

• Received
  03 Feb 2012
• Accepted
  15 Dec 2012