Application of a Rank-One Perturbation to Pendulum Systems

From a perturbation theory proposed by Mehl, et al., a study of the rank-one perturbation of the problems governed by pendulum systems is presented. Thus, a study of motion of the simple pendulum, double and triple pendulums with oscillating support, not coupled as coupled by a spring and double pendulum with fixed support is proposed. Finally (strong) stability and instability zones are calculated for each studied system.


Introduction
Consider the following Hill's equation d 2 x i dt 2 + ϕ i (t)x i = 0, i = 1, ..., N, where ϕ i are periodic functions and N ≥ 1. This type of equation is part of an important class of differential equations (see (Hoffmann & Stroobant, 2007), (Hsu, 1961) and (Yakubovich & Starzhinskii, 1975)). Hill's equations appear in several scientific fields such as: physics, mechanics and chemistry. In these fields, these types of equations are used more officially in the study of the stability of certain real phenomena such as the motion of a pendulum ( see in (Hsu, 1961) and (Lalanne, Berthier, & Der Hagopian, 1984)), the motion of an ion through a quadrupole analyzer (see (Hoffmann & Stroobant, 2007)), the vibratory motion of an elliptical membrane (see in (Mathieu, 1868)) and so on. These equations can be in the following Hamiltonian form (see for example (Dosso, 2006) and (Dosso & Coulibaly, 2014)): where t −→ H(t) is a piecewise continues and periodic matrix function, J is an invertible and skew-symmetric matrix and I 2N the identity matrix of order 2N.
Let W ≡ X(P) be the monodronic matrix of system (1). This matrix plays a very important role in the strong stability study of differential systems because its strong stability is equivalent to that of system (1) (see , (Dosso, 2006), (Dosso & Coulibaly, 2014) and (Yakubovich & Starzhinskii, 1975)). Thus, it suffices to know the nature of stability of this matrix to deduce that of system (1). Considering the spectrum of the monodronic matrix W, we have the following classification given by Godunov and Sadkane in (Godunov & Sadkane, 2001) and (Godunov & Sadkane, 2006).
Definition 1 Let λ be a semi-simple eigenvalue of a symplectic matrix W on the unit circle. We say that λ is of the red (respectively green) color or in short r-eigenvalue (respectively g-eigenvalue) if (S 0 x, x) > 0 (respectively (S 0 x, x) < 0) on the eigenspace associated with λ, where S 0 = 1 2 (JW) + (JW) T . If (S 0 x, x) = 0, we say that λ is of mixed color. In definition 1, the notation (., .) denoted an inner scalar product.
From this classification, the following proposition gives us the conditions of the strong stability of the monodromy of a symplectic matrix W.
Proposition 1 W is strongly stable if and only if: 1) all its eigenvalues are on the unit circle; 2) its eigenvalues are either red color or either green color; 3) the quantity δ S = min{|e iθ l − e iθ j | : e iθ l , e iθ j are r-and g-eigenvalues of W} should not be close to zero.

Theorem 1 The system (1) is strongly stable if and only if one of the following conditions is verified
1) the monodromy matrix W of system (1) is strongly stable.
2) the sequence of matrix average (S (n) ) n≥0 definied by converges to a positive definie symmetric constant matrix S (∞) and the quantity δ S = min{|e iθ l − e iθ j | : e iθ l , e iθ j are r-and g-eigenvalues of W} is not close to zero.
3) there exists ε > 0 such that any Hamiltonian system with P-periodic coefficients of the form In this paper, we are interested to the application of the perturbation theory proposed in  to pendulum systems. Thus, we recall the definition of the rank-one perturbation of a Hamiltonian system given in : Definition 2 We call rank-one perturbation of Hamiltonian system with periodic coefficients (1) any differential system of the form where u ∈ R 2N is a non-zero random vector. From this definition and results of , we give as a consequence of the strong stability of (1) on its rank-one perturbation as follows.
Proposition 2 If system (1) is strongly stable, then there exists ε > 0 such that for any vector u ∈ R 2N verifying uu T JW < ε, we have X(P) = (I + uu T J)W is stable, where W ≡ X(P).
The purpose of this present paper is to analyze the (strong) stability of some everyday problems governed by Hamiltonian systems with periodic coefficients using the perturbation theory introduced in (Dosso, Arouna, & Koua Brou, 2018). The paper is organized as follows: in section 2, we study the stability of the motion of a simple pendulum with oscillating support. The third and fourth sections are respectively dedicated to the study of the stability of the motion of the double and triple pendulum with oscillating supports. In each of these sections, our study is organized in two parts. In the first part, we study the case where the double and triple pendulums are not coupled and in the second part, we study the case where these pendulums are coupled by a spring. Finally, in the last section, we are interested in the study of the (strong) stability of the motion of a double pendulum with fixed supports.
Throughout this paper, the symbol . denotes the Euclidean norm of matrices or vectors. In the present figures, the zones in red, blue and white color denote respectively the zones of instability, stability and strong stability of the rank-one perturbation of (1).

Simple Pendulum With Oscillating Support
Consider the following simple pendulum (see Figure1) whose support is subjected to an oscillating motion f (t) defined by f (t) = α cos(Ωt) in (Hsu, 1961). According to (Hsu, 1961), the equation of motion of the simple pendulum is governed by where k 0 is the radius of gyration of the pendulum about its point of suspension and c the distance between the point of suspension and the center of the pendulum. This equation can be written as a Mathieu's equation of the form (see (Hsu, 1961)) where τ = Ωt, = cα k 2 0 and δ = cg k 2 0 Ω 2 . Using the change of variable given in (2) with N = 1, it is easy to see that (7) can be reduce to Hamiltonian form (1), with To analyze the (strong) stability of the motion of the pendulum, we perturb the solution X(τ, δ, ) of (1) by the following matrix of rank one: where u a = a 0 and a ∈ [0, 1[. Then the perturb motion of the pendulum is described by: X a (τ, δ, ) = I + u a u T a J X(τ, δ, ). According to , the equation of the pendulum's motion then can be written as: Journal of Mathematics Research Vol. 12, No. 5;2020 The spectral portrait and the (strong) stability zone of the matrix solution X a (τ, δ, ) of equation (8)  In Figure 2, we notice a small change in the spectral portrait of X a (τ, δ, ) for δ = 1 and = 0.8 whereas for δ = 1.93 and = 1.93, we don't observe any change in its spectral portrait in presence of this rank-one perturbation. In Figure 3, we observe that the stable region of X a (τ, δ, ) obtained in presence of the rank-one perturbation is smaller than that obtained in absence of this perturbation. In fact, S (n 0 ) (τ) takes much lager values when a is different to zero; and the region where δ S (τ) is represented in green color, is small when a is different to zero.  The Figure 4 represents the (strong) stability zone of X a (2π, , δ) in the plane of parameters (δ, ) ∈ [0, 1.98] × [0, 2]. In this Figure, we note the presence of two regions: a first zone in red color, representing the unstable zone and a second zone, in white, representing the strong stable zone. In presence of perturbation, we notice a widening of the unstable zone(see Figure on the left). This shows that the perturbation is a factor that increase an instability of the motion of the pendulum.

Double Pendulum With Oscillating Supports
We consider two identical simple pendulums attached to a support common (see Figures 5 and 9). In this part, we restrict our study to the case where the support of each pendulum is subjected to an oscillatory motion f (t) defined by f (t) = α cos(Ωt) (see in (Hsu, 1961)). Since the two pendulums are identical, according to (Hsu, 1961), the equation of the motion of the two pendulums will be the same. Then, the equation of motion is given by:

Uncoupled Double Pendulums With Oscillating Supports
where k 0 is the radius of gyration of the pendulum around its point of suspension, and c is the distance between the point of suspension and the center of the pendulum.
Using the change of variable τ = Ωt the equation of the system then becomes:  Vol. 12, No. 5;2020 Finally, using the following change of variables defined in (2) with N = 2, we obtained system (1) with In what follows, considering the rank-one perturbation of the fundamental solution X(τ, δ, ) of its corresponding Hamiltonian system by the following matrix of rank one where According to , its rank-one perturbation is and the equation of motion then becomes The figure below represents the spectral portrait of X a (τ, δ, ) for (δ, ) ∈ {(1, 0.8) , (1.93, 1.93)} and a ∈ {0; 0.35}, with τ ∈ [0, 2π]. In this figure, we note a small change in the spectral portrait of X a (τ, δ, ), due to the rank-one perturbation. For these parameters, the (strong) stability zone of X a (τ, δ, ) (τ ∈ [0, 2π]) is plotted in Figure 7. This figure shows a coarse widening of the unstable region and a narrowing of the stable region of X a (τ, δ, ) when the rank-one perturbation is taken into account. In fact, S (n 0 ) (τ) takes much lager values when a is different to zero; and the region where δ S (τ) is represented in green color, is small when a is different to zero. In Figure 8, we observe the presence of two regions in the (strong) stability zone: a first zone, in red color, representing the unstable zone and a second zone, in blue, representing the stable zone. When the motion of the two uncoupled pendulums is subjected to the rank-one perturbation, we notice a widening of the unstable zone (see Figure on the left). This shows again that the perturbation is a factor that increase an instability of the motion of the double uncoupled pendulum.

Coupled Double Pendulums With Oscillating Supports
In this part, the two above simple pendulums are coupled by a spring of constant stiffness k (see Figure 9) Figure 9. Model of the coupled double pendulum with oscillating supports According to (Hsu, 1961), the motion of the system is governed by the following differential equation: where with m the mass of each pendulum and b the distance between the point of suspension and the point of attachment of the coupling spring.
Using successively the following change of variables: the equation of motion of the system then becomes (see for example in (Hsu, 1961)): where Finally, using the change of variables given in (2)  In what follows, considering that the motion X(τ, δ, , e) of the system is subjected to a perturbation of the form where vector the equation of motion takes form (13).

Triple Pendulum With Oscillating Supports
Here, we consider three identical simple pendulums attached to a common support (see Figures 13 and 17). In this problem, we restrict our study to the case where the support of each pendulum is subjected to an oscillating motion f (t) defined by f (t) = α cos(Ωt) (see in (Hsu, 1961)).

Uncoupled Triple Pendulums With Oscillating Supports
In this first part, the three simple pendulums are uncoupled. Figure 13. Model of the uncoupled triple pendulums with oscillating supports Since the three pendulums are identical, according to (Hsu, 1961), the differential equation of motion will be the same for all three. Then, the equation of motion becomes: where k 0 and c are defined in section 2.
Using the fact that f (t) = α cos(Ωt), the above differential system can be written as According to (Hsu, 1961), using the change of variables τ = Ωt, the differential equation of motion of the triple pendulums may be reduced to: where = cα k 2 0 and δ = cg k 2 0 Ω 2 Introducing the change of variables given in (2) with N = 3, it is easy to see that the motion of the uncoupled system is gouverned by (1), with To study the (strong) stability of the motion of the triple pendulums, we perturb the motion of the uncoupled system by the following rank-one matrix: where From the above, we deduce that X a (τ, δ, ) can be rewritten as: X a (τ, δ, ) = I + u a u T a J X (τ, δ, ) and the equation of motion then becomes (see ) Figures 14 and 15 show respectively the spectral portrait of X a (τ, δ, ) and the (strong) stability region of X a (τ, δ, ) for τ ∈ [0, 2π] and (δ, ) ∈ {(1, 0.8) , (1.93, 1.93)}, with a ∈ {0, 0.35}.
In Figure 14, we note that the spectral portrait undergoes a slight modification in presence of the rank-one perturbation. In figure 15, we observe that in presence of the rank-one perturbation there is a coarse widening of the unstable region and a narrowing of the stable region. Since, S (n 0 ) (τ) takes much lager values when a is different to zero ; and the region where δ S (τ) is represented in green color, is small when a is different to zero.  Vol. 12, No. 5;2020 In figure 16, the results obtained are consistent with the observations made in figure 8 of the second problem of this paper.

Coupled Triple Pendulums With Oscillating Supports
In this part, we couple the three simple pendulums by two springs of identical constant of stiffness k. We know that the kinetic energy of each pendulum is of the form (see in (Timoshenko, Young, & Weaver, 1974)): From a result of (Timoshenko, Young, & Weaver, 1974), we obtain the total kinetic energy of the coupled system by: Since the gravitational potential energy of each pendulum is given by: The total gravitational potential energy of the coupled system is given by: with m the mass of each pendulum and b the distance from the point of suspension to the point where the coupling spring is attached.
Replacing f (t) by its new expression in equation (23), we get the following equation Using the change of variable τ = Ωt, the equation of the triple pendulums motion then becomes: With the change of variable given in (2), we get system (1) with and In this part, we assume that the motion of the coupled system is perturbed by a matrix of rank one of the form (16), where and a ∈ [0, 1[. Then according to , it is easy to see that the equation of the motion of the coupled system is governed by equation (21).
As expected, we can visualize respectively, the spectral portrait of X a (τ, δ, , e) and the (strong) stability zone of the motion of the system. In Figure 18, we observe small change in the spectral portrait of X a (τ, δ, , e). In figure 19, we note that in presence of the rank-one perturbation, the motion of the coupled system becomes very unstable. Because, the euclidian norm of S (n 0 ) (τ) takes much lager values when a is different to zero; and the region where δ S (τ) is represented in green color, is small when a is different to zero.  Vol. 12, No. 5;2020 In Figure 20, we note a slight difference between the two figures due to the small rank-one perturbation of the system described by our triple coupled pendulums.  According to (Yakubovich & Starzhinskii, 1975), the equation of the pendulum movement is given by: where y = φ ξ and P(τ, , δ) = Using the change of variables (2) with N = 2, we obtain the Hamiltonian system with 2π-periodic coefficients (1) with : To simplify the problem, we take: m 1 = 2g, m 2 = 5g, l 1 = 0.5m, l 2 = 1m, and Ω = 1 ; and we subject the motion of the system to a rank-one perturbation of the form (11). Then, the equation of the movement of the system can be put in the form (13). To study the (strong) stability of the motion of the pendulum for τ ∈ [0, 2π], numerical simulations were done with ( , δ) ∈ {(0.05, 0.5) , (0.2, 0.6)} and a = 0, 0.35.

Concluding Remark
In this article, we have applied the theory of rank-one perturbation introduced in  to some problems governed by pendulum systems. The systems concerned are the pendulum with oscillating supports and double pendulum with fixed supports. To do this work, firstly, we rewrite the motion of these systems in Hamiltonian form (1). Secondly, we contented ourselves with a study of (strong) stability introduced in (Dosso, 2006) to analyze the effect of the rank-one perturbation on the motion of these pendulum systems. The results obtained show that the presence of the perturbation on the motion of the pendulum system favors more the loss (strong) stability of the motion of systems.