Controlling the stability of periodic orbits of completely integrable systems

We provide a constructive method designed in order to control the stability of a given periodic orbit of a general completely integrable system. The method consists of a specific type of perturbation, such that the resulting perturbed system becomes a codimension-one dissipative dynamical system which also admits that orbit as a periodic orbit, but whose stability can be a-priori prescribed. The main results are illustrated in the case of a three dimensional dissipative perturbation of the harmonic oscillator, and respectively Euler's equations form the free rigid body dynamics.


Introduction
The main purpose of this work is to provide a constructive method of controlling the stability of periodic orbits of completely integrable systems. The method consists of a specific type of perturbation, such that the resulting perturbed system becomes a codimension-one dissipative dynamical system.
The controllability procedure is based on an explicit formula for the characteristic multipliers of a given periodic orbit of a general codimension-one dissipative dynamical system. Even if this explicit formula is not the main purpose of the paper, it can be considered as the core of the article, since all main results are actually based on it. Recall that the explicit knowledge of the characteristic multipliers of a periodic orbit, is extremely useful for the study of its stability (for details regarding the stability of periodic orbits see e.g., [5], [9]).
The second result of this work is a consequence of the explicit computation of the characteristic multipliers of a given periodic orbit, and consists of two stability results. More precisely, if there exists i 0 ∈ {1, . . . , p} such that T 0 h i 0 (γ(s))ds > 0, then the periodic orbit Γ is unstable.
The third result of this article, which is also the main result, provides a method to partially stabilize a given periodic orbit of a completely integrable dynamical system. More precisely, to a given completely integrable system, and respectively a given periodic orbit, we explicitly associate a dissipative dynamical system admitting the same periodic orbit, and moreover, this periodic orbit is orbitally phase asymptotically stable, relative to a certain dynamically invariant set. Note that dissipative perturbations were also used in stabilization procedures of equilibrium states of Hamiltonian systems, see e.g., [1], [2].
The structure of the paper is the following. In the second section, one recalls a characterization of codimension-one dissipative dynamical systems, that will be used in the next sections. The third section is dedicated to the explicit computation of the characteristic multipliers of a given periodic orbit of a general codimension-one dissipative dynamical system. The fourth section uses the results from the previous section, in order to provide sufficient conditions to guarantee the partial stability, and respectively the instability of periodic orbits of codimension-one dissipative dynamical systems. The last section contains an explicit method to stabilize (relatively to a certain dynamically invariant set) a given periodic orbit of a completely integrable dynamical system.

Codimension-one dissipative dynamical systems
In this short section we recall some results concerning the codimension-one dissipative dynamical systems. For more details regarding the characterization of general dissipative dynamical systems see e.g., [8].
Recall that by a codimension-one dissipative dynamical system (defined eventually on an open subset U ⊆ R n ), we mean a dynamical systemẋ = X(x), X ∈ X(U), for which there exist k, p ∈ N, with k +p = n−1, and some smooth functions I 1 , . . . , I k , D 1 , . . . , D p , h 1 , . . . , h p ∈ C ∞ (U, R), such that the vector field X conserves I 1 , . . . , I k , and dissipates D 1 , . . . , D p , with associated dissipation rates h 1 D 1 , . . . , h p D p , i.e., L X I 1 = · · · = L X I k = 0, and respectively L X D 1 = h 1 D 1 , . . . , L X D p = h p D p , where the notation L X stands for the Lie derivative along the vector field X. Note that L X f = X, ∇f , where f ∈ C ∞ (U, R) is an arbitrary smooth function, and ∇ states for the gradient operator associated to the standard inner product on R n , namely ·, · .
Let us recall now a result from [8] regarding the local structure of a codimension-one dissipative dynamical system. Theorem 2.1 ( [8]) Let k, p ∈ N be two natural numbers such that k + p = n−1, and let I 1 , . . . , I k , D 1 , . . . , D p , h 1 , . . . , h p ∈ C ∞ (U, R) be a given set of smooth functions defined on an open subset U ⊆ R n , such that {∇I 1 , . . . , ∇I k , ∇D 1 , . . . , ∇D p } ⊂ X(U) forms a set of pointwise linearly independent vector fields on U.
Then the smooth vector fields X ∈ X(U) which verify simultaneously the conditions are characterized as follows where ν ∈ C ∞ (U, R) is an arbitrary rescaling function, and "⋆" stands for the Hodge star operator for multivector fields.

Remark 2.2 ([8])
The vector field X 0 is itself a solution of the system (2.1), while the vector field ⋆ p j=1 ∇D j ∧ k l=1 ∇I l is a solution of the homogeneous system Let us now recall from [8] a consequence of the above theorem, which gives a local characterization of completely integrable systems.

Remark 2.3 ([8])
In the case when p = 0 (and consequently k = n − 1), the dynamical system generated by the vector field X will be completely integrable, and the conclusion of Theorem (2.1) becomes: The smooth vector fields X ∈ X(U) which verify the conditions L X I 1 = · · · = L X I n−1 = 0, are given by where ν ∈ C ∞ (U, R) is a smooth arbitrary function.
3 The characteristic multipliers of periodic orbits of codimension-one dissipative dynamical systems In this section we compute explicitly the characteristic multipliers of periodic orbits of codimension-one dissipative dynamical systems. Let us recall first that for a general dynamical systemẋ = X(x), generated by a smooth vector field X ∈ X(U), defined on an open subset U ⊆ R n , and respectively a given T −periodic orbit Γ := {γ(t) ⊂ U : 0 ≤ t ≤ T }, the characteristic multipliers of Γ are the eigenvalues of the fundamental matrix u(T ), where u is the solution of the variational equation and I n,n stands for the identity matrix of dimensions n × n. Recall that since Γ is a periodic orbit, 1 will be always a characteristic multiplier of Γ, (see e.g., [6]). Taking into account the complexity of the variational equation, the computation of characteristic multipliers in general is almost impossible, since there exist no general methods to solve explicitly the variational equation.
One of the main results of this paper is to complete this task for the class of codimension-one dissipative dynamical systems, if one knows an explicit parameterization of the periodic orbit to be analyzed. In order to do that we will use the following result from [4].
• there exists a (n − 1) × (n − 1) matrix k(x) of real functions satisfying: Using the above result, we will compute explicitly the characteristic multipliers of a given periodic orbit of a codimension-one dissipative dynamical system. Let us state now the main result of this section.
Then, the characteristic multipliers of the periodic orbit Γ are Proof. The first two conditions from the hypothesis of Theorem (3.1) obviously imply the first two conditions of this theorem. Let us construct now a matrix (n − 1) × (n − 1), k(x), which verifies the equivalent of equation (3.1), namely the equation where by (ID) ⊤ we mean the transpose of the matrix ID = (I 1 , . . . , I k , D 1 , . . . , D p ).
Hence, the matrix k is a solution of equation (3.3), since the system (3.4) is obviously equivalent to which is by hypothesis verified by the vector field X. Let us solve now the equation (3.2) associated to the matrix k. In order to solve the equation, let us split first the (n − 1) Using the same type of splitting, the initial condition matrix v(0) = I n−1,n−1 splits as follows where I k,k , I p,p stands for the identity matrices of dimensions k × k, and respectively p × p. Consequently, the equation (3.2) becomes Using standard ODE techniques, one obtains the unique solution Since from Theorem (3.1), the characteristic multipliers of the periodic orbit Γ are given by {1} ∪ σ(v(T )), we obtain the conclusion.
Note that if p = 0, we recover a classical result concerning the characteristic multipliers of periodic orbits of completely integrable systems. More precisely, if p = 0, then the dynamical systemẋ = X(x) from Theorem (3.2) becomes completely integrable, and using the conclusion of Theorem (3.2) we get that the characteristic multipliers of any periodic orbit of a completely integrable system, are all equal to one.
4 Some stability results regarding the periodic orbits of codimension-one dissipative dynamical systems This section has two main purposes, namely, the first purpose is to provide sufficient conditions to guarantee the partial orbital asymptotic stability with asymptotic phase of periodic orbits of a codimension-one dissipative dynamical system, whereas the second purpose is to give sufficient conditions to guarantee the instability of periodic orbits of a codimension-one dissipative dynamical system. Let us start by recalling some definitions and also some general results concerning the stability of the periodic orbits of a general dynamical system. In order to do that, letẋ = X(x) be a dynamical system generated by a smooth vector field X ∈ X(U), defined eventually on an open subset U ⊆ R n . Suppose Γ = {γ(t) ⊂ U : 0 ≤ t ≤ T } is a T −periodic orbit ofẋ = X(x).
• The periodic orbit Γ is called unstable if it is not orbitally stable.
• The periodic orbit Γ is called orbitally phase asymptotically stable, if it is asymptotically orbitally stable and there is a δ > 0 such that for each Let us now recall a classical result which gives some sufficient conditions to guarantee the stability/instability of a periodic orbit in terms of its characteristic multipliers. For more details regarding these results see e.g., [5], [9]. • If the characteristic multiplier one is simple (has multiplicity one) and the rest of the characteristic multipliers of the periodic orbit Γ have all of them the modulus strictly less then one, then the periodic orbit Γ is asymptotically orbitally stable with asymptotic phase.
• If there exists at least one characteristic multiplier of the periodic orbit Γ, whose modulus is strictly greater then one, then the periodic orbit Γ is unstable.
Let us now state the main result of this section, which is a generalization of the above result in the case when the characteristic multiplier one is not simple. Theorem 4.3 Letẋ = X(x) be a codimension-one dissipative dynamical system generated by a smooth vector field X ∈ X(U) defined eventually on an open subset U ⊆ R n , such that there exists k, p ∈ N with p > 0, k + p = n − 1, and respectively I 1 , . . . , I k , D 1 , . . . , D p , h 1 , . . . , h p ∈ C ∞ (U, R) such that L X I 1 = · · · = L X I k = 0, and • ∇I 1 (γ(t)), . . . , ∇I k (γ(t)), ∇D 1 (γ(t)), . . . , ∇D p (γ(t)), X(γ(t)) are linearly independent for each 0 ≤ t ≤ T .
Then, if moreover 0 ∈ R k is a regular value of the map I = (I 1 , . . . , I k ) : U ⊆ R n → R k , and if  By a classical result concerning the properties of characteristic multipliers in the presence of first integrals (see e.g., [6]) we have that, if the common level set of the first integrals I 1 , . . . , I k , containing Γ, is a smooth manifold, then the characteristic multipliers of Γ as a periodic orbit of the restriction of the vector field X to this dynamically invariant manifold are the following: 1 (due to the fact that Γ is a periodic orbit also for the restriction of X), and respectively the rest of n − k − 1 characteristic multipliers of Γ as a periodic orbit of X. Recall that Γ as a periodic orbit of X has k + 1 characteristic multipliers equal to one (k of them associated to the first integrals I 1 , . . . , I k , and one due to the fact that Γ is a periodic orbit), and respectively some other n−k −1 characteristic multipliers (possible some of them also being equal to one).
Consequently, if 0 ∈ R k is a regular value of the map I := (I 1 , . . . , I k ), then the dynamical systemẋ = X| I −1 ({0}) (x), given by the restriction of the vector field X to the dynamically invariant manifold I −1 ({0}), admits Γ as periodic orbit (by dynamical invariance), and the associated characteristic multipliers are Hence, one can apply the Theorem (4.2) for the dynamical system generated by the vector field X| I −1 ({0}) and respectively for the periodic orbit Γ, and conclude the corresponding stability/instability results. Consequently, by dynamical invariance, the same conclusions hold true also for the periodic orbit Γ associated to the original vector field X with respect to perturbations along the invariant manifold I −1 ({0}).
More precisely, if T 0 h 1 (γ(s))ds < 0, . . . , T 0 h p (γ(s))ds < 0, then the characteristic multipliers of the periodic orbit Γ of the vector field X| I −1 ({0}) have the following properties: the characteristic multiplier one is simple (its multiplicity is one), and the rest of characteristic multipliers have modulus strictly less then one, and hence the periodic orbit is orbitally phase asymptotically stable. Hence, because of dynamical invariance, the same conclusion holds in the case of the vector field X with respect to perturbations along the invariant manifold I −1 ({0}).
On the other hand (even if 0 ∈ R k it is not a regular value of the map I := (I 1 , . . . , I k )), if there exists i 0 ∈ {1, . . . , p} such that T 0 h i 0 (γ(s))ds > 0, we obtain directly from Theorem (4.2) that the periodic orbit Γ of the vector field X, it is unstable.
Let us illustrate now the results of the above theorem in the case of a three dimensional dissipative perturbation of the harmonic oscillator.
Example 4.4 Letẋ = X(x), x = (x, y, z) ∈ R 3 , be the dynamical system generated by the smooth vector field where h ∈ C ∞ (R 3 , R) is an arbitrary given smooth real function.
Note that the above defined dynamical system is a codimension-one dissipative system, associated with the following data since, L X I = 0 and L X D = hD.

Orbitally asymptotically stabilizing the periodic orbits of completely integrable dynamical systems
The purpose of this section is to apply the results from the previous section in order to partially orbitally asymptotically stabilize, a given periodic orbit of a completely integrable dynamical system. In order to do that, let us consider a completely integrable dynamical systemẋ = X(x), X ∈ X(U), defined eventually on an open subset U ⊆ R n (i.e., it admits a set of n−1 first integrals, I 1 , . . . , I k , D 1 , . . . , D p ∈ C ∞ (U, R), independent at least on an open subset V ⊆ U). Suppose that Γ = {γ(t) ⊂ V : 0 ≤ t ≤ T } is a T −periodic orbit of the systemẋ = X(x). The idea for the stabilization procedure is to perturb the completely integrable systemẋ = X(x), in such a way that the perturbed dynamical system becomes a dissipative dynamical system on V , which admits also Γ as a periodic orbit, and moreover verifies the hypothesis of Theorem (4.3). Note that using classical perturbation methods, the persistence of periodic orbits after perturbations, follows as a consequence of the implicit function theorem. The method introduced in this section, provide for the class of completely integrable dynamical system, an explicit perturbation which preserve (under reasonable conditions) an a-priori given periodic orbit.

Remark 5.2
In the hypothesis of the Theorem (5.1), note that: is a subset of the equilibrium points of the completely integrable vector field X.
Let us now illustrate the above stabilization result in the case of the harmonic oscillator. We consider this simple example in order to point out that different choices of I ′ s and D ′ s may generate different domains of definition for the perturbed vector field. Example 5.3 Let us consider the family of harmonic oscillators, described by the three dimensional vector field The induced dynamical system, admits a 2π−periodic orbit given by Γ = {γ(t) = (sin t, cos t, 0) : 0 ≤ t ≤ 2π}. Moreover, the system (5.1) is completely integrable, since it has two independent first integrals, namely In order to apply the Theorem (5.1), the candidates for the functions I and D are the first integrals I 1 and respectively I 2 . Consequently, we have two cases, namely I = I 1 and D = I 2 , and respectively I = I 2 and D = I 1 .
⋄ Let us now analyze the first case, namely I = I 1 and D = I 2 .
By straightforward computations we obtain that the vector field X 0 from Theorem (5.1), in this case has the expression X 0 (x, y, z) = zu(x, y, z)∂ z , (x, y, z) ∈ R 3 , and consequently it verifies the condition X 0 • γ = 0, for any smooth real function u ∈ C ∞ (R 3 , R).
Consequently, the perturbed systeṁ is a codimension-one dissipative dynamical system associated to I, D, u ∈ C ∞ (R 3 , R), i.e., L X+X 0 I = 0, and respectively L X+X 0 D = uD. Since X 0 • γ = 0, for any smooth real function u ∈ C ∞ (R 3 , R), we obtain that Γ is a periodic orbit of the dissipative systemẋ = X(x) + X 0 (x), for any smooth real function u ∈ C ∞ (R 3 , R).
Moreover, the rest of hypothesis of Theorem (5.1) are similar to those of Theorem (4.3), and were already been verified in Example (4.4) for the vector field X(x, y, z) + X 0 (x, y, z) = y∂ x − x∂ y + zu(x, y, z)∂ z .
⋄ ⋄ Let us now analyze the second case, namely I = I 2 and D = I 1 .
Hence, by Theorem (5.1), we obtain the following conclusions: • for any smooth function u ∈ C ∞ (V, R) such that 2π 0 u(sin t, cos t, 0)dt < 0, the periodic orbit Γ of the associated perturbed system (5.3) is orbitally phase asymptotically stable, with respect to perturbations along the plane I −1 ({0}); • for any smooth function u ∈ C ∞ (V, R) such that 2π 0 u(sin t, cos t, 0)dt > 0, the periodic orbit Γ of the associated perturbed system (5.3) is unstable.
Let us now illustrate the main result in the case of a mechanical dynamical system, namely Euler's equations from the free rigid body dynamics.