Controlling the stability of periodic orbits of completely integrable systems
aa r X i v : . [ m a t h . D S ] S e p Controlling the stability of periodic orbitsof completely integrable systems
R˘azvan M. Tudoran
Abstract
We provide a constructive method designed in order to control the stabilityof a given periodic orbit of a general completely integrable system. The methodconsists of a specific type of perturbation, such that the resulting perturbed systembecomes a codimension-one dissipative dynamical system which also admits thatorbit as a periodic orbit, but whose stability can be a-priori prescribed. The mainresults are illustrated in the case of a three dimensional dissipative perturbationof the harmonic oscillator, and respectively Euler’s equations form the free rigidbody dynamics.
AMS 2000 : 37C27; 37C75; 34C25; 37J35.
Keywords : dissipative dynamics; periodic orbits; characteristic multipliers; stabil-ity.
The main purpose of this work is to provide a constructive method of controlling thestability of periodic orbits of completely integrable systems. The method consists ofa specific type of perturbation, such that the resulting perturbed system becomes acodimension-one dissipative dynamical system.The controllability procedure is based on an explicit formula for the characteristicmultipliers of a given periodic orbit of a general codimension-one dissipative dynamicalsystem. Even if this explicit formula is not the main purpose of the paper, it can beconsidered 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 ofperiodic orbits see e.g., [5], [9]).Because of the local nature of the main results, one can suppose that we work on anopen subset U ⊆ R n . More precisely, let ˙ x = X ( x ), X ∈ X ( U ), be a given codimension-one dissipative dynamical system, i.e., there exists k, p ∈ N such that k + p = n − I , . . . , I k , D , . . . , D p , h , . . . , h p ∈ C ∞ ( U, R ) such that thevector field X conserves I , . . . , I k , and dissipates D , . . . , D p with associated dissipationrates h D , . . . , h p D p . Suppose that Γ := { γ ( t ) ⊂ U : 0 ≤ t ≤ T } is a T − periodic orbit1f ˙ x = X ( x ) such that Γ ⊂ ID − ( { } ), and moreover, 0 ∈ R n − is a regular value of themap ID := ( I , . . . , I k , D , . . . , D p ) : U ⊆ R n → R n − .In the above hypothesis, the first result states that, if ∇ I ( γ ( t )) , . . . , ∇ I k ( γ ( t )) , ∇ D ( γ ( t )) , . . . , ∇ D p ( γ ( t )) , X ( γ ( t ))are linearly independent for each 0 ≤ t ≤ T , then the characteristic multipliers of theperiodic orbit Γ are, 1 , . . . , | {z } k +1 times , exp (cid:16)R T h ( γ ( s )) ds (cid:17) , . . . , exp (cid:16)R T h p ( γ ( s )) ds (cid:17) .The second result of this work is a consequence of the explicit computation of thecharacteristic multipliers of a given periodic orbit, and consists of two stability results.More precisely, if there exists i ∈ { , . . . , p } such that R T h i ( γ ( s )) ds >
0, then theperiodic orbit Γ is unstable.On the other hand (supposing that 0 ∈ R k is a regular value of the map I :=( I , . . . , I k )), if R T h ( γ ( s )) ds < . . . , R T h p ( γ ( s )) ds <
0, then the periodic orbit Γ isorbitally phase asymptotically stable, with respect to perturbations along the invariantmanifold I − ( { } ).The third result of this article, which is also the main result, provides a method topartially stabilize a given periodic orbit of a completely integrable dynamical system.More precisely, to a given completely integrable system, and respectively a given periodicorbit, we explicitly associate a dissipative dynamical system admitting the same periodicorbit, and moreover, this periodic orbit is orbitally phase asymptotically stable, relativeto a certain dynamically invariant set. Note that dissipative perturbations were alsoused 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 char-acterization of codimension-one dissipative dynamical systems, that will be used in thenext sections. The third section is dedicated to the explicit computation of the char-acteristic multipliers of a given periodic orbit of a general codimension-one dissipativedynamical system. The fourth section uses the results from the previous section, inorder to provide sufficient conditions to guarantee the partial stability, and respectivelythe instability of periodic orbits of codimension-one dissipative dynamical systems. Thelast section contains an explicit method to stabilize (relatively to a certain dynamicallyinvariant set) a given periodic orbit of a completely integrable dynamical system. In this short section we recall some results concerning the codimension-one dissipativedynamical systems. For more details regarding the characterization of general dissipativedynamical systems see e.g., [8].Recall that by a codimension-one dissipative dynamical system (defined eventually onan open subset U ⊆ R n ), we mean a dynamical system ˙ x = X ( x ), X ∈ X ( U ), for whichthere exist k, p ∈ N , with k + p = n −
1, and some smooth functions I , . . . , I k , D , . . . , D p , h , . . . , h p ∈ C ∞ ( U, R ), such that the vector field X conserves I , . . . , I k , and dissipates2 , . . . , D p , with associated dissipation rates h D , . . . , h p D p , i.e., L X I = · · · = L X I k =0, and respectively L X D = h D , . . . , L X D p = h p D p , where the notation L X standsfor the Lie derivative along the vector field X . Note that L X f = h X, ∇ f i , where f ∈ C ∞ ( U, R ) is an arbitrary smooth function, and ∇ states for the gradient operatorassociated to the standard inner product on R n , namely h· , ·i .Let us recall now a result from [8] regarding the local structure of a codimension-onedissipative dynamical system. Theorem 2.1 ([8])
Let k, p ∈ N be two natural numbers such that k + p = n − , and let I , . . . , I k , D , . . . , D p , h , . . . , h p ∈ C ∞ ( U, R ) be a given set of smooth functions definedon an open subset U ⊆ R n , such that {∇ I , . . . , ∇ I k , ∇ D , . . . , ∇ D p } ⊂ X ( U ) forms aset of pointwise linearly independent vector fields on U .Then the smooth vector fields X ∈ X ( U ) which verify simultaneously the conditions (cid:26) L X I = · · · = L X I k = 0 , L X D = h D , . . . , L X D p = h p D p , (2.1) are characterized as follows X = X + ν " ⋆ p ^ j =1 ∇ D j ∧ k ^ l =1 ∇ I l ! , where ν ∈ C ∞ ( U, R ) is an arbitrary rescaling function, X = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p ^ i =1 ∇ D i ∧ k ^ j =1 ∇ I j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) − n − · p X i =1 ( − n − i h i D i Θ i , Θ i = ⋆ " p ^ j =1 ,j = i ∇ D j ∧ k ^ l =1 ∇ I l ∧ ⋆ p ^ j =1 ∇ D j ∧ k ^ l =1 ∇ I l ! , and ” ⋆ ” stands for the Hodge star operator for multivector fields. Remark 2.2 ([8])
The vector field X is itself a solution of the system (2.1) , while thevector field ⋆ (cid:16)V pj =1 ∇ D j ∧ V kl =1 ∇ I l (cid:17) is a solution of the homogeneous system (cid:26) L X I = · · · = L X I k = 0 , L X D = · · · = L X D p = 0 . Let us now recall from [8] a consequence of the above theorem, which gives a localcharacterization of completely integrable systems.3 emark 2.3 ([8])
In the case when p = 0 (and consequently k = n − ), the dynamicalsystem generated by the vector field X will be completely integrable, and the conclusionof Theorem (2.1) becomes:The smooth vector fields X ∈ X ( U ) which verify the conditions L X I = · · · = L X I n − = 0 , are given by X = ν [ ⋆ ( ∇ I ∧ · · · ∧ ∇ I n − )] , where ν ∈ C ∞ ( U, R ) is a smooth arbitrary function. In this section we compute explicitly the characteristic multipliers of periodic orbits ofcodimension-one dissipative dynamical systems. Let us recall first that for a generaldynamical system ˙ x = X ( x ), generated by a smooth vector field X ∈ X ( U ), defined onan 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 dudt = DX ( γ ( t )) u ( t ) , u (0) = I n,n , and I n,n stands for the identity matrix of dimensions n × n . Recall that since Γ is aperiodic orbit, 1 will be always a characteristic multiplier of Γ, (see e.g., [6]). Takinginto account the complexity of the variational equation, the computation of characteristicmultipliers in general is almost impossible, since there exist no general methods to solveexplicitly the variational equation.One of the main results of this paper is to complete this task for the class ofcodimension-one dissipative dynamical systems, if one knows an explicit parameteri-zation of the periodic orbit to be analyzed. In order to do that we will use the followingresult from [4]. Theorem 3.1 ([4])
Let
Γ = { γ ( t ) ⊂ U : 0 ≤ t ≤ T } be a T − periodic orbit of adynamical system ˙ x = X ( x ) . Consider a smooth function f : U ⊆ R n → R n − , f =( f , . . . , f n − ) ⊤ , such that: • Γ is contained in T n − i =1 { f i ( x ) = 0 } , • the crossing of all the manifolds { f i ( x ) = 0 } for i ∈ { , . . . , n − } are transversalover Γ , • there exists a ( n − × ( n − matrix k ( x ) of real functions satisfying: Df ( x ) X ( x ) = k ( x ) f ( x ) . (3.1)4 et v ( t ) be the ( n − × ( n − fundamental matrix solution of dvdt = k ( γ ( t )) v ( t ) , v (0) = I n − ,n − , (3.2) where I n − ,n − stands for the identity matrix of dimensions ( n − × ( n − . Then thecharacteristic multipliers of Γ are { } ∪ σ ( v ( T )) , where σ ( v ( T )) stands for the spectrumof v ( T ) . Using the above result, we will compute explicitly the characteristic multipliers ofa given periodic orbit of a codimension-one dissipative dynamical system. Let us statenow the main result of this section.
Theorem 3.2
Let ˙ x = X ( x ) be a codimension-one dissipative dynamical system gener-ated by a smooth vector field X ∈ X ( U ) defined eventually on an open subset U ⊆ R n ,such that there exist k, p ∈ N , k + p = n − , and respectively I , . . . , I k , D , . . . , D p , h , . . . , h p ∈ C ∞ ( U, R ) such that L X I = · · · = L X I k = 0 , and L X D = h D , . . . , L X D p = h p D p . Suppose that Γ = { γ ( t ) ⊂ U : 0 ≤ t ≤ T } is a T − periodic orbit of ˙ x = X ( x ) , such that the following conditions hold true: • Γ ⊂ ID − ( { } ) , and ∈ R n − is a regular value of the map ID = ( I , . . . , I k , D , . . . , D p ) : U ⊆ R n → R n − , • ∇ I ( γ ( t )) , . . . , ∇ I k ( γ ( t )) , ∇ D ( γ ( t )) , . . . , ∇ D p ( γ ( t )) , X ( γ ( t )) are linearlyindependent for each ≤ t ≤ T .Then, the characteristic multipliers of the periodic orbit Γ are , . . . , | {z } k +1 times , exp (cid:18)Z T h ( γ ( s )) ds (cid:19) , . . . , exp (cid:18)Z T h p ( γ ( s )) ds (cid:19) . Proof.
The first two conditions from the hypothesis of Theorem (3.1) obviously implythe first two conditions of this theorem. Let us construct now a matrix ( n − × ( n − k ( x ), which verifies the equivalent of equation (3.1), namely the equation D ( ID ) ⊤ ( x ) X ( x ) = k ( x )( ID ) ⊤ ( x ) , (3.3)where by ( ID ) ⊤ we mean the transpose of the matrix ID = ( I , . . . , I k , D , . . . , D p ).If we define k = (cid:20) O k,k O k,p O p,k H p,p (cid:21) , where O k,k , O k,p , O p,k stands for the null matrices ofdimensions k × k, k × p, p × k, and H p,p = diag[ h , . . . , h p ], then the equation (3.3) canbe equivalently written as follows (cid:26) h∇ I , X i = · · · = h∇ I k , X i = 0 , h∇ D , X i = h D , . . . , h∇ D p , X i = h p D p . (3.4)5ence, the matrix k is a solution of equation (3.3), since the system (3.4) is obviouslyequivalent to (cid:26) L X I = · · · = L X I k = 0 , L X D = h D , . . . , L X D p = h p D p , 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 theequation, let us split first the ( n − × ( n −
1) matrix v as follows v = (cid:20) v k,k v k,p v p,k v p,p (cid:21) , where the blocks v k,k , v k,p , v p,k , v p,p have dimension k × k , k × p , p × k , and respectively p × p . Using the same type of splitting, the initial condition matrix v (0) = I n − ,n − splitsas follows v (0) = (cid:20) v k,k (0) v k,p (0) v p,k (0) v p,p (0) (cid:21) = (cid:20) I k,k O k,p O p,k I p,p (cid:21) , 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 (cid:20) ˙ v k,k ( t ) ˙ v k,p ( t )˙ v p,k ( t ) ˙ v p,p ( t ) (cid:21) = (cid:20) O k,k O k,p O p,k H p,p ( γ ( t )) (cid:21) · (cid:20) v k,k ( t ) v k,p ( t ) v p,k ( t ) v p,p ( t ) (cid:21) , (cid:20) v k,k (0) v k,p (0) v p,k (0) v p,p (0) (cid:21) = (cid:20) I k,k O k,p O p,k I p,p (cid:21) , or equivalently ˙ v k,k ( t ) = O k,k , ˙ v k,p ( t ) = O k,p , ˙ v p,k ( t ) = H p,p ( γ ( t )) v p,k ( t ) , ˙ v p,p ( t ) = H p,p ( γ ( t )) v p,p ( t ) ,v k,k (0) = I k,k ,v k,p (0) = O k,p ,v p,k (0) = O p,k ,v p,p (0) = I p,p , where H p,p ( γ ( t )) = diag [ h ( γ ( t )) , . . . , h p ( γ ( t ))], and t ∈ [0 , T ].Using standard ODE techniques, one obtains the unique solution v : [0 , T ] → GL ( n − , R ) ,v ( t ) = (cid:20) I k,k O k,p O p,k v p,p ( t ) (cid:21) , where v p,p ( t ) = diag (cid:20) exp (cid:18)Z t h ( γ ( s )) ds (cid:19) , . . . , exp (cid:18)Z t h p ( γ ( s )) ds (cid:19)(cid:21) . Since from Theorem (3.1), the characteristic multipliers of the periodic orbit Γ aregiven by { } ∪ σ ( v ( T )), we obtain the conclusion.Note that if p = 0, we recover a classical result concerning the characteristic multi-pliers of periodic orbits of completely integrable systems. More precisely, if p = 0, then6he dynamical system ˙ x = X ( x ) from Theorem (3.2) becomes completely integrable, andusing the conclusion of Theorem (3.2) we get that the characteristic multipliers of anyperiodic orbit of a completely integrable system, are all equal to one. This section has two main purposes, namely, the first purpose is to provide sufficientconditions to guarantee the partial orbital asymptotic stability with asymptotic phase ofperiodic orbits of a codimension-one dissipative dynamical system, whereas the secondpurpose is to give sufficient conditions to guarantee the instability of periodic orbits ofa codimension-one dissipative dynamical system.Let us start by recalling some definitions and also some general results concerningthe stability of the periodic orbits of a general dynamical system. In order to do that,let ˙ x = 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 of ˙ x = X ( x ). Definition 4.1 • The periodic orbit Γ is called orbitally stable if, given ε > there exists a δ > such that dist( x ( t, x ) , Γ) < ε for all t > and for all x ∈ U such that dist( x , Γ) < δ . • The periodic orbit Γ is called unstable if it is not orbitally stable. • The periodic orbit Γ is called orbitally asymptotically stable if it is orbitallystable and (by choosing δ smaller if necessary), dist( x ( t, x ) , Γ) → as t → ∞ . • The periodic orbit Γ is called orbitally phase asymptotically stable , if it isasymptotically orbitally stable and there is a δ > such that for each x ∈ U with dist( x , Γ) < δ , there exists θ = θ ( x ) such that lim t →∞ k x ( t, x ) − γ ( t + θ ) k = 0 . Let us now recall a classical result which gives some sufficient conditions to guaranteethe stability/instability of a periodic orbit in terms of its characteristic multipliers. Formore details regarding these results see e.g., [5], [9].
Theorem 4.2
Suppose
Γ = { γ ( t ) ⊂ U : 0 ≤ t ≤ T } is a T − periodic orbit of thedynamical system ˙ x = X ( x ) generated by a smooth vector field X ∈ X ( U ) , definedeventually on an open subset U ⊆ R n . • If the characteristic multiplier one is simple (has multiplicity one) and the rest ofthe characteristic multipliers of the periodic orbit Γ have all of them the modulusstrictly less then one, then the periodic orbit Γ is asymptotically orbitally stablewith asymptotic phase. If there exists at least one characteristic multiplier of the periodic orbit Γ , whosemodulus 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 aboveresult in the case when the characteristic multiplier one is not simple.
Theorem 4.3
Let ˙ x = X ( x ) be a codimension-one dissipative dynamical system gen-erated 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 > , k + p = n − , and respectively I , . . . , I k , D , . . . , D p , h , . . . , h p ∈ C ∞ ( U, R ) such that L X I = · · · = L X I k = 0 ,and L X D = h D , . . . , L X D p = h p D p . Suppose Γ = { γ ( t ) ⊂ U : 0 ≤ t ≤ T } is a T − periodic orbit of ˙ x = X ( x ) , such that the following conditions hold true: • Γ ⊂ ID − ( { } ) , and ∈ R n − is a regular value of the map ID = ( I , . . . , I k , D , . . . , D p ) : U ⊆ R n → R n − , • ∇ I ( γ ( t )) , . . . , ∇ I k ( γ ( t )) , ∇ D ( γ ( t )) , . . . , ∇ D p ( γ ( t )) , X ( γ ( t )) are linearlyindependent for each ≤ t ≤ T .Then, if moreover ∈ R k is a regular value of the map I = ( I , . . . , I k ) : U ⊆ R n → R k , and if Z T h ( γ ( s )) ds < , . . . , Z T h p ( γ ( s )) ds < , then the periodic orbit Γ is orbitally phase asymptotically stable, with respect to pertur-bations along the invariant manifold I − ( { } ) .On the other hand, if there exists i ∈ { , . . . , p } such that R T h i ( γ ( s )) ds > , thenthe periodic orbit Γ is unstable. Proof.
Recall from Theorem (3.2) that the characteristic multipliers of the periodicorbit Γ of the vector field X are1 , . . . , | {z } k +1 times , exp (cid:18)Z T h ( γ ( s )) ds (cid:19) , . . . , exp (cid:18)Z T h p ( γ ( s )) ds (cid:19) . By a classical result concerning the properties of characteristic multipliers in thepresence of first integrals (see e.g., [6]) we have that, if the common level set of the firstintegrals I , . . . , I k , containing Γ, is a smooth manifold, then the characteristic multipliersof Γ as a periodic orbit of the restriction of the vector field X to this dynamicallyinvariant manifold are the following: 1 (due to the fact that Γ is a periodic orbit also forthe restriction of X ), and respectively the rest of n − k − X . Recall that Γ as a periodic orbit of X has k + 1 characteristicmultipliers equal to one ( k of them associated to the first integrals I , . . . , I k , and one dueto the fact that Γ is a periodic orbit), and respectively some other n − k − ∈ R k is a regular value of the map I := ( I , . . . , I k ), then thedynamical system ˙ x = X | I − ( { } ) ( x ) , given by the restriction of the vector field X tothe dynamically invariant manifold I − ( { } ), admits Γ as periodic orbit (by dynamicalinvariance), and the associated characteristic multipliers are1 , exp (cid:18)Z T h ( γ ( s )) ds (cid:19) , . . . , exp (cid:18)Z T h p ( γ ( s )) ds (cid:19) . Hence, one can apply the Theorem (4.2) for the dynamical system generated bythe vector field X | I − ( { } ) and respectively for the periodic orbit Γ, and conclude thecorresponding stability/instability results. Consequently, by dynamical invariance, thesame conclusions hold true also for the periodic orbit Γ associated to the original vectorfield X with respect to perturbations along the invariant manifold I − ( { } ).More precisely, if R T h ( γ ( s )) ds < . . . , R T h p ( γ ( s )) ds <
0, then the characteristicmultipliers of the periodic orbit Γ of the vector field X | I − ( { } ) have the following prop-erties: the characteristic multiplier one is simple (its multiplicity is one), and the restof characteristic multipliers have modulus strictly less then one, and hence the periodicorbit 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 perturbationsalong the invariant manifold I − ( { } ).On the other hand (even if 0 ∈ R k it is not a regular value of the map I :=( I , . . . , I k )), if there exists i ∈ { , . . . , p } such that R T h i ( γ ( s )) ds >
0, we obtain di-rectly 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 dimen-sional dissipative perturbation of the harmonic oscillator. Example 4.4
Let ˙ x = X ( x ) , x = ( x, y, z ) ∈ R , be the dynamical system generated bythe smooth vector field X ( x, y, z ) = y∂ x − x∂ y + zh ( x, y, z ) ∂ z , where h ∈ C ∞ ( R , R ) is an arbitrary given smooth real function.Then, the set Γ = { γ ( t ) = (sin t, cos t,
0) : 0 ≤ t ≤ π } is a π − periodic orbit of thedynamical system ˙ x = X ( x ) .Note that the above defined dynamical system is a codimension-one dissipative system,associated with the following data • I : R → R , I ( x, y, z ) = x + y − , • D : R → R , D ( x, y, z ) = z , • h : R → R ,since, L X I = 0 and L X D = hD .The hypothesis of the Theorem (4.3) are verified since Γ ⊂ ID − ( { (0 , } ) , • (0 , is a regular value of the map ID : R → R , ID ( x, y, z ) = ( x + y − , z ) , • ∇ I (sin t, cos t, , ∇ D (sin t, cos t, , X (sin t, cos t, are linearly independentfor each t ∈ [0 , π ] , • is a regular value of the map I .Hence, by Theorem (4.3) we obtain the following conclusions: • if R π h (sin t, cos t, dt < , then the periodic orbit Γ is orbitally phase asymptoti-cally stable, with respect to perturbations along the cylinder I − ( { } ) , • if R π h (sin t, cos t, dt > , then the periodic orbit Γ is unstable. The purpose of this section is to apply the results from the previous section in orderto partially orbitally asymptotically stabilize, a given periodic orbit of a completelyintegrable dynamical system. In order to do that, let us consider a completely integrabledynamical system ˙ x = X ( x ), X ∈ X ( U ), defined eventually on an open subset U ⊆ R n (i.e., it admits a set of n − I , . . . , I k , D , . . . , D p ∈ C ∞ ( U, R ), independentat least on an open subset V ⊆ U ). Suppose that Γ = { γ ( t ) ⊂ V : 0 ≤ t ≤ T } is a T − periodic orbit of the system ˙ x = X ( x ). The idea for the stabilization procedure is toperturb the completely integrable system ˙ x = X ( x ), in such a way that the perturbeddynamical system becomes a dissipative dynamical system on V , which admits also Γ asa periodic orbit, and moreover verifies the hypothesis of Theorem (4.3). Note that usingclassical perturbation methods, the persistence of periodic orbits after perturbations,follows as a consequence of the implicit function theorem. The method introduced inthis section, provide for the class of completely integrable dynamical system, an explicitperturbation which preserve (under reasonable conditions) an a-priori given periodicorbit. Theorem 5.1
Let ˙ x = X ( x ) be a completely integrable dynamical system generated bya smooth vector field X ∈ X ( U ) defined eventually on an open subset U ⊆ R n , andlet k, p ∈ N be two natural numbers, with k + p = n − , such that there exist n − first integrals I , . . . , I k , D , . . . , D p ∈ C ∞ ( U, R ) , independent on an open subset V ⊆ U .Suppose the system ˙ x = X ( x ) admits a T − periodic orbit Γ = { γ ( t ) ⊂ V : 0 ≤ t ≤ T } such that: • Γ ⊂ ID − ( { } ) , and ∈ R n − is a regular value of the map ID = ( I , . . . , I k , D , . . . , D p ) : V ⊆ R n → R n − , ∇ I ( γ ( t )) , . . . , ∇ I k ( γ ( t )) , ∇ D ( γ ( t )) , . . . , ∇ D p ( γ ( t )) , X ( γ ( t )) are linearlyindependent for each ≤ t ≤ T .If moreover, ∈ R k is a regular value of the map I = ( I , . . . , I k ) : V ⊆ R n → R k , then for any choice of smooth functions h , . . . , h p ∈ C ∞ ( V, R ) such that Z T h ( γ ( s )) ds < , . . . , Z T h p ( γ ( s )) ds < , Γ , as a periodic orbit of the dissipative dynamical system ˙ x = X ( x ) + X ( x ) , x ∈ V , X = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p ^ i =1 ∇ D i ∧ k ^ j =1 ∇ I j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) − n − · p X i =1 ( − n − i h i D i Θ i , Θ i = ⋆ " p ^ j =1 ,j = i ∇ D j ∧ k ^ l =1 ∇ I l ∧ ⋆ p ^ j =1 ∇ D j ∧ k ^ l =1 ∇ I l ! , is orbitally phase asymptotically stable, with respect to perturbations along the invariantmanifold I − ( { } ) .On the other hand, for any choice of smooth functions k , . . . , k p ∈ C ∞ ( V, R ) , suchthat there exists i ∈ { , . . . , p } for which Z T k i ( γ ( s )) ds > , Γ , as a periodic orbit of the dissipative dynamical system ˙ x = X ( x ) + X ( x ) , x ∈ V , X = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p ^ i =1 ∇ D i ∧ k ^ j =1 ∇ I j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) − n − · p X i =1 ( − n − i k i D i Θ i , Θ i = ⋆ " p ^ j =1 ,j = i ∇ D j ∧ k ^ l =1 ∇ I l ∧ ⋆ p ^ j =1 ∇ D j ∧ k ^ l =1 ∇ I l ! , is an unstable periodic orbit. Proof.
Let us show first that the perturbed system ˙ x = X ( x ) + X ( x ), is a codimension-one dissipative dynamical system. In order to do that, will be enough to prove that thevector field X + X conserves I , . . . , I k and dissipates D , . . . , D p . For a unified notation,let us denote for each j ∈ { , . . . , p } , u j = h j , for the fist hypothesis of Theorem (5.1),and respectively u j = k j , for the second hypothesis of Theorem (5.1).Recall from Remark (2.2) that: (cid:26) L X I = · · · = L X I k = 0 , L X D = u D , . . . , L X D p = u p D p . i ∈ { , . . . , k } and respectively j ∈ { , . . . , p } , we obtain (cid:26) L X + X I i = L X I i + L X I i = 0 + 0 = 0 , L X + X D j = L X D j + L X D j = 0 + u j D j = u j D j , and consequently, ˙ x = X ( x ) + X ( x ), x ∈ V , is a codimension-one dissipative dynamicalsystem.Recall that Γ = { γ ( t ) ⊂ V : 0 ≤ t ≤ T } is a periodic orbit of the dynamical system˙ x = X ( x ) + X ( x ) too, since the hypothesis Γ ⊂ ID − ( { } ) implies that D i ◦ γ = 0, forevery i ∈ { , . . . , p } , and consequently X ( γ ( t )) = 0, for every t ∈ [0 , T ].Note that since ( X + X )( γ ( t )) = X ( γ ( t )), for every t ∈ [0 , T ], the conditionthat ∇ I ( γ ( t )) , . . . , ∇ I k ( γ ( t )) , ∇ D ( γ ( t )) , . . . , ∇ D p ( γ ( t )) , X ( γ ( t )) are linearly indepen-dent for each 0 ≤ t ≤ T , is obviously equivalent with the condition that ∇ I ( γ ( t )) , . . . , ∇ I k ( γ ( t )) , ∇ D ( γ ( t )) , . . . , ∇ D p ( γ ( t )) , ( X + X )( γ ( t ))are linearly independent for each 0 ≤ t ≤ T .Now the conclusions of Theorem (5.1) follow by applying the Theorem (4.3) for thecodimension-one dissipative dynamical system ˙ x = X ( x )+ X ( x ), x ∈ V , and respectivelythe T − periodic orbit Γ = { γ ( t ) ⊂ V : 0 ≤ t ≤ T } . Remark 5.2
In the hypothesis of the Theorem (5.1) , note that: • a consequence of the Remark (2.3) is that the set of points x ∈ U such that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p ^ i =1 ∇ D i ( x ) ∧ k ^ j =1 ∇ I j ( x ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n − = 0 , is a subset of the equilibrium points of the completely integrable vector field X . • the condition Γ ⊂ ID − ( { } ) implies that for any choice of smooth functions h , . . . , h p ∈ C ∞ ( V, R ) , the control vector field X ∈ X ( V ) , given by X = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p ^ i =1 ∇ D i ∧ k ^ j =1 ∇ I j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) − n − · p X i =1 ( − n − i h i D i Θ i , Θ i = ⋆ " p ^ j =1 ,j = i ∇ D j ∧ k ^ l =1 ∇ I l ∧ ⋆ p ^ j =1 ∇ D j ∧ k ^ l =1 ∇ I l ! , verifies that X ( γ ( t )) = 0 , for every t ∈ [0 , T ] ; • each of the smooth functions h , . . . , h p ∈ C ∞ ( V, R ) might be chosen of the typee.g., h ( x ) = − ( ψ ( x ) + c ) , x ∈ V , with ψ ∈ C ∞ ( V, R ) and c ∈ (0 , ∞ ) , since Z T h ( γ ( s )) ds = − Z T ψ ( γ ( s )) ds − T c ≤ −
T c < the smooth function k i ∈ C ∞ ( V, R ) might be chosen of type e.g., k ( x ) = φ ( x ) + c , x ∈ V , with φ ∈ C ∞ ( V, R ) and c ∈ (0 , ∞ ) , since Z T k ( γ ( s )) ds = Z T φ ( γ ( s )) ds + T c ≥ T c > . Let us now illustrate the above stabilization result in the case of the harmonic oscil-lator. 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 threedimensional vector field X ( x, y, z ) = y∂ x − x∂ y ∈ X ( R ) . The induced dynamical system, ˙ x = X ( x ) , x = ( x, y, z ) ∈ R , (5.1) admits a π − periodic orbit given by Γ = { γ ( t ) = (sin t, cos t,
0) : 0 ≤ t ≤ π } .Moreover, the system (5.1) is completely integrable, since it has two independent firstintegrals, namely I ( x, y, z ) = x + y − , I ( x, y, z ) = z. In order to apply the Theorem (5.1) , the candidates for the functions I and D arethe first integrals I and respectively I . Consequently, we have two cases, namely I = I and D = I , and respectively I = I and D = I . ⋄ Let us now analyze the first case, namely I = I and D = I . By straightforward computations we obtain that the vector field X from Theorem (5.1) , in this case has the expression X ( x, y, z ) = zu ( x, y, z ) ∂ z , ( x, y, z ) ∈ R , and consequently it verifies the condition X ◦ γ = 0 , for any smooth real function u ∈ C ∞ ( R , R ) .Consequently, the perturbed system ˙ x = X ( x ) + X ( x ) , x = ( x, y, z ) ∈ R , (5.2) is a codimension-one dissipative dynamical system associated to I, D, u ∈ C ∞ ( R , R ) ,i.e., L X + X I = 0 , and respectively L X + X D = uD .Since X ◦ γ = 0 , for any smooth real function u ∈ C ∞ ( R , R ) , we obtain that Γ is aperiodic orbit of the dissipative system ˙ x = X ( x ) + X ( x ) , for any smooth real function u ∈ C ∞ ( R , 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 ( x, y, z ) = y∂ x − x∂ y + zu ( x, y, z ) ∂ z . Hence, by Theorem (5.1) , we obtain the following conclusions: for any smooth function u ∈ C ∞ ( R , R ) such that R π u (sin t, cos t, dt < , theperiodic orbit Γ of the associated perturbed system (5.2) is orbitally phase asymp-totically stable, with respect to perturbations along the cylinder I − ( { } ) ; • for any smooth function u ∈ C ∞ ( R , R ) such that R π u (sin t, cos t, dt > , theperiodic orbit Γ of the associated perturbed system (5.2) is unstable. ⋄ ⋄ Let us now analyze the second case, namely I = I and D = I . By straightforward computations we obtain that the vector field X from Theorem (5.1) , in this case has the expression X ( x, y, z ) = u ( x, y, z )( x + y − x + y ) ( x∂ x + y∂ y ) , ( x, y, z ) ∈ V := R \{ (0 , , z ) : z ∈ R } , and consequently it verifies the condition X ◦ γ = 0 , for any smooth real function u ∈ C ∞ ( V, R ) .Consequently, the perturbed system ˙ x = X ( x ) + X ( x ) , x = ( x, y, z ) ∈ V, (5.3) is a codimension-one dissipative dynamical system associated to I, D, u , i.e., L X + X I =0 , and respectively L X + X D = uD .Since X ◦ γ = 0 , for any smooth real function u ∈ C ∞ ( V, R ) , we obtain that Γ is aperiodic orbit of the dissipative system ˙ x = X ( x ) + X ( x ) , for any smooth real function u ∈ C ∞ ( V, R ) .Moreover, the rest of hypothesis of Theorem (5.1) are obviously verified, since theyare similar with those from the previous case. Note that in this case, the perturbed vectorfield is given by X ( x, y, z ) + X ( x, y, z ) = (cid:20) u ( x, y, z ) x ( x + y − x + y ) + y (cid:21) ∂ x + (cid:20) u ( x, y, z ) y ( x + y − x + y ) − x (cid:21) ∂ y . Hence, by Theorem (5.1) , we obtain the following conclusions: • for any smooth function u ∈ C ∞ ( V, R ) such that R π u (sin t, cos t, dt < , theperiodic orbit Γ of the associated perturbed system (5.3) is orbitally phase asymp-totically stable, with respect to perturbations along the plane I − ( { } ) ; • for any smooth function u ∈ C ∞ ( V, R ) such that R π u (sin t, cos t, dt > , theperiodic 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.14 xample 5.4
Let us recall that Euler’s equations in terms of the rigid body angularmomenta are generated by the vector field X ( x, y, z ) = (cid:18) I − I (cid:19) yz∂ x + (cid:18) I − I (cid:19) zx∂ y + (cid:18) I − I (cid:19) xy∂ z ∈ X ( R ) , where I , I , I are the moments of inertia in the principal axis frame of the rigid body.We will suppose in the following that I > I > I > .The induced dynamical system, ˙ x = X ( x ) , x = ( x, y, z ) ∈ R , (5.4) is completely integrable, since it has two independent first integrals, namely F ( x, y, z ) = 12 (cid:18) x I + y I + z I (cid:19) , F ( x, y, z ) = 12 (cid:0) x + y + z (cid:1) . Recall that there exists an open and dense subset of the image of the map ( F , F ) : R → R , such that each fiber of any element ( h, c ) from this set, corresponds to periodicorbits of Euler’s equations. Moreover, any such element is a regular value of ( F , F ) ,as well as its components for the corresponding maps, F and respectively F . For moredetails see, e.g., [3].Let ( h, c ) ∈ R belongs to the above mention set, and let us denote J ( x, y, z ) := F ( x, y, z ) − h, J ( x, y, z ) := F ( x, y, z ) − c. In the above conditions, the set
Γ = ( γ ( t ) = ( γ ( t ) , γ ( t ) , γ ( t )) : 0 ≤ t ≤ K √ I I I p I − I )( hI − c ) ) , where γ ( t ) = s I ( c − hI ) I − I · cn s I − I )( hI − c ) I I I · t ; k ,γ ( t ) = s I ( c − hI ) I − I · sn s I − I )( hI − c ) I I I · t ; k ,γ ( t ) = − s I ( − c + hI ) I − I · dn s I − I )( hI − c ) I I I · t ; k ,k = s ( hI − c )( I − I )( hI − c )( I − I ) ,K = Z dt p (1 − t )(1 − k t ) , s a K √ I I I p I − I )( hI − c ) − periodic orbit of the dynamical system (5.4) , which belongsto the common zero level set of the first integrals J and J . See for details, e.g., [3].Recall that the above parameterization of Γ is given in terms of Jacobi elliptic functions.By straightforward computations we get that ∇ J ( x ) , ∇ J ( x ) , X ( x ) , are linearly de-pendent vectors if and only if x is an equilibrium point of the dynamical system (5.4) .Hence, the vectors ∇ J ( γ ( t )) , ∇ J ( γ ( t )) , X ( γ ( t )) , are linearly independent for each t ∈ " , K √ I I I p I − I )( hI − c ) . In order to apply the Theorem (5.1) , the candidates for the functions I and D are thefirst integrals J and respectively J . Consequently, we have two cases, namely I = J and D = J , and respectively I = J and D = J . ⋄ Let us now analyze the first case, namely I = J and D = J . By straightforward computations we obtain that the vector field X from Theorem (5.1) , in this case has the expression X ( x, y, z ) = u ( x, y, z ) (cid:20)
12 ( x + y + z ) − c (cid:21)(cid:20) xy (cid:18) I − I (cid:19)(cid:21) + (cid:20) yz (cid:18) I − I (cid:19)(cid:21) + (cid:20) xz (cid:18) I − I (cid:19)(cid:21) · { x (cid:20) I (cid:18) I − I (cid:19) y + 1 I (cid:18) I − I (cid:19) z (cid:21) ∂ x + y (cid:20) I (cid:18) I − I (cid:19) x + 1 I (cid:18) I − I (cid:19) z (cid:21) ∂ y + z (cid:20) I (cid:18) I − I (cid:19) x + 1 I (cid:18) I − I (cid:19) y (cid:21) ∂ z } , ( x, y, z ) ∈ V, where V := R \ {{ ( x, ,
0) : x ∈ R } ∪ { (0 , y,
0) : y ∈ R } ∪ { (0 , , z ) : z ∈ R }} (note that V is exactly the complement of the set of equilibrium states of Euler’s equations).Recall that the vector field X verifies the condition X ◦ γ = 0 , for any smooth realfunction u ∈ C ∞ ( V, R ) , since Γ belongs to the zero level set of the first integral J .Consequently, the perturbed system ˙ x = X ( x ) + X ( x ) , x = ( x, y, z ) ∈ R , (5.5) is a codimension-one dissipative dynamical system associated to I, D, u , i.e., L X + X I =0 , and respectively L X + X D = uD .Since X ◦ γ = 0 , for any smooth real function u ∈ C ∞ ( V, R ) , we obtain that Γ is aperiodic orbit of the dissipative system ˙ x = X ( x ) + X ( x ) , for any smooth real function u ∈ C ∞ ( V, R ) .Since all the hypothesis of Theorem (5.1) are verified, we obtain the following con-clusions. If one denotes T := 4 K √ I I I p I − I )( hI − c ) , then for any smooth function u ∈ C ∞ ( V, R ) such that R T u ( γ ( t ) , γ ( t ) , γ ( t )) dt < , theperiodic orbit Γ of the associated perturbed system (5.5) is orbitally phase asymp-totically stable, with respect to perturbations along the ellipsoid I − ( { } ) = (cid:26) ( x, y, z ) : 12 (cid:18) x I + y I + z I (cid:19) − h = 0 (cid:27) ; • for any smooth function u ∈ C ∞ ( V, R ) such that R T u ( γ ( t ) , γ ( t ) , γ ( t )) dt > , theperiodic orbit Γ of the associated perturbed system (5.5) is unstable. ⋄ ⋄ Let us now analyze the second case, namely I = J and D = J . By straightforward computations we obtain that the vector field X from Theorem (5.1) , in this case has the expression X ( x, y, z ) = u ( x, y, z ) (cid:20) (cid:18) x I + y I + z I (cid:19) − h (cid:21)(cid:20) xy (cid:18) I − I (cid:19)(cid:21) + (cid:20) yz (cid:18) I − I (cid:19)(cid:21) + (cid:20) xz (cid:18) I − I (cid:19)(cid:21) · { x (cid:20)(cid:18) I − I (cid:19) y + (cid:18) I − I (cid:19) z (cid:21) ∂ x + y (cid:20)(cid:18) I − I (cid:19) x + (cid:18) I − I (cid:19) z (cid:21) ∂ y + z (cid:20)(cid:18) I − I (cid:19) x + (cid:18) I − I (cid:19) y (cid:21) ∂ z } , ( x, y, z ) ∈ V, where V := R \ {{ ( x, ,
0) : x ∈ R } ∪ { (0 , y,
0) : y ∈ R } ∪ { (0 , , z ) : z ∈ R }} (note that V is exactly the complement of the set of equilibrium states of Euler’s equations).Recall that the vector field X verifies the condition X ◦ γ = 0 , for any smooth realfunction u ∈ C ∞ ( V, R ) , since Γ belongs to the zero level set of the first integral J .Consequently, the perturbed system ˙ x = X ( x ) + X ( x ) , x = ( x, y, z ) ∈ R , (5.6) is a codimension-one dissipative dynamical system associated to I, D, u , i.e., L X + X I =0 , and respectively L X + X D = uD .Since X ◦ γ = 0 , for any smooth real function u ∈ C ∞ ( V, R ) , we obtain that Γ is aperiodic orbit of the dissipative system ˙ x = X ( x ) + X ( x ) , for any smooth real function u ∈ C ∞ ( V, R ) .Since all the hypothesis of Theorem (5.1) are verified, we obtain the following con-clusions. If one denotes T := 4 K √ I I I p I − I )( hI − c ) , then • for any smooth function u ∈ C ∞ ( V, R ) such that R T u ( γ ( t ) , γ ( t ) , γ ( t )) dt < , theperiodic orbit Γ of the associated perturbed system (5.6) is orbitally phase asymp-totically stable, with respect to perturbations along the sphere I − ( { } ) = (cid:26) ( x, y, z ) : 12 (cid:0) x + y + z (cid:1) − c = 0 (cid:27) ;17 for any smooth function u ∈ C ∞ ( V, R ) such that R T u ( γ ( t ) , γ ( t ) , γ ( t )) dt > , theperiodic orbit Γ of the associated perturbed system (5.6) is unstable. Acknowledgment
This work was supported by a grant of the Romanian National Authority for ScientificResearch, CNCS-UEFISCDI, project number PN-II-RU-TE-2011-3-0103.
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R.M. Tudoran
West University of Timi¸soaraFaculty of Mathematics and Computer ScienceDepartment of MathematicsBlvd. Vasile Pˆarvan, No. 4 1800223 - Timi¸soara, Romˆania.E-mail: [email protected]@math.uvt.ro