Dynamical systems with a prescribed attracting set and applications to conservative dynamics

We provide an explicit method to construct dynamical systems which admit an a-priori prescribed attracting set. As application, we provide a method to construct perturbations of conservative dynamical systems, which admit an a-priori prescribed leafwise attracting set.


Introduction
The aim of this article is to provide an explicit method to construct dynamical systems which admit an arbitrary a-priori prescribed attracting set, i.e., a closed and invariant set which attracts every bounded positive orbit of the dynamical system. As application, we give an answer to the following problem: given a conservative n−dimensional dynamical system (i.e., a dynamical system which admits a (k + p)−dimensional vector type first integral, where k + p < n) and an invariant set S (given as the level set of a k−dimensional first integral defined by some k−dimensional projection of the original (k + p)−dimensional first integral), construct a curve of dynamical systems starting from the original system, such that each system on this curve is still conservative (admitting the p−dimensional first integral which together with the k−dimensional first integral, forms the original (k + p)−dimensional first integral), keeps invariant the set S ∩ Mrk (where Mrk is the open set consisting of the points where the rank of the (k + p)−dimensional first integral is maximal), and moreover, the intersection of S ∩ Mrk with each level set (corresponding to regular values) of the p−dimensional first integral, is an attracting set of each system on the curve (except for the original system) restricted to the corresponding level set.
More precisely, in the second section of this article we recall a result from [4] which provides an explicit method to construct the class of smooth vector fields (defined on a smooth Riemannian manifold) which admit an a-priori given set of first integrals and, in the same time, dissipate a given set of scalar quantities, with a-priori defined dissipation rates. The third section represent the main part of this work, and gives an explicit method to construct a dynamical system which admits an a-priori defined attracting set. The only requirement needed in order to construct the vector field associated to a given closed subset of R n , is to know a representation of this set as a level set of some smooth function. Consequently, since any closed subset of R n can be expressed as a level set of some smooth function, this method makes possible the construction of a vector field which have as attracting set a general a-priori prescribed closed subset of R n . The fourth section gives an application of the results given in the previous section, to construct leafwise attracting sets for perturbations of conservative dynamical systems. More precisely, let S be a given dynamical system (defined on an open subset U ⊆ R n ) which admits k +p smooth first integrals, I 1 , . . . , I k , D 1 , . . . , D p (or equivalently, it admits two vector type first integrals, I := (I 1 , . . . , I k ) and D := (D 1 , . . . , D p )). Let Σ D 1 ,...,Dp d 1 ,...,dp be a dynamically invariant set of S, given by the level set of the vector type first integral D corresponding to some (regular or singular) value d := (d 1 , . . . , d p ) ∈ Im(D). Starting with these data, we construct a smooth family of dynamical systems {S λ } λ≥0 (defined on the open subset Mrk((D, I)) ⊆ U consisting of the points of maximum rank of the smooth function (D, I)), such that S 0 = S| Mrk((D,I)) , and for all λ > 0, the associated dynamical system, S λ , admits also the vector type first integral I| Mrk((D,I)) , keeps dynamically invariant the set Σ D 1 ,...,Dp d 1 ,...,dp ∩Mrk((D, I)) and moreover, the invariant set Σ D 1 ,...,Dp d 1 ,...,dp ∩(I| Mrk((D,I)) ) −1 ({µ}) (if not void) is an attracting set of S λ | (I| Mrk((D,I)) ) −1 ({µ}) , for every regular value µ ∈ Im(I| Mrk((D,I)) ). In particular, if µ is a regular value of I| Mrk((D,I)) such that the intersection of some connected components of (I| Mrk((D,I)) ) −1 ({µ}) with the invariant set Σ D 1 ,...,Dp d 1 ,...,dp ∩ Mrk((D, I)), contain a single orbit of the dynamical system S, e.g., equilibrium point, periodic orbit, homoclinic or heteroclinic cycle (if any such µ exists), then these orbits preserve their nature as orbits of the dynamical system S λ (for each λ > 0), and moreover they attract every bounded positive orbit of the dynamical system S λ | (I| Mrk((D,I)) ) −1 ({µ}) sharing the same connected component. In the case of equilibrium points, these become asymptotically stable, as equilibrium states of the dynamical system S λ | (I| Mrk((D,I)) ) −1 ({µ}) . The aim of the last section of this article is to present the correspondents of the main results of the previous section, in the case when the invariant set Σ D 1 ,...,Dp d 1 ,...,dp ∩ Mrk((D, I)) is foliated by the level sets of regular values of the vector type first integral D p ′ → | Mrk((D,I)) := (D p ′ +1 | Mrk((D,I)) , . . . , D p | Mrk((D,I)) ), where p ′ is a natural number, such that 0 < p ′ < p.

Dynamical systems with prescribed conserved and dissipated scalar quantities
In this section we recall a result from [4] which provides a constructive method to obtain the class of smooth vector fields defined on a smooth Riemannian manifold, which admits an a-priori given set of first integrals and, in the same time, dissipate a given set of scalar quantities, with a-priori defined dissipation rates.
More precisely, we have the following result, which is the key ingredient to obtain the main results of this article. Theorem 2.1 ([4]) Let (M, g) be an n-dimensional smooth Riemannian manifold, and fix k, p ∈ N two natural numbers such that p > 0 and 0 < k + p ≤ n. Let h 1 , . . . , h p ∈ C ∞ (U, R) be a given set of smooth functions defined on an open subset U ⊆ M, and respectively let I 1 , . . . , I k , D 1 , . . . , D p ∈ C ∞ (U, R) be given, such that form a set of pointwise linearly independent vector fields on U.
Then the set of solutions X ∈ X(U) of the system locally generated by the set of vector fields and respectively the set of locally defined vector fields forms a moving frame. The notation ⋆ stands for the Hodge star operator on multivector fields, and L X F := g(∇ g F, X) stands for the Lie derivative of the smooth function F ∈ C ∞ (U, R) along the vector field X.
Note that in contrast with the vector fields ∇ g I 1 , . . . , ∇ g I k , ∇ g D 1 , . . . , ∇ g D p , which are globally defined on U, the vector fields Z 1 , . . . , Z n−(k+p) exist in general only locally around each point x ∈ U, in some open neighborhood U x ⊆ U. Nevertheless, the equations (2.1) have a globally defined particular solution in U, given by the vector field X 0 . Moreover, if X is a vector field which conserves I 1 , . . . , I k , D 1 , . . . , D p (i.e., X is a solution of the homogeneous system associated to (2.1)), then X +X 0 is a global solution of (2.1). More precisely, we have the following result from [4]. Theorem 2.2 ([4]) Letẋ = X(x) be the dynamical system generated by a vector field X ∈ X(U) which conserves the smooth functions I 1 , . . . , I k , D 1 , . . . , D p ∈ C ∞ (U, R). Assume that ∇ g I 1 , . . . , ∇ g I k , ∇ g D 1 , . . . , ∇ g D p are pointwise linearly independent on some open subset V ⊆ U. Then the perturbed dynamical systeṁ with X 0 ∈ X(V ) given in Theorem (2.1), is a dissipative dynamical system, generated by the dissipative vector field X + X 0 ∈ X(V ) which conserves I 1 , . . . , I k (i.e., L X+X 0 I 1 = · · · = L X+X 0 I k = 0) and dissipates D 1 , . . . , D p with (corresponding) dissipation rates

Dynamical systems with prescribed attracting set
This section is the main part of this paper and gives an explicit method to construct a dynamical system which admits an a-priori defined attracting set. The only requirement needed in order to construct the vector field associated to a given closed subset of R n , is to know a representation of this set as the level set of some smooth function. Consequently, since any closed subset of R n can be expressed as a level set of some smooth function, this method makes possible the construction of a vector field which have as attracting set a general a-priori prescribed closed subset of R n . Let us start by fixing a natural number 1 ≤ p ≤ n, and a closed subset of R n given by Σ ..,dp is a smooth (n−p)-dimensional submanifold of R n , and hence for every x ∈ Σ D 1 ,...,Dp d 1 ,...,dp , the vectors ∇D 1 (x), . . . , ∇D p (x) are linearly independent, where ∇ stands for the gradient operator associated with respect to the canonical inner product on R n . By Sard's theorem we know that almost all points in the image of D are regular values, i.e., the set of singular values of D is a set of Lebesgue measure zero in R p . Let us denote by Mrk(D) ⊆ U the set of maximal rank points of D, i.e., the points x ∈ U such that the vectors ∇D 1 (x), . . . , ∇D p (x) are linearly independent. Recall that Mrk(D) is an open subset of U contained in the set of regular points of D. Recall that a point x 0 ∈ U is a regular point of D if there exists an open neighborhood U x 0 ⊆ U such that rank(dD(x)) = rank(dD(x 0 )), for all x ∈ U x 0 . Recall also that the set of regular points of D is an open dense subset of U in contrast with Mrk(D) which is open but not necessarily dense. The rank of dD(·) is constant on each connected component of the set of regular points of D. Concerning the set Σ D 1 ,...,Dp d 1 ,...,dp , if (d 1 , . . . , d p ) is a regular value of D, then Σ D 1 ,...,Dp d 1 ,...,dp ⊂ Mrk(D). Before stating the main theorem of this section, let us recall briefly some terminology and also some classical results we shall need in the sequel. For details see e.g., [1], [5].
In order to do that, let us consider a smooth vector field X ∈ X(U) defined on an open set U ⊆ R n . Then, for each x ∈ U we shall denote by t ∈ I x ⊆ R → x(t; x) ∈ U the integral curve of X starting from x at t = 0, i.e., the solution of the Cauchy problem dx/dt = X(x(t)), x(0) = x, defined on the maximal domain I x ⊆ R, where I x is an open interval of R containing the origin. For each x ∈ U we associate the set O + x := {y ∈ U : y = x(t; x), t ≥ 0}, called the positive orbit of x. Consequently, a subset S ⊆ U is called positively invariant if for every x ∈ S we have that O + x ⊆ S. If a set S is positively invariant, then so are the sets S andS. Let us recall that if O + x is contained in some compact subset of U, then the solution x(t; x) is defined for all t ∈ [0, ∞). If one denotes by ω( , and ω(x) = ω(x(t; x)), for all t ≥ 0. Note that for all points y ∈ ω(x), we have that O + y ⊆ ω(x), and hence the ω−limit set of x can be expressed as ω(x) = {O + y : y ∈ O + x }. Moreover, for every x ∈ U such that the set {x(t; x) : t ≥ 0} is bounded, the associated ω−limit set, ω(x), is a nonempty, invariant, compact and connected subset of U, and x(t; x) approaches ω(x) for t → ∞, i.e., x(t; x) → ω(x) for t → ∞. Recall that given a closed and invariant set S ⊂ U, we say that the solution starting from a point x ∈ U approaches the set S (and we denote x(t; x) → S), if for every ε > 0 there exists T > 0 such that dist(x(t; x), S) < ε, for all t > T , where for every point p ∈ U, dist(p, S) := inf x∈S dist (p, x). In what follows, in order to show that a solution starting from a point x ∈ U approaches some closed and invariant set S as t → ∞, we shall prove that ω(x) ⊆ S, and then using the attracting property of an ω−limit set, i.e., x(t; x) → ω(x) for t → ∞, we get that x(t; x) → S for t → ∞. Definition 3.1 Let U ⊂ R n be an open subset of R n and let X ∈ X(U) be a smooth vector field that admits at least one bounded positive orbit. A closed and invariant subset A ⊂ U will be called attracting set of the dynamical system generated by X if for every point x ∈ U such that the set {x(t; x) : t ≥ 0} is bounded, the integral curve of X starting from x approaches A as t → ∞, i.e., x(t; x) → A for t → ∞.
Next result points out an important property of attracting sets.

Remark 3.2 Let
A ⊂ U be an attracting set of the dynamical system generated by a smooth vector field X ∈ X(U). Then for every positively invariant compact set K ⊂ U (if any), and every point x ∈ K, the integral curve of X starting from x approaches A as t → ∞.
In the above settings, we shall construct a smooth vector field X defined on the open set Mrk(D) ⊆ U, whose set of equilibrium points equals Σ D 1 ,...,Dp d 1 ,...,dp ∩ Mrk(D), and moreover, this set is an attracting set of X, i.e., for every x ∈ Mrk(D), such that the set is a regular value of D = (D 1 , . . . , D p ), then Σ D 1 ,...,Dp d 1 ,...,dp ⊂ Mrk(D), and hence in this case, the vector field X admits the attracting set Σ D 1 ,...,Dp d 1 ,...,dp .
In order to do that, let us fix a strictly positive real number λ > 0. Then using the Theorem (2.1), we construct a smooth vector field X ∈ X(Mrk(D)) such that Note that by construction, Σ D 1 ,...,Dp d 1 ,...,dp ∩ Mrk(D) is a dynamically invariant set of X. By Theorem (2.1), a particular solution of the equation (3.1) with maximal domain of definition, is given by the vector field X λ 0 ∈ X(Mrk(D)) defined as follows: Let us state now the main result of this section.
Then for each real number λ > 0, one can associate a smooth vector field X λ 0 ∈ X(Mrk(D)) given by such that the following statements hold true.
(a) The set of the equilibrium states of the vector field X λ Proof. Let us define the smooth function F : Mrk(D) → [0, ∞) given by Let x ∈ Mrk(D) be given, and let t ∈ I x ⊆ R → x(t; x) ∈ Mrk(D) be the integral curve of the vector field X λ 0 ∈ X(Mrk(D)) such that x(0; x) = x, where I x ⊆ R stands for the maximal domain of definition of the solution x(·; x).
The item (a) follows directly from Theorem (3.3) and the Remark (3.2). In order to prove the statement (b) it is enough to construct a strict Lyapunov function associated to each isolated equilibrium state x e ∈ Σ D 1 ,...,Dp ..,dp ∩ Mrk(D)). In order to do that, pick an isolated point x e ∈ Σ Since by hypothesis we have that U xe ∩ (Σ D 1 ,...,Dp d 1 ,...,dp ∩ Mrk(D)) = {x e }, and the set of zeros of F in U xe is the set U xe ∩ (Σ D 1 ,...,Dp d 1 ,...,dp ∩ Mrk(D)), it follows that x e is the unique solution of the equation and consequently F is a strict Lyapunov function associated to the equilibrium point x e .
(a) The set of the equilibrium states of the vector field X λ Using the properties of ω−limit sets and the Theorem (3.5) we get the following result. ..,dp contains isolated points, then each such a point x ∈ Σ D 1 ,...,Dp d 1 ,...,dp is an asymptotically stable equilibrium point of X λ 0 .

Application to conservative dynamics
The aim of this section is to apply the main results of the above section in order to provide an answer to the following problem: given a conservative n−dimensional dynamical system (i.e., a dynamical system which admits a (k + p)−dimensional vector type first integral, where k + p < n; for a brief introduction see, e.g., [2], [3]) and an invariant set S (given as the level set of a k−dimensional first integral defined by some k−dimensional projection of the original (k + p)−dimensional first integral), construct a curve of dynamical systems starting from the original system, such that each system on this curve is still conservative (admitting the p−dimensional first integral which together with the k−dimensional first integral, forms the original (k + p)−dimensional first integral), keeps invariant the set S ∩ Mrk (where Mrk is the open set consisting of the points where the rank of the (k +p)−dimensional first integral is maximal), and moreover, the intersection of S ∩ Mrk with each level set (corresponding to regular values) of the p−dimensional first integral, is an attracting set of each system on the curve (excepting the original system).

Application to conservative dynamics with a prescribed foliated invariant set
The aim of this section is to present the correspondents of the main results of the previous section, in the case when the invariant set S ∩ Mrk((D, I)) = Σ D 1 ,...,Dp d 1 ,...,dp ∩ Mrk((D, I)) is foliated by the level sets of regular values of the vector type first integral D p ′ → | Mrk((D,I)) := (D p ′ +1 | Mrk((D,I)) , . . . , D p | Mrk((D,I)) ), where p ′ is a natural number, such that 0 < p ′ < p. More precisely, we consider a dynamical system S generated by a smooth vector field X ∈ X(U) defined on an open subset U ⊆ R n , which admits k + p smooth first integrals (1 < k + p < n, k ≥ 0), I 1 , . . . , I k , D 1 , . . . , D p ∈ C ∞ (U, R). Let S be an invariant set of S, given by the level set of the vector type first integral D corresponding to some regular (or singular) value d := (d 1 , . . . , d p ) ∈ Im(D), i.e., S = Σ D 1 ,...,Dp d 1 ,...,dp := D −1 ({(d 1 , . . . , d p )}). Let us fix some p ′ ∈ N such that 0 < p ′ ≤ p. Then using the Theorem (2.1), a particular solution of the system of equations is given by the vector field X = X λ;p ′ 0 ∈ X(Mrk((D, I))) defined by where Note that by construction, X λ;p ′ 0 ∈ X(Mrk((D, I))) admits the vector type first integrals I| Mrk((D,I)) := (I 1 | Mrk((D,I)) , . . . , I k | Mrk((D,I)) ) and dynamically invariant, as well as the set for each regular value µ of the vector type first integral I| Mrk((D,I)) , and respectively for each regular value ν of the vector type first integral D p ′ → | Mrk((D,I)) . Let us state now a theorem which points out some of the main properties of the vector field X λ;p ′ 0 ∈ X(Mrk((D, I))).
(a) The set of the equilibrium states of the vector field X λ;p ′ 0 is given by I)).