On the fixed points set of differential systems reversibilities
OOn the fixed points set of differential systemsreversibilities
Marco Sabatini ∗ October 5, 2015
Abstract
We extend a result proved in [7] for mirror symmetries of planar sys-tems to measure-preserving non-linear reversibilities of n -dimensional sys-tems, dropping the analyticity and nondegeneracy conditions. Keywords : ODE, center, reversibility, divergence
Let us consider a planar differential system˙ z = F ( z ) , (1)where F ( z ) = ( F ( z ) , F ( z )) ∈ C (Ω , IR ), Ω ⊂ IR open and connected, z ∈ Ω.We denote by φ ( t, z ) the local flow defined by (1) in Ω. We assume the origin O to be a critical point of (1). One of the classical problems considered in thestudy of such systems is the so-called center-focus problem for a rotation point.It consists in distinguishing among the two following possibilities, • O is an attractor; • O is an accumulation point of cycles, in particular a center.Poincar´e and Lyapunov developped a procedure that applies to analytic systemswith a non-degenerate critical point O , allowing to discern among the onlypossible cases for analytic systems, i. e. a center or a focus.If the system is non-analytic or the critical point is degenerate, in order toprove that a system has a center at O one can either look for a first integralwith an isolated extremum at O , or apply a symmetry argument first used byPoincar´e, looking for a mirror symmetry of the solution set. The symmetry ∗ Dip. di Matematica, Univ. di Trento, I-38123 Povo, (TN) - Italy.Phone: ++39(0461)281670, Fax: ++39(0461)281624, Email: [email protected] -This paper was partially supported by the GNAMPA,
Gruppo Nazionale per l’Analisi Matem-atica, la Probabilit`a e le loro Applicazioni . a r X i v : . [ m a t h . D S ] O c t ethod is easy to apply if one knows the symmetry line, since after a suitableaxes rotation the symmetry conditions reduce to a parity verification on thecomponents of the vector field. On the other hand, if such a line is unknownthe question is obviously more complex, depending on a free parameter [4, 8].In [7], assuming the system to be analytic and the critical point O to be non-degenerate, it was proved that if a symmetry line exists, then it is contained inthe zero-divergence set. This allows for a fast analysis of the possible mirrorsymmetries, since they are determined by the lowest order non-linear terms of F ( z ). In [7] this was applied to study quadratic and cubic polynomial systems,also considered in [17].The existence of a center can also be proved finding a non-linear symmetry.We say that a diffeomorphism σ ∈ C (Ω , Ω) of Ω onto itself is an involution if σ ( z ) = σ ( σ ( z )) = z ∀ z ∈ Ω . An involution σ is said to be a reversibility of the system (1) if φ ( t, σ ( z )) = σ ( φ ( − t, z )) , for all t ∈ IR and z ∈ Ω such that both sides are defined. The non-lineargeneralization of the symmetry line of a mirror symmetry is the set δ consistingof the σ -fixed points, { z ∈ Ω : z = σ ( z ) } . Poincar´e’s argument can be easilyextended to reversibilities, proving that an orbit intersecting δ is a cycle. As aconsequence, if δ is a curve containing a rotation point of a reversible system,then such a point is a center. Reversibilities can be revealed by examining thesymmetry properties of the vector field. In fact, if σ ∈ C (Ω , Ω), then the σ -reversibility of (1) can be checked by verifying the relationship V ( σ ( z )) = − J σ ( z ) · V ( z ) , (2)where J σ ( z ) is the Jacobian matrix of σ at z . Finding a non-linear reversibilityis obviously much more difficult than proving mirror symmetry with respect toa line. Moreover the fixed-points set δ of a non-linear reversibility σ is not, ingeneral, a line. The existence of non-linear reversibilities in a neighbourhood ofa nilpotent critical point was studied in [16, 3, 15]. Other results on reversibility,or using reversibility can be found in [1, 5, 6, 10, 9, 11, 12, 13, 18].We say that σ is measure-preserving if µ ( σ ( A )) = µ ( A ) , where µ is the Lebesgue measure and A ⊂ Ω a measurable set. Measure-preserving diffeomorphisms appear frequently in different areas of mathematics.For instance, the flow defined by a Hamiltonian system is measure-preserving.Also, the maps involved in the celebrated Jacobian Conjecture are area-preserving[2]. In fact, they are canonical transformations of the plane onto itself. The def-initions of involution, reversibility and measure-preserving involution can begiven in the same terms in dimension n . The characterization (2) holds as wellfor n -dimensional systems. 2n this paper we prove that the fixed-points set of a measure-preserving re-versibility of a n -dimensional system is contained in the zero-divergence set ofthe vector field. We neither assume the system to be analytic, nor a criticalpoint to be non-degenerate. In fact, our proof holds even for systems withoutcritical points, hence it is also applicable to prove the existence of period annulinot contained in central regions. We apply our result to give an integrabilitycondition for a class of planar y -quadratic systems. Without changing notation, in this section we denote again by˙ z = F ( z ) , (3)a differential system in an open and connected set Ω ⊂ IR n , where F ( z ) =( F ( z ) , . . . , F n ( z )) ∈ C (Ω , IR n ), z = ( z , . . . , z n ). Similarly, we write φ ( t, z ) forthe local flow defined by (1) in Ω. Let us denote by div F ( z ) the divergence ofthe vector field F ( z ). We set∆ = { z ∈ Ω : div F ( z ) = 0 } . In [14] it was proved, for planar σ -reversible systems, that every arc of orbitcontaining both z and σ ( z ), contains a σ -fixed point, too. Such a proof extendswithout changes to n -dimensional systems. In next theorem we show that if σ is a measure-preserving reversibility, then every arc of orbit containing both z and σ ( z ), contains a zero-divergence point, too. As a consequence, we showthat a regular σ -fixed point is a zero-divergence point. Theorem 1
Let σ ∈ C (Ω , Ω) be a measure-preserving reversibility of (1).Theni) if there exist z ∈ Ω and t z ∈ IR such that σ ( z ) = φ ( t z , z ) (cid:54) = z , then the arcof orbit γ z connecting z to σ ( z ) contains a point z ∈ ∆ .;ii) if z is a regular σ -fixed point, then z ∈ ∆ .Proof. i) Possibly exchanging z and σ ( z ), we may assume t z > γ z con-necting z to σ ( z ). Then div F ( z ) has constant sign, say div F ( z ) >
0, and bythe compactness of such an arc, γ z has an open neighbourhood U such thatdiv F ( z ) > U . By the continuity of φ ( t, · ) there exists a measureable neigh-bourhood A of z such that µ ( A ) > φ ( t, A ) ⊂ U for all t ∈ [0 , t z ]. ByLiouville theorem, in a positive divergence region the measure increases alongthe local flow, hence t , t ∈ [0 , t z ] , t < t ⇒ µ ( φ ( t , A )) < µ ( φ ( t , A )) .
3n the other hand, for t ∈ (0 , t z ] one has µ ( A ) < µ ( φ ( t, A )) = µ ( σ ( φ ( t, A ))) = µ ( φ ( − t, σ ( A ))) ≤ µ ( σ ( A )) = µ ( A ) , contradiction.ii) Since F ( z ) (cid:54) = 0, there exists ε > φ ( t, z ) is a simple (i. e.injective) regular curve defined in the interval [ − ε, ε ]. Then σ ( φ ( − ε, z )) = φ ( ε, σ ( z )) = φ ( ε, z ) , The hypotheses of point i ) hold, since the orbit through z contains the distinctpoints φ ( − ε, z ) and σ ( φ ( − ε, z )) = φ ( ε, z ), hence such an arc contains a point of∆ . Similarly, every arc φ ([ − η, η ] , z ), with 0 < η < ε , contains a point of ∆ .By the continuity of div F , this implies that div F ( z ) = 0. ♣ If the orbit γ z is a cycle, then both its arc going from z to σ ( z ) (cid:54) = z , andthe second arc going from σ ( z ) to z contain a point of ∆ . In [14] it was provedthat every every cycle γ of a σ -reversible system satisfying σ ( γ ) = γ containsexactly two σ -fixed points.In general the converse to the statement of theorem 1 is not true, thereexist reversibilities with zero-divergence points that are not fixed points of thereversibility. For instance, the system (cid:26) ˙ x = y ˙ y = − x is hamiltonian, hence has identically zero divergence, and most of its regularpoints are not fixed points of the reversibility σ ( x, y ) = ( − x, y ).Next corollary’s proof is immediate. Corollary 1
Let σ ∈ C (Ω , Ω) be a mirror symmetry of (1). Then the symme-try hyperplane is contained in ∆ . y -quadraticsystems In this section we consider a class of planar y -quadratic systems, (cid:26) ˙ x = r ( x ) y + s ( x )˙ y = − g ( x ) − f ( x ) y − h ( x ) y . (4)If r ( x ) (cid:54) = 0, such a system is orbitally equivalent to a second order scalar O.D.E.Its zero-divergence set has a very simple form, since it is the graph of a one-variable function, y = α ( x ). Since a class of area-preserving involutions isprovided by the family of triangular maps, σ ( x, y ) = ( x, α ( x ) − y ) , (5)4aving the curve y = α ( x ) as fixed curve, we look for conditions under which(5) is a reversibility for (4). This suggests to perform a change of variables,taking y = α ( x ) into one axis, and check the reversibility conditions in the newvariables. It comes out that, under the hypotheses of next theorem, the newconditions are equivalent to the classical mirror symmetry of a vector field. Theorem 2
Let r, s, f, g, h ∈ C ( a, b ) , with a < < b . If for all x ∈ ( a, b ) , r ( x ) (cid:54) = 0 , r (cid:48) ( x ) − h ( x ) (cid:54) = 0 , and r ( x ) f ( x ) − h ( x ) s ( x ) − s (cid:48) ( x ) r ( x ) + s ( x ) r (cid:48) ( x ) = 0 , (6) then the system (4) is σ -reversible, with σ ( x, y ) = (cid:18) x, f ( x ) − s (cid:48) ( x ) r (cid:48) ( x ) − h ( x ) − y (cid:19) = (cid:18) x, − s ( x ) r ( x ) − y (cid:19) . Proof.
Setting F ( x, y ) = ( r ( x ) y + s ( x ) , − g ( x ) − f ( x ) y − h ( x ) y ), one hasdiv F ( x, y ) = r (cid:48) ( x ) y + s (cid:48) ( x ) − f ( x ) − h ( x ) y. One has div F ( x, y ) = 0 on the curve y = α ( x ), where α ( x ) = f ( x ) − s (cid:48) ( x ) r (cid:48) ( x ) − h ( x ) . The change of variables ( u, v ) = Λ( x, y ) = ( x, y − α ( x )) takes the system (4)into the system (cid:26) ˙ u = rv + rα + s ˙ v = − g − f α − hα − αα (cid:48) r − α (cid:48) s − ( f + 2 hα + α (cid:48) r ) v − hv , (7)omitting the dependence on u . The curve y = α ( x ) is taken by Λ into the u axis.The flow φ defined by (4) is σ -reversible if and only if the flow φ Λ definedby (7) is reversible with respect to the involution σ Λ ( u, v ) = Λ( σ (Λ − ( u, v ))) = ( u, − v ) . Hence the system (4) is σ -reversible if and only if the components of the system(7) satisfy the usual parity properties for mirror reversibility with respect to the u axis, ˙ u ( u, − v ) = − ˙ u ( u, v ) , ˙ v ( u, − v ) = ˙ v ( u, v ) . This is equivalent to the conditions (cid:26) s + rα = 0 f + 2 hα + rα (cid:48) = 0 . (8)5he condition (6) can be written as s = r ( f − s (cid:48) )2 h − r (cid:48) , hence s + rα = r ( f − s (cid:48) )2 h − r (cid:48) + r ( f − s (cid:48) ) r (cid:48) − h = 0 , which shows that the first equation in (8) is satisfied. As for the second one,from the first equation at the points where r ( x ) (cid:54) = 0 one has α = − sr . Hence one can write the second equation as follows f + 2 hα + rα (cid:48) = f − hsr − r s (cid:48) r − r (cid:48) sr = f r − hs − s (cid:48) r + r (cid:48) sr = 0 , because the numerator vanishes by (6). ♣ As a consequence of the above theorem, on has the following application tothe existence of centers. The proof is immediate.
Corollary 2
Under the hypotheses ot theorem 2, if the origin O is an isolatedcritical point of rotation type, then O is a center. The inequalities in the hypotheses of theorem 2 are not very restrictive. Infact, from the first equation of (8) follows that under the hypotheses of theorem2, r ( x ) = 0 implies s ( x ) = 0. In this case the line x = x is an invariant linefor the system (4), hence its points do not belong to a central region.Moreover, if r (cid:48) ( x ) − h ( x ) = 0 and f ( x ) − s (cid:48) ( x ) (cid:54) = 0, then the curve y = α ( x ) has a vertical asymptote at x , hence the transformation Λ cannot beextended continuously to a transformation defined also at x .If r (cid:48) ( x ) − h ( x ) ≡ I , then the system divergence in theset I × IR depends only on x , hence ∆ is a family of lines parallel to the y axis. Since ∆ contains the reversibility curve, this is not compatible with atriangular reversibility of the form σ ( x, y ) = ( x, α ( x ) − y ).Finally, we observe that under the hypotheses of theorem 2 the reversibilitycurve is the vertical isocline.We give two examples of σ -reversible systems equivalent to second orderdifferential equations. Both satisfy the hypotheses of theorem 2 and have acenter at the origin.In this example we choose r ( x ) ≡ s ( x ) = x , g ( x ) = x , f ( x ) = 2 x + 2 x , h ( x ) ≡
1. In this case one has α ( x ) = − x . (cid:26) ˙ x = y + x ˙ y = − x − xy (1 + x ) − y . (9)6igure 1: Some orbits of system (9)The local phase portrait is sketched in figure 1, where the dash-dotted curve isthe σ -fixed point curve.In next example we choose r ( x ) ≡ s ( x ) = sin x , g ( x ) = x , f ( x ) =2 sin x + 2 x cos x , h ( x ) ≡
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