A note on an open conjecture in rational dynamical system
aa r X i v : . [ m a t h . D S ] J a n A NOTE ON AN OPEN CONJECTURE IN RATIONALDYNAMICAL SYSTEM
A Preprint
Zeraoulia Ra๏ฌk โ Email:[email protected]
January 21, 2021
Abs tract
This note is an attempt with the open conjecture proposed by the authors of[1] which states :Assume ๐ผ, ๐ฝ, ๐ โ [ , โ) . Then every positive solution of the di๏ฌerence equation : ๐ง ๐ + = ๐ผ + ๐ง ๐ ๐ฝ + ๐ง ๐ โ ๐๐ง ๐ โ , ๐ = , , . . . is bounded if and only if ๐ฝ = ๐ .We will use a construction of sub-energy function and properties of Toddโs di๏ฌerence equation todisprove that conjecture in general. K eywor ds Di๏ฌerence equation; boundedness properties ยท Toddโs equation ยท sub-energy function MSC :39A10; 39A22
A major project in the ๏ฌeld of rational di๏ฌerence equations [3] has been to determine the boundedness properties of allthird-order equations of the form : ๐ฅ ๐ + = ๐ผ + ๐ฝ๐ฅ ๐ + ๐พ๐ฅ ๐ โ + ๐ฟ๐ฅ ๐ โ ๐ด + ๐ต๐ฅ ๐ + ๐ถ๐ฅ ๐ โ + ๐ท๐ฅ ๐ โ (1)with nonnegative parameters ๐ผ, ๐ฝ, ๐พ, ๐ฟ, ๐ด, ๐ต, ๐ถ and ๐ท , one wishes to show that either the solutions remain bounded forall positive initial conditions, or there exist positive initial conditions so that the solutions are unbounded .Dynamics ofThird-Order Rational Di๏ฌerence Equations with Open Problems and Conjectures [4] treats the large class of di๏ฌerenceequations described by Equation (1),Some open problems related to (1) in which the boundedness properties were notknown was recently solved in [2], By the following assumption ๐ฟ = ๐ด = ๐ต = ๐ถ = with the variable change ๐ฅ ๐ โ ๐ฅ ๐ ๐ท ,with ๐ผ โฅ , ๐ฝ > , ๐พ > , ๐ท > equation (1) reduces to the following form : ๐ฅ ๐ + = ๐ท๐ผ + ๐ฅ ๐ ๐ฝ + ๐ฅ ๐ โ ๐พ๐ฅ ๐ โ , ๐ = , , . . . , ๐ท๐ผ = ๐ผ โฒ (2)It is shown in a paper by Lugo and Palladino [5] that there exist unbounded solutions of (2)in the case that โค ๐ผ < and < ๐ฝ < .Ying Sue Huang and Peter M. Knopf showed in [3] for ๐ผ โฒ โฅ , ๐ฝ > and if ๐ฝ โ there exist positive initialconditions such that the solutions are unbounded except for the case ๐ผ โฒ = and ๐ฝ > , Question related to Boundednessof solutions of (2) in the case ๐ฝ = ๐พ is the folllowing conjecture which it is proposed as eight open conjecture in thispaper [1] by G. LADAS, G. LUGO AND F. J. PALLADINO, In our present paper we disprove in general the only ifpart of the conjecture 8 in [1] using sub-energy function and some properties of Toddโs di๏ฌerence equation [6] and [7] โ not university de Batna.2 preprint - January 21, 2021 Assume ๐ผ, ๐ฝ, ๐ โ [ , โ) . Then every positive solution of the di๏ฌerence equation : ๐ง ๐ + = ๐ผ + ๐ง ๐ ๐ฝ + ๐ง ๐ โ ๐๐ง ๐ โ , ๐ = , , . . . (3)is bounded if and only if ๐ฝ = ๐ Suppose that ๐ฝ = ๐ > . Let ๐ฅ ๐ : = ๐ง ๐ / ๐ฝ and ๐ : = ๐ผ / ๐ฝ . Then the dynamics (3) can be rewritten as ๐ฅ ๐ + = ๐ + ๐ฅ ๐ + ๐ฅ ๐ โ ๐ฅ ๐ โ (4)(say for ๐ = , , . . . ), just with one parameter ๐ โฅ , The dynamic (4)is exactly the Toddโs di๏ฌerence equation ,In thiscase the equation is generally referred to by the cognomen โToddโs equationโ and possesses the invariant : ( ๐ + ๐ฅ ๐ + ๐ฅ ๐ โ + ๐ฅ ๐ โ ) (cid:18) + ๐ฅ ๐ (cid:19) (cid:18) + ๐ฅ ๐ โ (cid:19) (cid:18) + ๐ฅ ๐ โ (cid:19) = constant (5)The invariants of di๏ฌerence equations play an important role in understanding the stability and qualitative behaviorof their solutions. To be more precise, if the invariant is a bounded mainfold [8], then the solution is also bounded,Recently Hirota et al [9] found two conserved quantities ๐ป ๐ and ๐ป ๐ for the third- order Lyness equation , Note thatLyness equation is a special case of equation (4) such that ๐ = ,The two quantities are independents and One ofthe conserved quantities is the same form as that of (5) ,Both of two conserved quantities formula were derived fromdiscretization of an anharmonic oscillator namely using its equation of its motion see the ๏ฌrst equation here [9], wemay consider those conserved quantities as conserved sub- energy of anharmonic oscillator this means that (5) presenta sub energy function of that anharmonic oscillator , To prove the "if" part of the conjecture it would be enough toconstruct for each nonnegative ๐ , a "sub-energy" function [12] ๐ ๐ : ( , โ) โ R such that : ๐ ๐ ( ๐ฅ , ๐ฅ , ๐ฅ ) โ โ as ๐ฅ + ๐ฅ + ๐ฅ โ โ (6)Note that the sub-energy function is the invariant of the third di๏ฌerence equation ,namely the dynamics (4) , if weassume that : ๐ ๐ ( ๐ฅ ๐ , ๐ฅ ๐ โ , ๐ฅ ๐ โ ) = ( ๐ + ๐ฅ ๐ + ๐ฅ ๐ โ + ๐ฅ ๐ โ ) (cid:18) + ๐ฅ ๐ (cid:19) (cid:18) + ๐ฅ ๐ โ (cid:19) (cid:18) + ๐ฅ ๐ โ (cid:19) = constant , ๐ โฅ (7)then the condition (6) is satis๏ฌed in (7) .see Lemma2 in ([14].p.4) .For RHS of (7) see also Theorem2.1 in ([6]p.31),And Since the invariant of the dynamic of (4) is constant then ๐ ๐ could be referred to as the conservation of energyalong the path of the dynamical system.For some natural ๐ and all ๐ฅ = ( ๐ฅ , ๐ฅ , ๐ฅ ) โ ( , โ) one has the "sub-energy"inequality ๐ ๐ ( ๐ ๐ ๐ฅ ) โค ๐ ๐ ( ๐ฅ ) , where ๐๐ฅ : = ( ๐ฅ , ๐ฅ , ๐ฅ ) , with ๐ฅ = ๐ + ๐ฅ + ๐ฅ ๐ฅ , according to the dynamics. Of course, ๐ ๐ isthe ๐ th power of the operator ๐ . For ๐ = , the sub-energy inequality is the functional inequality ๐ ๐ (cid:16) ๐ฅ , ๐ฅ , ๐ + ๐ฅ + ๐ฅ ๐ฅ (cid:17) โค ๐ ๐ ( ๐ฅ , ๐ฅ , ๐ฅ ) for all positive ๐ฅ , ๐ฅ , ๐ฅ , (8)To construct a sub-energy function, one might want to start with some easy function ๐ ๐, such that ๐ ๐, ( ๐ฅ , ๐ฅ , ๐ฅ ) โ โ as ๐ฅ + ๐ฅ + ๐ฅ โ โ , and then consider something like ๐ ๐, โจ ( ๐ ๐, โฆ ๐ ๐ ) โจ ( ๐ ๐, โฆ ๐ ๐ ) โจ . . . ,Inequality (8) can beobviously restated in the following more symmetric form: ๐ฅ ๐ฅ = ๐ + ๐ฅ + ๐ฅ = โ ๐ ๐ ( ๐ฅ , ๐ฅ , ๐ฅ ) โค ๐ ๐ ( ๐ฅ , ๐ฅ , ๐ฅ ) (9)for all positive real ๐ฅ , ๐ฅ , ๐ฅ , ๐ฅ . The condition ๐ฅ + ๐ฅ + ๐ฅ โ โ in (6) can be replaced by any one of the following(stronger) conditions: (i) ๐ฅ โ โ or (ii) ๐ฅ โ โ or (iii) ๐ฅ โ โ ; this of course will replace condition (6) by aweaker condition, which will make it easier to construct a sub-energy function ๐ ๐ ,Here are details: Suppose that (8)holds for some function ๐ ๐ such that ๐ ๐ ( ๐ฅ , ๐ฅ , ๐ฅ ) โ โ as ๐ฅ โ โ . Suppose that, nonetheless, a positive sequence ( ๐ฅ , ๐ฅ , . . . ) satisfying condition (4) is unbounded, so that, as ๐ โ โ , one has ๐ฅ ๐ ๐ โ โ for some sequence ( ๐ ๐ ) ofnatural numbers. Then ๐ ๐ ( ๐ฅ ๐ ๐ , ๐ฅ + ๐ ๐ , ๐ฅ + ๐ ๐ ) โ โ as ๐ โ โ . This contradicts (6), which implies, by induction, that ๐ ๐ ( ๐ฅ ๐ , ๐ฅ + ๐ , ๐ฅ + ๐ ) โค ๐ ๐ ( ๐ฅ , ๐ฅ , ๐ฅ ) for all natural ๐ . Quite similarly one can do with (ii) ๐ฅ โ โ or (iii) ๐ฅ โ โ in placeof (i) ๐ฅ โ โ . 2 preprint - January 21, 2021 Also, instead of the dynamics of the triples ( ๐ฅ ๐ , ๐ฅ + ๐ , ๐ฅ + ๐ ) one can consider the corresponding dynamics (in ๐ ) of theconsecutive ๐ -tuples ( ๐ฅ ๐ , . . . , ๐ฅ ๐ โ + ๐ ) for any ๏ฌxed natural ๐ .Also, instead of inequality ๐ ๐ ( ๐ฅ , ๐ฅ , ๐ฅ ) โค ๐ ๐ ( ๐ฅ , ๐ฅ , ๐ฅ ) in (6), one may consider a weaker inequality like ๐ ๐ ( ๐ฅ , ๐ฅ , ๐ฅ ) โค ๐ ๐ ( ๐ฅ , ๐ฅ , ๐ฅ ) โจ ๐ ๐ ( ๐ฅ , ๐ฅ , ๐ฅ ) for all positive ๐ฅ , . . . , ๐ฅ satisfying condition (4), Thanks to the invariant of Toddโsdi๏ฌerence equation (7) which it is de๏ฌned in our case to be a sub-energy function such that it is easy to see that the "if"part of the conjecture would follow since the sub-energy ๐ ๐ is always found .In ([6], p.35) Authors showed that everypositive solution of dynamics (4) using invariant are bounded and persist this result is the a๏ฌrmation that invariantmust be a constant sub-energy function which it is always found for all positive initial conditions [15] One can try to dothe "only if" part in a similar manner. Suppose that < ๐ฝ โ ๐ > . Let ๐ข ๐ : = ๐ง ๐ /โ ๐ฝ๐ , ๐ : = ๐ผ /( ๐ฝ๐ ) , and ๐ : = p ๐ฝ / ๐ โ .Then the dynamics (4) can be rewritten as: ๐ข ๐ + = ๐ + ๐๐ข ๐ + ๐ข ๐ โ / ๐๐ข ๐ โ , (10)just with two parameters, ๐ โฅ and ๐ > . Suppose one can construct, for each pair ( ๐, ๐ ) โ [ , โ) ร (cid:0) ( , โ) \ { } (cid:1) and some ๐ = ๐ ๐,๐ โ ( , โ) , a " ๐ -super-energy" function ๐ = ๐ ๐,๐ ; ๐ : ( , โ) โ ( , โ) such that ๐ is bounded on eachbounded subset of ( , โ) and ๐ (cid:16) ๐ข , ๐ข , ๐ + ๐๐ข + ๐ข / ๐๐ข (cid:17) โฅ ๐ ๐ ( ๐ข , ๐ข , ๐ข ) for all positive ๐ข , ๐ข , ๐ข . (11)Then, by induction, ๐ ( ๐ข ๐ , ๐ข + ๐ , ๐ข + ๐ ) โฅ ๐ ๐ ๐ ( ๐ข , ๐ข , ๐ข ) โ โ as ๐ โ โ , for any sequence ( ๐ข ๐ ) satisfying (10). Thereforeand because ๐ is bounded on each bounded subset of ( , โ) , it would follow that the sequence ( ๐ข ๐ ) is unbounded.For any pair ( ๐, ๐ ) โ [ , โ) ร ( , โ)) and any ๐ โ ( , โ) , there is no " ๐ -super-energy" function ๐ : ( , โ) โ ( , โ) .This follows because the point ( ๐ข ๐,๐ , ๐ข ๐,๐ , ๐ข ๐,๐ ) with ๐ข ๐,๐ : = + ๐ + โ ๐ + ๐ ๐ + ๐ + ๐ is a ๏ฌxed point (in fact, theonly ๏ฌxed point) of the map ๐ given by the formula ๐ ( ๐ข , ๐ข , ๐ข ) = (cid:16) ๐ข , ๐ข , ๐ + ๐๐ข + ๐ข / ๐๐ข (cid:17) . (If ๐ โ , then this pointis the only ๏ฌxed point [13] of the map ๐ as well.)This also disproves, in general, the "only if" part of the conjecture de๏ฌned in (3)However, One may now try to amend this conjecture by excluding the initial point ( ๐ข ๐,๐ , ๐ข ๐,๐ , ๐ข ๐,๐ ) . Then, ac-cordingly, the de๏ฌnition of a " ๐ -super-energy" function would have it de๏ฌned on a subset (say ๐ ) of the set ( , โ) \ {( ๐ข ๐,๐ , ๐ข ๐,๐ , ๐ข ๐,๐ )} , instead of ( , โ) ; such a subset may be allowed to depend on the choice of the ini-tial point ( ๐ข , ๐ข , ๐ข ) , say on its distance from the ๏ฌxed point ( ๐ข ๐,๐ , ๐ข ๐,๐ , ๐ข ๐,๐ ) , and one would then have to also provethat ๐ is invariant under the map ๐ , M. R. S. KULENOVIC showed in ([10] .p.4 ) that the construction of lyaponovfunction is possible for third-order generalizations of Lynessโ equation ,namely Toddโs equation which it is given by : ๐ ( ๐ฅ, ๐ฆ, ๐ง ) = ๐ผ ( ๐ฅ, ๐ฆ, ๐ง ) โ ๐ผ ( ๐, ๐, ๐ ) = ๐ผ ( ๐ฅ, ๐ฆ, ๐ง ) โ ( ๐ + ) ๐ (12)Where ๐ is the equilibrum of Toddโs equation or the dynamics de๏ฌned in ( ) Consequently, ๐ is stable, Here ๐ผ ( ๐ฅ, ๐ฆ, ๐ง ) is the invariant of toddโs equation which it is de๏ฌned in (7), Assume ๐ = ๐ท ๐ = ( , โ) \ {( ๐ข ๐,๐ , ๐ข ๐,๐ , ๐ข ๐,๐ )} is theneighborhood of ๐ . Since the lyaponove function of the dynamics ( ) exists [12] and well de๏ฌned this means that allthe following three conditions are satis๏ฌed .See ([11].p.177): โข ๐ ( ๐ ) = ๐ผ ( ๐ ) โ ๐ผ ( ๐ ) = โข ๐ ( ๐ ) = ๐ผ ( ๐ ) โ ๐ผ ( ๐ ) > , for ๐ฅ โ ๐ โข ( ๐๐ ) ( ๐ ) = ๐ผ ( ๐ ( ๐ )) โ ๐ผ ( ๐ ) = ๐ผ ( ๐ ) โ ๐ผ ( ๐ ) = ๐ ( ๐ ) we use the fact that ๐ผ is an invariant implies ๐ is invariant under the map ๐ using both of conditions ) and ) Conclusion
We disproved the only if part of the titled conjecture using both of sub energy function and boundednessof Toddโs di๏ฌerence equation then the conjecture is true only for the if part but not in general ,Existence of subenergy function implies strongly energy conservation along path of dynamical system which indicate the stability andboundedness of dynamics ,Conversely the dynamics would be chaotic .
Acknowledgements
The Author thanks Iosif Pinelis from Department of Mathematical Sciences Michigan Technolog-ical University for his help to contribute the best to this paper .3 preprint - January 21, 2021
Funding
No funding supported this research .
Data Availability
The data used to support the ๏ฌndings of this study are available from the corresponding author upon request
Competing interor the ests
The author declare that they have no competing interests.
Authorsโ contributions
Author contributed to the writing of the present article. He also read and approved the ๏ฌnal manuscript
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University of Batna2.Algeria ,. 53, Route de Constantine. Fรฉsdis, Batna 05078.De-partement of mathematicsHigh school Hamla3. Ben๏ฌis Taher
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