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 Raο¬k β 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 πΌ, π½, π β [ , β) . Then every positive solution of the diο¬erence equation : π§ π + = πΌ + π§ π π½ + π§ π β ππ§ π β , π = , , . . . is bounded if and only if π½ = π .We will use a construction of sub-energy function and properties of Toddβs diο¬erence equation todisprove that conjecture in general. K eywor ds Diο¬erence equation; boundedness properties Β· Toddβs equation Β· sub-energy function MSC :39A10; 39A22
A major project in the ο¬eld of rational diο¬erence equations [3] has been to determine the boundedness properties of allthird-order equations of the form : π₯ π + = πΌ + π½π₯ π + πΎπ₯ π β + πΏπ₯ π β π΄ + π΅π₯ π + πΆπ₯ π β + π·π₯ π β (1)with nonnegative parameters πΌ, π½, πΎ, πΏ, π΄, π΅, πΆ and π· , 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 Diο¬erence Equations with Open Problems and Conjectures [4] treats the large class of diο¬erenceequations 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 πΏ = π΄ = π΅ = πΆ = with the variable change π₯ π β π₯ π π· ,with πΌ β₯ , π½ > , πΎ > , π· > equation (1) reduces to the following form : π₯ π + = π·πΌ + π₯ π π½ + π₯ π β πΎπ₯ π β , π = , , . . . , π·πΌ = πΌ β² (2)It is shown in a paper by Lugo and Palladino [5] that there exist unbounded solutions of (2)in the case that β€ πΌ < and < π½ < .Ying Sue Huang and Peter M. Knopf showed in [3] for πΌ β² β₯ , π½ > and if π½ β there exist positive initialconditions such that the solutions are unbounded except for the case πΌ β² = and π½ > , Question related to Boundednessof solutions of (2) in the case π½ = πΎ 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 Toddβs diο¬erence equation [6] and [7] β not university de Batna.2 preprint - January 21, 2021 Assume πΌ, π½, π β [ , β) . Then every positive solution of the diο¬erence equation : π§ π + = πΌ + π§ π π½ + π§ π β ππ§ π β , π = , , . . . (3)is bounded if and only if π½ = π Suppose that π½ = π > . Let π₯ π : = π§ π / π½ and π : = πΌ / π½ . Then the dynamics (3) can be rewritten as π₯ π + = π + π₯ π + π₯ π β π₯ π β (4)(say for π = , , . . . ), just with one parameter π β₯ , The dynamic (4)is exactly the Toddβs diο¬erence equation ,In thiscase the equation is generally referred to by the cognomen βToddβs equationβ and possesses the invariant : ( π + π₯ π + π₯ π β + π₯ π β ) (cid:18) + π₯ π (cid:19) (cid:18) + π₯ π β (cid:19) (cid:18) + π₯ π β (cid:19) = constant (5)The invariants of diο¬erence 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 π» π and π» π for the third- order Lyness equation , Note thatLyness equation is a special case of equation (4) such that π = ,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 ο¬rst 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 π , a "sub-energy" function [12] π π : ( , β) β R such that : π π ( π₯ , π₯ , π₯ ) β β as π₯ + π₯ + π₯ β β (6)Note that the sub-energy function is the invariant of the third diο¬erence equation ,namely the dynamics (4) , if weassume that : π π ( π₯ π , π₯ π β , π₯ π β ) = ( π + π₯ π + π₯ π β + π₯ π β ) (cid:18) + π₯ π (cid:19) (cid:18) + π₯ π β (cid:19) (cid:18) + π₯ π β (cid:19) = constant , π β₯ (7)then the condition (6) is satisο¬ed 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 π π could be referred to as the conservation of energyalong the path of the dynamical system.For some natural π and all π₯ = ( π₯ , π₯ , π₯ ) β ( , β) one has the "sub-energy"inequality π π ( π π π₯ ) β€ π π ( π₯ ) , where ππ₯ : = ( π₯ , π₯ , π₯ ) , with π₯ = π + π₯ + π₯ π₯ , according to the dynamics. Of course, π π isthe π th power of the operator π . For π = , the sub-energy inequality is the functional inequality π π (cid:16) π₯ , π₯ , π + π₯ + π₯ π₯ (cid:17) β€ π π ( π₯ , π₯ , π₯ ) for all positive π₯ , π₯ , π₯ , (8)To construct a sub-energy function, one might want to start with some easy function π π, such that π π, ( π₯ , π₯ , π₯ ) β β as π₯ + π₯ + π₯ β β , and then consider something like π π, β¨ ( π π, β¦ π π ) β¨ ( π π, β¦ π π ) β¨ . . . ,Inequality (8) can beobviously restated in the following more symmetric form: π₯ π₯ = π + π₯ + π₯ = β π π ( π₯ , π₯ , π₯ ) β€ π π ( π₯ , π₯ , π₯ ) (9)for all positive real π₯ , π₯ , π₯ , π₯ . The condition π₯ + π₯ + π₯ β β in (6) can be replaced by any one of the following(stronger) conditions: (i) π₯ β β or (ii) π₯ β β or (iii) π₯ β β ; this of course will replace condition (6) by aweaker condition, which will make it easier to construct a sub-energy function π π ,Here are details: Suppose that (8)holds for some function π π such that π π ( π₯ , π₯ , π₯ ) β β as π₯ β β . Suppose that, nonetheless, a positive sequence ( π₯ , π₯ , . . . ) satisfying condition (4) is unbounded, so that, as π β β , one has π₯ π π β β for some sequence ( π π ) ofnatural numbers. Then π π ( π₯ π π , π₯ + π π , π₯ + π π ) β β as π β β . This contradicts (6), which implies, by induction, that π π ( π₯ π , π₯ + π , π₯ + π ) β€ π π ( π₯ , π₯ , π₯ ) for all natural π . Quite similarly one can do with (ii) π₯ β β or (iii) π₯ β β in placeof (i) π₯ β β . 2 preprint - January 21, 2021 Also, instead of the dynamics of the triples ( π₯ π , π₯ + π , π₯ + π ) one can consider the corresponding dynamics (in π ) of theconsecutive π -tuples ( π₯ π , . . . , π₯ π β + π ) for any ο¬xed natural π .Also, instead of inequality π π ( π₯ , π₯ , π₯ ) β€ π π ( π₯ , π₯ , π₯ ) in (6), one may consider a weaker inequality like π π ( π₯ , π₯ , π₯ ) β€ π π ( π₯ , π₯ , π₯ ) β¨ π π ( π₯ , π₯ , π₯ ) for all positive π₯ , . . . , π₯ satisfying condition (4), Thanks to the invariant of Toddβsdiο¬erence equation (7) which it is deο¬ned 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 π π 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 aο¬rmation 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 < π½ β π > . Let π’ π : = π§ π /β π½π , π : = πΌ /( π½π ) , and π : = p π½ / π β .Then the dynamics (4) can be rewritten as: π’ π + = π + ππ’ π + π’ π β / ππ’ π β , (10)just with two parameters, π β₯ and π > . Suppose one can construct, for each pair ( π, π ) β [ , β) Γ (cid:0) ( , β) \ { } (cid:1) and some π = π π,π β ( , β) , a " π -super-energy" function π = π π,π ; π : ( , β) β ( , β) such that π is bounded on eachbounded subset of ( , β) and π (cid:16) π’ , π’ , π + ππ’ + π’ / ππ’ (cid:17) β₯ π π ( π’ , π’ , π’ ) for all positive π’ , π’ , π’ . (11)Then, by induction, π ( π’ π , π’ + π , π’ + π ) β₯ π π π ( π’ , π’ , π’ ) β β as π β β , for any sequence ( π’ π ) satisfying (10). Thereforeand because π is bounded on each bounded subset of ( , β) , it would follow that the sequence ( π’ π ) is unbounded.For any pair ( π, π ) β [ , β) Γ ( , β)) and any π β ( , β) , there is no " π -super-energy" function π : ( , β) β ( , β) .This follows because the point ( π’ π,π , π’ π,π , π’ π,π ) with π’ π,π : = + π + β π + π π + π + π is a ο¬xed point (in fact, theonly ο¬xed point) of the map π given by the formula π ( π’ , π’ , π’ ) = (cid:16) π’ , π’ , π + ππ’ + π’ / ππ’ (cid:17) . (If π β , then this pointis the only ο¬xed point [13] of the map π as well.)This also disproves, in general, the "only if" part of the conjecture deο¬ned in (3)However, One may now try to amend this conjecture by excluding the initial point ( π’ π,π , π’ π,π , π’ π,π ) . Then, ac-cordingly, the deο¬nition of a " π -super-energy" function would have it deο¬ned on a subset (say π ) of the set ( , β) \ {( π’ π,π , π’ π,π , π’ π,π )} , instead of ( , β) ; such a subset may be allowed to depend on the choice of the ini-tial point ( π’ , π’ , π’ ) , say on its distance from the ο¬xed point ( π’ π,π , π’ π,π , π’ π,π ) , and one would then have to also provethat π is invariant under the map π , M. R. S. KULENOVIC showed in ([10] .p.4 ) that the construction of lyaponovfunction is possible for third-order generalizations of Lynessβ equation ,namely Toddβs equation which it is given by : π ( π₯, π¦, π§ ) = πΌ ( π₯, π¦, π§ ) β πΌ ( π, π, π ) = πΌ ( π₯, π¦, π§ ) β ( π + ) π (12)Where π is the equilibrum of Toddβs equation or the dynamics deο¬ned in ( ) Consequently, π is stable, Here πΌ ( π₯, π¦, π§ ) is the invariant of toddβs equation which it is deο¬ned in (7), Assume π = π· π = ( , β) \ {( π’ π,π , π’ π,π , π’ π,π )} is theneighborhood of π . Since the lyaponove function of the dynamics ( ) exists [12] and well deο¬ned this means that allthe following three conditions are satisο¬ed .See ([11].p.177): β’ π ( π ) = πΌ ( π ) β πΌ ( π ) = β’ π ( π ) = πΌ ( π ) β πΌ ( π ) > , for π₯ β π β’ ( ππ ) ( π ) = πΌ ( π ( π )) β πΌ ( π ) = πΌ ( π ) β πΌ ( π ) = π ( π ) we use the fact that πΌ is an invariant implies π is invariant under the map π using both of conditions ) and ) Conclusion
We disproved the only if part of the titled conjecture using both of sub energy function and boundednessof Toddβs diο¬erence 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 ο¬ndings of this study are available from the corresponding author upon request
Competing interor the ests
The author declare that they have no competing interests.
Authorsβ contributions
Author contributed to the writing of the present article. He also read and approved the ο¬nal manuscript
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University of Batna2.Algeria ,. 53, Route de Constantine. FΓ©sdis, Batna 05078.De-partement of mathematicsHigh school Hamla3. Benο¬is Taher
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