Joural of Mathematical Scieces (4) -5 Betty Joes & Sisters Publishig Period Fuctio of a Lieard Equatio Khalil T Al-Dosary Departmet of Mathematics, Uiversity of Sharjah, Sharjah 77, Uited Arab Emirates Abstract: We study the period fuctio of the quadratic Lieard equatio of a certai type i order to give ecessary ad sufficiet coditios for mootoicity ad isochroicty of the period fuctio We apply this result to idetify the regio of mootoicity of the period fuctio of particular cases Key words: Cete isochroous cete Lieard equatio, mootoicity, period fuctio troductio this paper we study the mootoicity property of the period fuctio of the quadratic Lieard equatio of the type + f( ) + g ( ) + h ( ) = () We shall look for a appropriate equivalet differetial system such that the required computatios ca be actually performed This equivalet differetial system is of the form = y a ( ) by ( ) () y = c ( ) ay ( ) by ( ) where a, b, ad c are fuctios of class C defied o a ope eighborhood of the origi A sigular poit of system () is called a ceter if there is a deleted eighborhood of the poit which cosists etirely of closed trajectories surroudig that poit We say that equatio () has a ceter if ay oe of the equivalet systems, ad the all, has a ceter f system () has a ceter at the origi O, we call U the largest ope coected regio covered with cycles surroudig O Defie a fuctio P : U R, by associatig to every (, y) U the miimal period of the cycle passig through (, y) P is called the period fuctio of O Correspodig author: Khalil T Al-Dosary, PhD, professo research fields: scietific computig, mathematical aalysis, cotrol, optimizatio E-mail: dosary@sharjahacae The period fuctio has bee etesively studied by umber of differet authors [-6] Let N be a ivariat coected subset of U We say that P is icreasig (strictly icreasig) i N if, for every couple of cyclesδ, δ N, with δ cotaied i the iterior of δ, we have P( δ) P( δ ) ( P ( δ) P( δ ) ) We say that O is a isochroous ceter if P is costat i a eighborhood of O The geeral approach of the article has bee itroduced before by M Sabatii [7] i his work with the case f ( =, accordigly with a differet plaar system Sice we are applyig Theorem of M Sabatii [7] i our work, ad for the sake of competece, we state that Theorem here Theorem of [7]: Let () have a ceter at O Assume that there eist a star-shaped set R such that ω( r, for all (, the, P ) f there eists a zero-measure set Z [,π ) such that, for all θ [, π ) \ Z, the fuctio r ω( is icreasig (decreasig) i (, r ( ), the P is decreasig (icreasig) i N ; ) f poit ) holds, ad for every orbit δ i a eighborhood V of O there eists a poit (r δ, θ δ ) δ such that r ω( θδ ) is strictly icreasig (strictly decreasig) at r δ, the P is strictly decreasig (strictly icreasig) i N ;
Period Fuctio of a Lieard Equatio ) f there eists a zero-measure set Z [, π ), such that, for all θ [, π ) \ Z, the r ω( costat i (, r ( ), the P is costat i N ad C where (, R) is defied as follows: δ (, F( e Q ( ) a ( ) = g (), = b ( ) = β ( F(, e f (), = ( ) = ( ), F ( s) β ( ) =, F g s ds e ds γ () = (), F( s) sf s e ds F( = e by itegratig by parts β ( =, Ω ( = s) b( s) ds b( F( s) adδ () = sg() s Ω() s e ds The followig Lemmas are stated without proofs sice the details of the proofs are log We skip it, ad it ca be set o a request Lemma f ad if f C (, R, the b is cotiuous, f C (, the f () f () b C (, b () = 6 Lemma f ad, if f, g C (, R, the a is cotiuous, g C R f C (, ),, the g () a C (, a () = Lemma Let f, gh, C ( R, ), the the fuctio, c ( ) = Q ( )[ h ( ) a( Q ) ( )] is cotiuous, ad if, is,,, the f C g C h C c C ad c() = h() =, c () = h () Lemma 4 Q ( ) >, for every R Utilizig these Lemmas, oe ca prove the et Lemma Sice Lemma 5 f f, g, h C (, h() =, the system () is of class C i a eighborhood of the origi ad equivalet to the equatio () The Mai Results order to state the first theorem, we defie the followig fuctio σ as: σ = + 4 4 ( h ( ) h( f( h ) ( ) [ bh ( ) ( ) Theorem 4 4 g( a ) ( )] Q ( ) 4 a( Q ) ( + f f, g, h C (, g() = h() = Let the origi be a ceter of () f c( ) > for, ad: ) ( ( σ ( ) σ for decreasig (icreasig) iu ; ) ( i U, the P is σ, the P is costat i Proof The agular speed of () has the form c( y + y Sice c ( > for, the the agular speed is egative i polar coordiates, for almost π π all values of,,π amely θ,, the θ [ ) agular speed is cosθc( r cosθ ) θ = ω( r, = si θ r Hece ω r cos θc ( r cosθ ) cosθc( r cosθ ) = r
Period Fuctio of a Lieard Equatio c ( c( = ( + y ) This yield c ( c( = σ ( Therefore ω = σ ( Sice ω( r, for ( r cosθ, r siθ ), the ω( = ω(, so ( ω( = σ ( ( + y ) From Lemma 4, we have Q ( ) > for all, hece ω( if σ ( the, ad cosequetly, the fuctio r ω( is icreasig The applyig Theorem of [7] completes the proof Theorem f the origi of system () is cete ad f, g, h C (, g() = h() =, c ( > for, ad σ ( ( σ ( ) for, ad there eists a sequece, with σ ( ) ( σ ( ) > ), the P is strictly decreasig (strictly icreasig) iu Proof Let Λ be a cycle of the system () cotaied iw The Λ meets the lie = at some poits, y ), correspodig to r, i polar ( ( coordiates Sice σ ( ) ( σ ( ) > ), we have, from the detailed computatio i the proof of Theorem ω( that > ( ) at ( r, θ ) The applyig Theorem of [7] completes the proof Corollary f the origi of system () is cete ad f, g, h C (, g() = h() =, ad h ( ) > The the statemet of the Theorem holds i a suitable subiterval of Remark With some computatio oe ca fid 5 τ ( σ ( = [ ] 4 where 4 τ ( ) = a ( Q ( h( + h () Corollary Let f, g, h C (, g() = h() =, h ( ) >, τ (, the the origi is the uique sigular poit of system () Defie a fuctio µ ( as, h( a ( Q ( + g (), µ ( = = Lemma 6 ) For, µ ( = h ( c( ) f f, g, h C (, the µ ( C (, R) Theorem Let the origi be a cete f, g, h be odd aalytic fuctios o ( η, η) for some η > c ( > for, the, ad ) P is strictly decreasig (strictly icreasig) at the origi if ad oly of τ ( has a proper maimum (Proper miimum) at the origi; ) The origi is a isochroous ceter if ad oly if τ ( i a eighborhood of the origi Proof 5 We have σ ( ) = µ (, ad Q ( > for all, theσ ( >, if ad oly if µ ( > O the 4 other had, we have τ ( ) = µ (, the τ ( > if ad oly if ( > µ fact, σ µ, are eve aalytic fuctios, the the origi is either proper miimum of σ, or it is proper maimum, or is idetically zero By Theorem, P is strictly
4 Period Fuctio of a Lieard Equatio icreasig, strictly decreasig, or costat, respectively Vice versa, if P is strictly icreasig, the σ caot be costat, otherwise, by Theorem, the ceter would be isochroous Moreove if P is strictly icreasig, the σ caot have a proper maimum at, that would imply P to be strictly decreasig by Theorem Hece, if P is strictly icreasig, the σ has a proper miimum The other cases ca be treated similarly Therefore, P is strictly icreasig at (strictly decreasig at, costat), if ad oly if, σ ( has a proper miimum at (has a proper maimum at, costat) Now we shall show that σ has a proper maimum at (has a proper miimum at, costat) if ad oly if τ has a proper maimum at (has a proper miimum at, costat) as follows: Sice τ is eve, aalytic ad τ ( ) = µ () =, the there are oly three possibilities for τ ca occur i a eighborhood of : A proper maimum at, a proper miimum at, or idetically zero f is a proper miimum ofσ, the σ ( ) = µ ( > for small, that is, also τ has a proper miimum at Similarly we ca prove that if is a proper maimum ofσ, ad the is proper maimum ofτ f σ (, the µ (, hece µ (, ad τ ( Vice-versa, if τ has a proper miimum at, the by what above, σ caot have a proper maimum, or be costat, hece it has a proper miimum at The other cases follow similarly The proof completes Corollary Let the origi be a cete f, g, h be odd aalytic fuctios o ( η, η) for some η >, ad c ( > for, the ) P is strictly decreasig (strictly icreasig) i N if ad oly of σ ( has a maimum at the origi; ) P is costat i N, if ad oly ifσ ( Now we cosider the equatio () with liear restorig term, + f ( + g( + = () This equatio with the case f (, has bee cosidered i [8], meas the Lieard equatio with liear restorig term, + g( + = (4) where its mootoicity at the origi is proved by computig the costat period fuctio [7], the author provides a estimate of the regio of mootoicity of P for the equatio (4) The et Corollary provides a estimate of the regio of mootoicity of P but with f (, meas the equatio () Corollary 4 Let the origi be a ceter of () f f, g, are aalytic, g ( ) the P is strictly icreasig i N, where ( α, β), α if { : b( a ( } β = sup { : b( a ( > } Eample = = >, The followig is a eample ehibitig a applicatio of the work for equatios with liear restorig term of a Rayleigh equatio type of the form + + m + = where, m costat to be determet ad accordigly fid out the iterval cotaiig the origi o which the period fuctio P is icreasig Cosider the sladered system = y, y = y my This system has a uique sigular poit (, ) Echagig variables ad multiplyig the vector field by -, the system becomes
Period Fuctio of a Lieard Equatio 5 = y + + m, y = This is a system of the type of equatio () with h( m =, f (, g( = As a cosequece of eigevalue aalysis, the origi is a ceter if Now, we apply Corollary 4 Sice f( ) =, the F( Therefore b ( ) =, Q ( ) = Hece a( = m 4 As a cosequece of Corollary 4, P is strictly icreasig i N, = ( α, β ) α = > 4 if : m β = > 4 sup : m >, 4 But ( m ), is equivalet 4 to + m The 4 4 m m So, if m, the there is o iterval o which P is icreasig This meas that the system has o icreasig period fuctio f m >, the P is icreasig o the set of cycles cotaied i the vertical strip ( y, ) defied by the iequality 4, m >, m Ackowledgmet The author would like to epress his thak to the Uiversity of Sharjah for its valued support Refereces [] J Car SN Chow, JK Hale, Abelia itegrals ad bifurcatio theory, Differetial Equatios 59 () () -56 [] SN Chow, JA Saders, O the umber of critical poits of the period, Differetial Equatios 64 (986) 5-66 [] SN Chow, D Wag, O the mootoicity of the period fuctio of some secod order equatios, Casopis Pest Math(986) 4-5 [4] WS Loud, Period solutio of + c + g( = εf (, Mem Amer Math Soc (959) -57 [5] J Smolle Wasserma, Global bifurcatio of steady-state solutios, Differetial Equatios 9 (98) 69-9 [6] D Wag, O the eistece of π -periodic solutios of differetial equatio + g( = p(, Chiese A Math A (5) (984) 6-7 [7] M Sabatii, O the period fuctio of Lieard systems, Differetial Equatios, 5 (999) 467-487 [8] A Cima, A Gasull, V Maosa, F Maosas, Algebraic properties of the liapuov ad period costat, Rocky Mout Jour Math 7 (997) 47-5