( ) ( t) ( 0) ( ) dw w. = = β. Then the solution of (1.1) is easily found to. wt = t+ t. We generalize this to the following nonlinear differential

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1 Periodic oluion of van der Pol differenial equaion. by A. Arimoo Deparmen of Mahemaic Muahi Iniue of Technology Tokyo Japan in Seminar a Kiami Iniue of Technology January 8 9. Inroducion Le u conider a imple differenial equaion for w w dw (.) w + w= w = d dw w = d wih iniial condiion w( ) α w = : = = β. Then he oluion of (.) i eaily found o be αco βin w = +. We generalize hi o he following nonlinear differenial equaion which i well known a van der Pol equaion: w w w + w= (.) ε where < ε <. If we ake (.3) W = AW Γ W w = w we can wrie (.) a he following way: where A = ε Γ=Γ =. ww ( ). ( ) The marix A ha wo eigen value r = ε + i and r = ε i ( ha a pecral repreenaion A = rp+ rp I = P+ P PP = PP =. Hence from he relaion rp = A rp = A r I P = ε ) and we have

2 P = A r r r P = ( r A) r r Uing hee we will be able o wrie he exponenial funcion of A :. { } e = e P + e P = e e A + re re r r A r r r r r r We can ge r r e e ε in = e r r So we have (.4) and r r re re ε in = e co ε r r. A in in e e ε A co ε = + ε in = e U Σ where in co U U( ) = = and Σ= in co ε.. Lemma.. We have (.5) UU = U + (.6) U = I (.7) in ΣU Σ= U Σ where U U ( ) in co = = and in co Σ= ε. I i ell known ha a oluion of (.3) aifie A (.8) ( ) A W = e W e Γ d Hence uing (.4) and (.7) we have (.9) ε in e W U W = Σ ( ) in e U Σ d w( ) w ( )

3 =ΣU ΣW e ΣU Σ d. w ( ) w ( ) Theorem.. I =Σ Σ + ε I (.) e W( ) U ( ) W S C ( ) ( ) where Proof in IS ( ) = e w ( ) w ( ) d = co I e w w d C Since U Σ = U we have Σ = I and ww ( ) ww ( ) I e ΣU Σ d =Σ U e U d =ΣU w ( ) w ( ) w( ) w ( ) I c. Hence we have from (.9) he deired reul (.). ( ) Theorem.3. If e W( ) lim hen here exi lim lim c co I I e w w d C = = in I( ) = IS ( ) = e w ( ) w ( ) d and (.) Σ W + ε = where Σ= ε ε Proof : From (.) we have e U ( ) W( ) W ( ) ( ) I S Σ =Σ + ε 3

4 and leing in he la equaion we have (.). Theorem.4 W( ) W( ) if and only if + = for ome > (.) ε e U = e U I ΣW Proof: Subiuing + ino of (.) ( ) ( ) e W( ) =ΣU Σ W + ε I C we have ( + ) ( + ) ε + e W( + ) =ΣU Σ W + ε I C.. Hence W( ) W( ) (.3) + = i equivalen o e U Σ W + ε =Σ W + ε IC. which lead o (.). Remark: W = W( ) if and only if W W or (.3) i auonomou. A a reul we have =. Thi i becaue our yem (.) Theorem.5 The following are equivalen W W (.4) = for ome > ε ( e U ) W (.5) Σ = ε Proof Taking = in (.) we have (.5). Hence we ee ha (.5) i equivalen o W = W. Hence we have (.4) by he propery of auonomou in he yem. 4

5 Theorem.6 The following are equivalen W W (.6) = for ome > (.7) ( + ) ( + ) ( ) ( ) e U = Proof (.6) (.7): Since from (.6) we have W = W( ) we have ε e U = ( e U I) ΣW which i for = ε = ( e U I) Σ W. Hence we have (.7). (.7) (.6) Puing in (.7) we have ε( I e U ) = ε. From (.) we have ε = Σ W o we have ε ( I e U ) W Σ = ε. The la formula i equivalen o (.6).. Properie of he periodic oluion of van der Pol differenial equaion If w ( + ε ) = w ( ε) wih he period = ( ε) of rajecory ( ) we have from Theorem.5 (.) ε ( e U ) W Σ = ε ha i wε hen 5

6 (.) w( ε )( e co ) + ( w( ε ) ε w ( ε )) in ( ) = ε e w ε w ε d (.3) e in w ε e in + w ε ε w ε e co ( ) = ε e co w ε w ε d Dividing boh ide of hee equaion by ε we have (.4) e co e in w( ε) + ( w( ε) ε w ( ε) ) ε ε in = e ( w ( ε) w ( ε) ) d (.5) w e in ( ( ) ( )) e ε w w co + ε ε ε ε ε ( ) = e co w ε w ε d Since we have known he rajecorie wε ( ) on he limi cycle are bounded uniformly in by aking ε in he equaion (.) we would have (.6) lim W( ε) = U limw ( ε) ε ε U co in = in co ha i he ame: (.7) ε ( ε) = = α + β lim w w co in lim w ε = w = α in + β co ε where α = lim w( ε) and β lim w( ε) ε ε limi of he period ( ε ) of he limi cycle i ( ε) equaion (.4) and (.5) we would have =. I could be conidered o how ha he lim = π. By aking ε in he ε 6

7 (.8) π ( ) πα + β + in w w d = which i π + + =. 4 (.9) πα β α ( α β ) Alo (.) π ( ) βπ+ α + co w w d = which i π = 4 (.) βπ α β ( α β ) where we have hown he exience of in ε = lim which i equivalen o he ε ε differeniabiliy of ε. When we olve he imulaneou equaion (.9) and (.) we have = and α + =. Hence we have w ( ε) w ( ε) β 4 for each by he propery of auonomou yem. ε { } lim + = 4 3. Anoher approach Le u conider one more van der Pol equaion x x x + x= (3.) µ where µ > dx x = d d x x = d inegraing hem we have ( ) x x x x d + = µ. Muliplying x boh ide of (3.) and from which if we apply an iniial condiion α = x β = x ( ) x β + x α = µ x x d. 7 hen we have

8 Le be he period of he limi cycle of hi yem. Then we have aking ( ) = x x d. A in he previou ecion if we ake µ we will be having π and x( ) x ( ) = α co+ β in. So we mu have = π ( ) = x x d. ( ( α β ) )( α β ) π d. = co + in in + co π π ( α in + β co ) = ( α in + β co ) ( α co + β in ) d From which we again obain µ { x( ) x( ) } α + = ha i we have β 4 lim + = 4 by arbirarily chooing he iniial ime inead of =. 8