Factorization method for some inhomogeneous Lienard equations
aa r X i v : . [ n li n . S I] J a n Factorization method for some inhomogeneous Li´enard equations
O. Cornejo-P´erez † , S. C. Mancas ⋄ , H. C. Rosu ⋆ , C. A. Rico-Olvera †∗ † Facultad de Ingenier´ıa, Universidad Aut´onoma de Queretaro,Centro Universitario Cerro de las Campanas, 76010 Santiago de Queretaro, Mexico ⋄ Department of Mathematics, Embry-Riddle Aeronautical University, Daytona Beach, FL 32114-3900, USA ⋆ IPICYT- Instituto Potosino de Investigaci´on Cient´ıfica y Tecnol´ogica,Camino a la Presa San Jos´e 2055, Col. Lomas 4a Secci´on, San Luis Potos´ı, 78216 S.L.P., Mexico (Dated: 18 January 2021)We obtain closed-form solutions of several inhomogeneous Li´enard equations by the factorizationmethod. The two factorization conditions involved in the method are turned into a system offirst-order differential equations containing the forcing term. In this way, one can find the forcingterms that lead to integrable cases. Because of the reduction of order feature of factorization,the solutions are simultaneously solutions of first-order differential equations with polynomialnonlinearities. The illustrative examples of Li´enard solutions obtained in this way generically haverational parts and consequently display singularities.
Keywords : Factorization, Inhomogeneous, Li´enard equation, Abel equation, Riccati equation
PACS numbers: 02.30.Hq; 02.30.Ik
I. INTRODUCTION
The exact solutions of nonlinear ordinary differential equations (ODEs) describe the behavior of a great variety ofphysical, chemical, biological, and engineering systems. Widespread systems in these vast areas of research can bedescribed by homogeneous Li´enard equations which have been intensively studied along the years, see, e.g., [1] andthe recent review [2]. On the other hand, the same type of inhomogeneous equations received relatively less attentiondespite the remarkable leap forward brought by the discovery of an irregular noise, later termed deterministic chaos,in the case of sinusoidally driven triode circuits by van der Pol and van der Mark in 1927 [3]. Our focus in this shortpaper is on inhomogeneous Li´enard type equations of the form¨ u + G ( u ) ˙ u + F ( u ) = I ( t ) , (1)where the dot denotes the time derivative, d/dt , G ( u ) and F ( u ) are arbitrary, but usually polynomial, functions of u ,and the forcing term I ( t ) is an arbitrary continuous function of time.The main goal of the present article is to show how the factorization method developed in [4–6] and the factorizationconditions thereof can be used to obtain some integrable inhomogeneous Li´enard equations for specific forcing terms.The key point is that the factorization method helps to reduce the inhomogeneous Li´enard equations to first-ordernonlinear equations, such as Abel and Riccati equations, which are presumably easier to solve in some cases. Werecall here that the reduction to Riccati equations of the linear Schr¨odinger equations has been extensively used insupersymmetric quantum mechanics, and in older factorization methods as reviewed in [7, 8]. II. THE NONLINEAR FACTORIZATION
As in [4–6], we consider the factorization of (1) (cid:20) ddt − f ( u ) (cid:21)(cid:20) ddt − f ( u ) (cid:21) u = I ( t ) (2) ∗ Electronic address: [email protected]; Electronic address: [email protected]; Electronic address: [email protected]; Electronicaddress: [email protected] under the conditions f + d ( f u ) du = − G ( u ) (3) f f u = F ( u ) , (4)adding the scheme proposed in [9], where one assumes [ d/dt − f ( u )] u = Ω( t ). This yields the following coupled ODEsfor (2), ˙Ω − f ( u )Ω = I ( t ) (5)˙ u − f ( u ) u = Ω( t ) , (6)which we further simplify by taking the second factorizing function as a constant, f = a ≡ const. ,˙Ω − a Ω = I ( t ) (7)˙ u − f ( u ) u = Ω( t ) . (8)In addition, using constant f , conditions (3) and (4) imply a relationship between functions F and G given by F ( u ) = − a (cid:18) c + a u + Z u G ( u ) du (cid:19) , (9)where c stands for the integration constant, or equivalently G ( u ) = − (cid:18) a dFdu + a (cid:19) . (10)Denoting I ( t ) = R t e − a t I ( t ) dt , the solution to (7) isΩ( t ) = e a t (cid:2) c + I ( t ) (cid:3) , (11)where c is an integration constant given by c = Ω(0). This allows to rewrite (8) in the form˙ u = 1 a F ( u ) + e a t (cid:2) c + I ( t ) (cid:3) , (12)whose general solution is also the solution of the Li´enard equation (1), while further particular solutions can beobtained by setting c = 0.Viceversa, one can say that (12) is a first-order nonlinear reduction of forced Li´enard equations of the form¨ u − (cid:18) a dFdu + a (cid:19) ˙ u + F ( u ) = I ( t ) . (13)Thus, integrable cases of (12) can provide Li´enard solutions in closed form. Since among the most encountered forcedLi´enard equations are those having F ( u ) in the form of cubic and quadratic polynomials, in the rest of the paper, weaddress the applications of this solution method to some cases of these types. III. THE INHOMOGENEOUS DUFFING-VAN DER POL OSCILLATOR
We choose here the particular cubic case F ( u ) = Au + Cu because it corresponds to the forced Duffing-van derPol oscillator [10] ¨ u − [( a + A/a ) + 3( C/a ) u ] ˙ u + Au + Cu = I ( t ) . (14)This equation admits the factorization (cid:20) ddt − a (cid:21)(cid:20) ddt − ( α + γu ) (cid:21) u = I ( t ) , (15)where α = A/a and γ = C/a .The corresponding first-order equation is the Abel equation˙ u = γu + αu + Ω( t ) . (16)The change of variables u = ye αt , x = γ α e αt , (17)turns (16) into the normal form dydx = y + N ( x ) , (18)with invariant N ( x ) = 1 γ e ( a − α ) t ( x ) (cid:2) c + I ( t ( x )) (cid:3) . (19)Unfortunately, this formula shows that inhomogeneous Abel equations in this category are not integrable by separationof variables because N ( x ) cannot be made constant as required by this type of integrability. Only in the force-freeparticular case I ( t ) = 0, the invariant can be reduced to the constant N = c γ . (20)By separation of variables, the solution is given by the implicit relationln " ( √N + y ) N / − √N y + y − √ − " − √N y √ = 6 N / ( x + c ) . (21)This solution has been obtained previously in [10]. IV. QUADRATIC INHOMOGENEOUS LI´ENARD EQUATIONS
If we set F ( u ) = Au + Bu , then the first order equivalent equation is the Riccati equation˙ u = βu + αu + Ω( t ) , β = B/a . (22)Equation (22) can be transformed into the normal form [11]˙ z = z + N ( t ) , (23)where z ( t ) = βu ( t ) + α , N ( t ) = β Ω( t ) − α . (24)For integrable cases of separable type, one should have N ( t ) as an arbitrary real constant that we choose p / t ) = (cid:0) p + α (cid:1) / β also a constant, as well as a constant driving force I ( t ) = − a β (cid:18) p + α (cid:19) . (25)In this very simple case we obtain a Li´enard solution of (13) of the form u ( t ) = − α β (cid:20) − pα tan (cid:16) p t + c ) (cid:17) (cid:21) . (26) A. Linear polynomial source term
After the constant driving, it is orderly to consider the source term as the linear polynomial I ( t ) = t + δ , where δ is an arbitrary constant. We set a = 1 and c = 0, and we obtain the Riccati equation˙ u = βu + αu − ( t + ˜ δ ) , ˜ δ = δ + 1 (27)with solution given by u ( t ) = − α β (cid:20) β α k Ai ′ (˜ t ) + Bi ′ (˜ t ) k Ai (˜ t ) + Bi (˜ t ) (cid:21) , (28)where ˜ t = β / [ α / β + ( t + ˜ δ ], the prime denotes the ˜ t derivative, and k is an integration constant. However, thepresence of the rational term in Airy functions turns singular such Li´enard solutions. B. Quadratic polynomial source term
Let the source term be the quadratic polynomial of type I ( t ) = a βt + ( a α − β ) t − ( a + α ). According toEqs. (11) and (22), and by setting c = 0, we have the Riccati equation˙ u = βu + αu − βt − αt + 1 . (29)This equation has the particular solution u ( t ) = t , while the general solution is given by u ( t ) = t − e t ( α + βt ) k β + e t ( α + βt ) √ β F (cid:16) α +2 βt √ β (cid:17) , (30)where F ( x ) = e − x R x e y dy is the Dawson integral, and k is an integration constant. Again, because of the rationalterm this solution is singular at − e t ( α + βt ) β − / F (cid:16) α +2 βt √ β (cid:17) = k . C. Exponential source term
For the source term of exponential form, I ( t ) = κe λt , and for c = 0, the Riccati equation is˙ u = βu + αu + κλ − a e λt . (31)The solution is given by u ( t ) = α β (cid:20) λα k Γ(1 − αλ )˜ tJ − αλ (2˜ t ) − ˜ t αλ ¯ F (˜ t ) k Γ(1 − αλ ) J − αλ (2˜ t ) + Γ(1 + αλ ) J αλ (2˜ t ) (cid:21) , (32)where k is an integration constant, ˜ t = 2 √ κβλ √ λ − a e λt/ , and ¯ F (˜ t ) is the following combination of hypergeometricfunctions ¯ F (˜ t ) = ˜ t F (cid:16) ; 2 + αλ ; − ˜ t (cid:17) + αλ F (cid:16) ; 1 + αλ ; − ˜ t (cid:17) + F (cid:16) ; αλ ; − ˜ t (cid:17) . For k = 0, we have the simpler solution u ( t ) = − α β (cid:20) λα ˜ t F (cid:0) ; 2 + αλ ; − ˜ t (cid:1) + F (cid:0) ; αλ ; − ˜ t (cid:1) F (cid:0) ; 1 + αλ ; − ˜ t (cid:1) (cid:21) . (33)The case corresponding to α = − u ( t ) = e t √ β tan hp β ( e t + k ) i . (34) D. Back to the constant source case
We return to the constant source term case since we wish to point out the interesting feature that it is more generalthan the exponential case. Indeed, let us take the source term as I ( t ) = ǫ , an arbitrary constant, and a = 1. Thisleads to the Riccati equation ˙ u = βu + αu + c e t − ǫ , (35)which is similar to the Riccati equation for the exponential case unless for ǫ . The general solution of (35) is a rationalexpression in Bessel functions given by u ( t ) = α β (cid:20) mα k ( α − m )Γ( − m ) J − m (˜ t ) − ( α + m )Γ( m ) J m (˜ t ) k Γ(1 − m ) J − m (˜ t ) + Γ(1 + m ) J m (˜ t ) + ˜ tα k Γ(1 − m ) J − m (˜ t ) + m Γ( m ) J m (˜ t )) k Γ(1 − m ) J − m (˜ t ) + Γ(1 + m ) J m (˜ t ) (cid:21) , (36)where m = p α + 4 βǫ , ˜ t = 2 √ βc e t/ , and k an integration constant. Obviously, it displays singularities at thezeros of its denominators.When k = 0, this solution takes the simpler form u ( t ) = − α β (cid:20) (cid:16) mα (cid:17) − ˜ tα J m +1 (˜ t ) J m (˜ t ) (cid:21) . (37)Notice that in the particular case of c = 0, the exponential scaling of time is annihilated and the Riccati equationis of constant coefficients having the well known regular kink solution u ( t ) = − α β (cid:20) mα tanh (cid:16) m t + k ) (cid:17) (cid:21) , (38)which is also a Li´enard kink. If in the expression for the parameter m we substitute ǫ by (25) for a = 1, we obtain m = ip , and (38) becomes the solution (26). V. CONCLUSION
The nonlinear factorization method developed in [4–6, 9] has been used to obtain closed-form solutions of certaintypes of inhomogeneous Li´enard equations. The conditions imposed upon the nonlinear coefficients of the equationsby the factorization method and the insertion of the forcing term in the factorization scheme act as designing toolsof specific forms of the forcing terms to generate integrable cases by these means. The illustrative examples havebeen chosen from the class of polynomial (up to cubic) and exponential forcing terms similarly to a recent study ofinhomogeneous Airy equations [12]. However, the obtained Li´enard solutions have rational parts which makes themprone to the presence of singularities. The only regular solutions that we have obtained by employing this simplefactorization method are the usual tanh kinks. Finally, the scheme presented here is bounded to constant factorizationfunctions f since only in this case equation (5) can be turned into the linear equation (7) in the independent variable t . [1] M. Lakshmanan and S. Rajasekar, Nonlinear Dynamics: Integrability, Chaos, and Patterns , Springer, Heidelberg 2003.[2] T. Harko and S.-D. Liang, Exact solutions of the Li´enard and generalized Li´enard type ordinary nonlinear differentialequations obtained by deforming the phase space coordinates of the linear harmonic oscillator,
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