Exact solutions to cubic duffing equation for a nonlinear electrical circuit Álvaro H. Salas S. * Jairo E. Castillo H. ** subitted date: Noveber 3 received date: Deceber 3 accepted date: April 4 Abstract: This work provides an exact solution to a cubic Duffing oscillator equation with initial conditions and bounded periodic solutions. This solution is expressed in ters of the Jacobi elliptic function (cn). This exact solution is used as a seed to give a good analytic approxiate solution to a nonlinear equation that describes a nonlinear electrical circuit. This last equation is solved nuerically and copared with the analytic solution obtained fro solving the cubic Duffing equation. It is suggested that the ethodology used herein ay be useful in the study of other nonlinear probles described by differential equations of the for z = F( z), F( z ) being an odd function in the sense that ( ) z ay be approxiated by an appropriate solution to a cubic Duffing oscillator equation. In particular, the exact solution ay be applied in the study of the cubic nonlinear Schrodinger equation, which is reduced to a cubic Duffing oscillator equation by eans of a travelling wave transforation. Key words Duffing equation, cubic Duffing oscillator equation, third-order Duffing odel, analytic solution, Jacobi elliptic functions. * B.Sc. In Matheatics, State University of Moldova (URSS). M.Sc. In Matheatics, Universidad Nacional de Colobia (Colobia). Research group FIZMAKO, Universidad de Caldas (Colobia). Current position: professor at Universidad de Caldas and Universidad Nacional de Colobia, Manizales-Nubia (Colobia). E-ail: asalash@yahoo.co ** B.Sc. In Physics and M.Sc. In Physics, Universidad rusa de la aistad de los pueblos (URSS). Research group FIZMAKO, Universidad Distrital Francisco José de Caldas, Bogotá (Colobia). Current position: professor at Universidad Distrital Francisco José de Caldas, Bogotá (Colobia). E-ail: jcastillo@udistrital.edu.co 46 Universidad Distrital Francisco José de Caldas - Technological Faculty
exact solutions to cubic duffing equation for a nonlinear electrical circuit Introduction The cubic Duffing equation is a differential equation with third-power nonlinearity. There are any probles in physics and engineering that lead to the nonlinear differential cubic Duffing equation: fro the oscillations of a siple pendulu, including nonlinear electrical circuits, to various applications in iage processing [,,3]. The search for new ethods that lead to the solution of the Duffing equation of different orders: third, fifth, seventh, and higher orders, is of vital iportance since these solutions ay be applied to a nonlinear cubic Schrodinger equation which has various applications in nonlinear optics, plasa physics, fluid echanics and Bose-Einstein condensates [4,5]. By using the gauge transforation, the nonlinear cubic Schodinger equation ay be reduced to a nonlinear ordinary differential cubic Duffing oscillator equation. This work is aied at finding, by eans of the Jacobi elliptic functions, the exact periodic solution of the differential cubic Duffing equation with initial conditions that describe the behaviour of a nonlinear electrical circuit.. Nonlinear electrical circuit Let us consider a capacitor of two terinals as a dipole in which a functional relationship between the electric charge, the voltage and the tie has the following for: f(, qut,) = () A nonlinear capacitor is said to be controlled by charge when it is possible to express the tension as a function of charge: u uq ( ) () As an exaple of nonlinear electrical circuit, let us consider the circuit shown in Figure. Figure. LC circuit This circuit consists of a linear inductor in series with a nonlinear capacitor. The relationship between the charge of the nonlinear capacitor and the voltage drop across it ay be approxiated by the following cubic equations [6,7,8] : u = sq + aq c 3, (3) where u c is the potential across the plates of the nonlinear capacitor, q is the charge, and s and a are constants. The equation of the circuit ay be written as di 3 L sq aq, + + = (4) where L is the inductance of the coil. Dividing dq by L and considering that i =, we obtain the cubic Duffing equation in the for dq 3 αq βq, + + = (5) being α = s / L = / LC and β = a / L = / LCq constants. A RESEARCH VISION Electronic Vision - year 7 nuber pp. 46-53 january -june of 4 47
Álvaro H. Salas S. - Jairo e. castillo H.. A brief introduction to Jacobi elliptic functions cn, sn and dn It is worth reebering that this work provides an exact solution to equation (5) in ters of the Jacobi elliptic function cn. This function is defined as follows: dθ cn( t, ) = cos ϕ, where t=. sin θ ϕ (6) There are other two iportant elliptic Jacobi functions, naely sn and dn, which are defined by: sn( t, ) = sin ϕ and dn( tk, ) = sin ϕ. (7) sn (, t) + cn (, t) =, dn (, t) = sn (, t) (8) lisn( t, ) = sin t, li cn( t, ) = cos t, li dn( t, ) =. (9) lisn( t, ) = tanh t, li cn( t, ) = sec h t, li dn( t, ) = sec h t. () These functions are derivable and so: d sn( t, ) = cn( t, )dn( t, ), The nuber ( < < ) is called elliptic odulus and the nuber ϕ is called Jacobi aplitude, and it is denoted by a (t, ). Thus, ϕ = a( t, ), sin( ϕ) = sin(a( t, )) = sn( t, ) d cn( t, ) = sn( t, )dn( t, ), d dn( t, ) = sn( t, )cn( t, ). () For these equations, the following identities hold: The graph of functions sn and cn are shown in Figure for =. 4 Figure. Graph of sn=sn ( t,/ 4) and cn=cn ( t,/ 4) on the interval t 8. 48 Universidad Distrital Francisco José de Caldas - Technological Faculty
exact solutions to cubic duffing equation for a nonlinear electrical circuit Fro Figure, it can be observed that functions sn and cn are periodic. They have a coon period equal to 4 K(/4) = 4 K ( ), where K= K ( ) is called the elliptic K function for odulus. In our case, K(/ 4).5964. Equations (9) indicate that functions sn and cn generalize the sine and cosine functions, respectively. Jacobi elliptic functions cn( t, ), sn( t, ) and dn( t, ) ay also be calculated for coplex values of t = u+ v as follows : dn( u, )cn( v, ')dn( v, ') dn( u+ v, ) = dn ( u, ) sn ( v, ') sn( u, )cn( u, )sn( v, ') dn ( u, ) sn ( v, ') dn( v, ') dn( v, ) = = cn( v, ') dc( v, '), ' =. < < (4) A RESEARCH VISION sn( u, )dn( v, ') sn( u + v, ) = + dn ( u, ) sn ( v, ') When >, these functions are defined by the following equations : cn( u, )dn( u, )sn( v, ')cn( v, ') dn ( u, ) sn ( v, ') cn( t, )=dn( t, ) sn( t, )= sn( t, ) sn( v, ') sn( v, ) = = cn( v, ') sn( v, '), ' =. < < () cn( u, )cn( v, ') cn( u + v, ) = dn ( u, ) sn ( v, ') dn( t, )=cn( t, ) > The following is also defined: cn( t, )= dn( t, ) nd( t, ) = < (5) (6) sn( u, )dn( u, )sn( v, ') dn( v, ') dn ( u, ) sn ( v, ') cn( v, ) = = cn( v, ') nc( v, '), ' =. < < (3) In soe cases, it is necessary to deal with a function of the for cn ( wt, k ), where w is any nuber (real or coplex) and k is a pure iaginary nuber, say k =, where is a real nuber. We ay define the expression cn( wt, k) = cn( wt, ) ) in a consistent way by using the forula: Electronic Vision - year 7 nuber pp. 46-53 january -june of 4 49
Álvaro H. Salas S. - Jairo e. castillo H. wt = cn( +, ) cn( wt, )= dn( + wt, ) for any constants c and c. Therefore the general solution to equation (5) is obtained by solving the syste cd( wt, ), where + = + (7) ( ) α = ω. ω β =. c () In the special case when both w and k are pure iaginary nubers, say w = ω and k = (ω and are real nubers), it is possible to define cn( wtk, ) = cn( ω t, ) as follows: which yields: c β ω = α + c β and =. ( β) α + c () cn( t, )=nd( t, ω + ω ) = + dn( + ω, ) + t 4. Exact solutions for the nonlinear odel (8) The values of c and c are deterined fro the initial conditions q() = q and dq i = t= = i. If q '() =, then c = q and c =. Thus, the solution to proble: dq 3 + αq+ β q =, q() = q, q () =. () Is In the linear case, the general solution to equation y () t + α yt () = is yt ( ) = ccos( αt+ c), where c and c are the constants of integration which are deterined fro the initial conditions y() = y and y () = y. When a cubic ter is added we obtain the nonlinear equation 3 y () t + αyt () + βy() t = and the solution cannot be expressed in ters of the cosine function. In this case, the Jacobi cn funtion solves this nonlinear equation. Indeed, direct calculations using equations () show that function y= c cn ( ω t+ c, ) satisfies the following differential equation ω y () t + ω yt () + y() t = 3 ( ) c (9) q qt ( ) = β qcn( α + qβt, ), α + qβ. ( α + q β) β q ( α+ βq ) (3) If =, then α β and since q li cn( ω t, ) = sech( ω t) the solution to (3) results in: qt ( ) = q sech( αt) (4) It is worth noting that solution (9) is bounded and non-periodic. This situation is typical for solitons. Soe iportant non-linear differential equations adit soliton solutions that can be expressed in ters of sech. 5 Universidad Distrital Francisco José de Caldas - Technological Faculty
exact solutions to cubic duffing equation for a nonlinear electrical circuit q On the other hand, if = β then ( α+ βq ) β and because li cn ( ωt, ) = cos( ωt) (3) reduces to: This function is periodic with period T.894. The corresponding graph in the interval [, T ] is shown in Figure 4. A RESEARCH VISION qt ( ) = q cos( αt) (5) Which is the solution to linear equation y () t + α yt () =. This result is consistent with the theory already known for the linear case. Figure 4. Graph of qt ( ) =.cd 9,.73..5 ( 3.36647 t ) The behaviour of solution (6) depends on paraeters a and b as well as on the initial condition q. In this paper we will consider five cases:.5.5..5..5 3. 3.5 First Case: α + q β > and β >. Let α =, β = and q =. Forula (3) yields qt ( ) =.cn(.499 t,.9999995). This function is periodic with period T 6.83. The corresponding graph in the interval [, T ] is shown in Figure 3. Figure 3. Graph of qt ( ) =.cn(.499 t,.9999995).. Third Case : α + q β < and β <. Let α =, β = 6 and q =. By coputing forulas (8) and (3), qt ( ) =. nd(.4435t,.73 ). This function is unbounded and periodic with period T 6.49. The corresponding graph in the interval [,4] is shown in Figure 5..5 3 4 5 6 Figure 5. Graph of qt ( ) =. 5,.73 nd(.443 t ).5. Second Case : α + q β > and β <. Let α =, β = 33 and q =. By coputing forulas (7) and (3), qt ( ) =.cd 9,.73 ( 3.36647 t ).3.. 3 4...3.4 Electronic Vision - year 7 nuber pp. 46-53 january -june of 4 5
Álvaro H. Salas S. - Jairo e. castillo H. Fourth Case: α + q β < and β >. Let α =, β = 3 and q =. By coputing forulas (8) and (3), qt ( ) =.nd(.4435 t,.5495 ). This function is bounded and periodic with period T.46. The corresponding graph in the interval [, 4] is shown in Figure 6. Figure 6. Graph of qt ( ) =.nd(.4435 t,.5495 ) For L =,8H, C = 9 pf, qo = C, i =, we 3 33 have that α = / LC = 4 and β = / LCq = 4. The nuerical solution for the Runge - Kutta ethod and the analytical solution we obtain fro (3), i.e. 6 q= qt ( ) = cn(8.9447,.77678865475) are copared (graphically) in Figure 7. Figure 7. Nuerical solution (dashed) vs analytical solution (solid).4.3.. 3 4 Fifth Case : α + q β =. In this case it is not possible to apply forula (3). Equation () then reduces to dq q q q 3 q q q β + β =, () =, () = Direct calculations show that function q= qt ( ) = q sech( q β ( t+ C)), (6) where C= arcsech( ) (7) q β Finally, let us consider a nonlinear circuit consisting of a linear inductor in series with a nonlinear capacitor, as shown in Figure. 5. Conclusions A way to study non-linear circuits by eans of the cubic Duffing differential equation has been shown, obtaining an exact solution in ters of a Jacobi elliptic function cn. It was possible to establish conditions under which the solution is periodic and haronic. This is very iportant in the study of electric resonators and their applications in engineering. On the other hand, the results (already know) for the linear case were generalized for the nonlinear case in a consistent and natural way. It ay be stated that the Jacobi elliptic functions are an essential way to solve any nonlinear probles. Solutions (4) and (7) turned out to be very interesting because they represent a type of soliton solution (for a ore thorough study of these solutions see [9]). 5 Universidad Distrital Francisco José de Caldas - Technological Faculty
exact solutions to cubic duffing equation for a nonlinear electrical circuit References [] a.n. Nayfeth, D.T. Mook, Non-linear oscillations. New York: John Wiley,973. [] a.i. Maiistov, Propagation of an ultiately short electroagnetic pulse in a nonlinear ediu described by the fifthorder Duffing odel, Opt. Spectros, 94(), 3, pp.5-57 [3] S. Morfa, J.C. Cote, "A nonlinear oscilators netwok devoted to iage processing", International Journal of Bifurcation and Chaos, vol. 4, No. 4 pp. 385-394, 9. [4] Yi Fang Liu, Guo Rong Li, "Matter wave soliton solution of the cubicquintic nonlinear Schrodinger equation with an anharonic potential", Applied Matheatics and Coputation, Vol 9, pp. 4847-485, 3. [5] Y Geng, J Li, L Zhang, "Exact explicit traveling wave solutions for two nonlinear Schrodinger type equatios" Applied Matheatics and Coputation, pp. 59-5,. [6] r.e. Mickens. "Nonlinear oscillations", Cabridge University Press 98, page 7. [7] E. Gluskin, "A nonlinear resistor and nonlinear inductor using a nonlinear capacitor", Journal of the Franklin Institute, 999, pp. 35-47. [8] W. A. Edson, "Vacuu tube oscillators", New York: Wiley, pp 48-4, 953. [9] D.S. Wang, X. Zeng, Y.Q. Ma, "Exact vortex solitons in a quasi-two-diensional Bose-Einstein condensate with spatially inhoogeneous cubic-quintic nonlinearity", Phys. Letters A, 376(),pp.367-37. A RESEARCH VISION Electronic Vision - year 7 nuber pp. 46-53 january -june of 4 53