On the complete integrability and linearization of certain second-order nonlinear ordinary differential equations

Abstract

A method of finding general solutions of second-order nonlinear ordinary differential equations by extending the Prelle-Singer (PS) method is briefly discussed. We explore integrating factors, integrals of motion and the general solution associated with several dynamical systems discussed in the current literature by employing our modifications and extensions of the PS method. In addition to the above we introduce a novel way of deriving linearizing transformations from the first integrals to linearize the second order nonlinear ordinary differential equations to free particle equation. We illustrate the theory with several potentially important examples and show that our procedure is widely applicable.

On the complete integrability and linearization of nonlinear ordinary differential equations. IV. Coupled second-order equations

“…However, after making these modifications, S 4 and U 4 remain the same. A motivation to perform this type of modification came from our earlier work on the applicability of the PS procedure to scalar second-order ODEs (Chandrasekar et al 2005a). However, unlike the scalar case, here we have to choose two functions, namely F and G, appropriately, such that the compatible forms ofR andK can be fixed.…”

“…In addition to the above, in this paper, we also introduce a new method to transform two coupled second-order ODEs to two uncoupled second-order ODEs. Thus, the PS procedure inherits several remarkable features both at the theoretical foundations and in the range of applications, which we have listed already in Chandrasekar et al (2005a). Finally, we have carefully fixed the examples so that the basic features associated with this method and the results which it leads to could be explained in an efficient way.…”

“…In this part of our study on the integrability and linearization of nonlinear ordinary differential equations (ODEs), we focus our attention on the theoretical formulation and applications of the modified Prelle-Singer (PS) procedure (Prelle & Singer 1983;Duarte et al 2001;Chandrasekar et al 2005aChandrasekar et al , 2006) to a set of two coupled second-order ODEs. The need for this demonstration is due to the fact that classifying and studying two-degrees-of-freedom dynamical systems are highly non-trivial problems in the theory of nonlinear dynamical systems.…”

A nonlocal connection between certain linear and nonlinear ordinary differential equations – Part II: Complex nonlinear oscillators

“…[10][11][12][13][14][15][16][17]. On the other hand the restriction f (x) = (α + βx 2 ) in (1) gives us generalized force-free Duffing-van der Pol oscillator (DVP) equation…”

“…The question becomes easier when one has a specific equation. In this case one may rewrite the given complex nonlinear ODE as a set of real ODEs by appropriately splitting the complex variable into real variables and then start to investigate the integrability of these real ODEs through any one of the analytical or geometrical methods, namely Painlevé test [7], Lie symmetry analysis [8], Prelle-Singer procedure [16], Darboux method [9] and so on. In this paper we extend the idea given in our previous paper [1] for the case of real ODEs to the case of complex ODEs and identify the integrable equations in a direct manner using nonlocal transformations.…”