Symmetries of nonlinear ordinary differential equations: the modified Emden equation as a case study

Lie symmetry analysis is one of the powerful tools to analyze nonlinear ordinary differential equations. We review the effectiveness of this method in terms of various symmetries. We present the method of deriving Lie point symmetries, contact symmetries, hidden symmetries, nonlocal symmetries, $\lambda$-symmetries, adjoint symmetries and telescopic vector fields of a second-order ordinary differential equation. We also illustrate the algorithm involved in each method by considering a nonlinear oscillator equation as an example. The connections between (i) symmetries and integrating factors and (ii) symmetries and integrals are also discussed and illustrated through the same example. The interconnections between some of the above symmetries, that is (i) Lie point symmetries and $\lambda$-symmetries and (ii) exponential nonlocal symmetries and $\lambda$-symmetries are also discussed. The order reduction procedure is invoked to derive the general solution of the second-order equation.Comment: 31 pages, To appear in the proceedings of NMI workshop on nonlinear integrable systems and their applications which was held at Centre for Nonlinear Dynamics, Tiruchirappalli, Indi

Influence of field-like torque in synchronization of spin torque oscillators

The magnetization dynamics of two parallelly coupled spin torque oscillators, destabilization of steady states and removal of multistability, are investigated by taking into account the influence of field-like torque. It is shown that the existence of such torque can cancel the effect of damping and can, therefore, cause the oscillators to exhibit synchronized oscillations in response to direct current. Further, our results show that the presence of field-like torque enhances the power and Q-factor of the synchronized oscillations. The validity of the above results is confirmed by numerical and analytical studies based on the stochastic Landau-Lifshitz-Gilbert-Slonczewski equation.Comment: 10 pages, 10 figures, Accepted for Publication in IEEE Transactions on Magnetic