Modelling of phenomena in applied sciences like physics, biology or engineering often leads to equations depending on parameters. These parameters describe how the environment influences a system. Thus, the knowledge of the values of the parameters at which the qualitative behaviour of the system changes has great importance. In this talk we study the structure of equations of the form $F(x,\lambda)=0,$ where $F$ is a nonlinear operator between Banach spaces and $\lambda$ represents the parameters.
We present a new approach to the construction of first integrals for second order autonomous systems without invoking a Lagrangian or Hamiltonian reformulation. We show and exploit the analogy between integrating factors of first order equations and their Lie point symmetry and integrating factors of second order autonomous systems and their dynamical symmetry. We connect intuitive and dynamical symmetry approaches through one-to-one correspondence in the framework proposed for first order systems. Conditional equations for first integrals are written out, as well as equations determining symmetries. The equations are applied on the simple harmonic oscillator and a class of nonlinear oscillators to yield integrating factors and first integrals.
A Lie algebraic approach is also used to study universal equations. To be universal, a differential equation must have a solution with which an arbitrary function can be uniformly approximated. A universal differential equation must be form invariant under translation and scaling. We present a way to construct universal equations.