It is a very interested question

What stability are you interested in: Linear or Nonlinear?

In either case, follow:

1. Fixed points,

2. Perturb around the above,

3. Order-by-order analyses, and hence,

4. Results.

Thank you, Mr Karmakar

The stability means the Lyapunov stability, that is, we suppose that x(t) is a stable signal in Lyapunov concept and the goal is how to analyze the Lyapunov stability of y(t) ?

Please have a careful look for the Lyapunov exponent calculations and stability at

[1] H. González-Aguilar and E. Ugalde (eds), Nonlinear Dynamics New Directions: Theoretical Aspects (Springer, Switzerland, 2015).

[2] M. Lakshamanan and S. Rajasekhar, Nonlinear Dynamics: Integrability, Chaos, and Patterns (Springer, New York, 2002).

Thank you, Mr Karmakar.