This study investigates the use of continuous-time dynamical systems for sparse signal recovery. The proposed dynamical system is in the form of a nonlinear ordinary differential equation (ODE) derived from the gradient flow of the Lasso objective function. The sparse signal recovery process of this ODE-based approach is demonstrated by numerical simulations using the Euler method. The state of the continuous-time dynamical system eventually converges to the equilibrium point corresponding to the minimum of the objective function. To gain insight into the local convergence properties of the system, a linear approximation around the equilibrium point is applied, yielding a closed-form error evolution ODE. This analysis shows the behavior of convergence to the equilibrium point. In addition, a variational optimization problem is proposed to optimize a time-dependent regularization parameter in order to improve both convergence speed and solution quality. The deep unfolded-variational optimization method is introduced as a means of solving this optimization problem, and its effectiveness is validated through numerical experiments.