Abstract <p>Solvers considering time-varying parameters are more suitable for addressing a variety of time-varying problems, whereas traditional fixed-parameter neural networks are somewhat insufficient for efficiently and quickly solving these problems. Many existing zeroing neural networks ensure rapid convergence using the infinite-valued AFs. For solving time-varying nonlinear equations, this paper proposes a finitely-activated variable parameter noise tolerant zeroing neural network (VPNTZNN), applied to trajectory tracking of redundant robotic arms. The designed variable parameters are error-dependent, enabling adaptive adjustment to optimal values as errors fluctuate, thereby ensuring faster convergence of the proposed VPNTZNN. And the constructed variable parameters and activation functions (AFs) do not escalate infinitely over time. Affected by the above variable parameters, the proposed finitely-activated VPNTZNN achieves rapid finite-time convergence with strong noise suppression. Simulation results validate the effectiveness of our method in solving time-variant nonlinear equations and in trajectory tracking of redundant manipulators. Moreover, this approach employs a finite-valued activation function to design a variable-parameter neural network, thereby avoiding the difficulties of practical implementation.</p> Graphical abstract <p></p>