Regardless of the marked differences between biological and artificial neural systems, one fundamental similarity is that they are essentially dynamical systems that can learn to imitate other dynamical systems whose governing equations are unknown. The brain is able to learn the dynamic nature of the physical world via experience; analogously, artificial neural systems such as reservoir computing networks (RCNs) can learn the long-term behavior of complex dynamical systems from data. Recent work has shown that the mechanism of such learning in RCNs is invertible generalized synchronization (IGS). Yet, whether IGS is also the mechanism of learning in biological systems remains unclear. To shed light on this question, we draw inspiration from features of the human brain to propose a general and biologically feasible learning framework that utilizes IGS. To evaluate the framework's relevance, we construct several distinct neural network models as instantiations of the proposed framework. Regardless of their particularities, these neural network models can consistently learn to imitate other dynamical processes with a biologically feasible adaptation rule that modulates the strength of synapses. Further, we observe and theoretically explain the spontaneous emergence of four distinct phenomena reminiscent of cognitive functions: (i) learning multiple dynamics; (ii) switching among the imitations of multiple dynamical systems, either spontaneously or driven by external cues; (iii) filling-in missing variables from incomplete observations; and (iv) deciphering superimposed input from different dynamical systems. Collectively, our findings support the notion that biological neural networks can learn the dynamic nature of their environment through the mechanism of IGS.