Generative Dynamical Synthesis: A Behavioral Learning Framework for Autonomous Neural Dynamical Systems from Bifurcation Diagrams

Abstract

Many machine-learning approaches to nonlinear dynamical systems focus on trajectory prediction or equation discovery. This work introduces the Generative Dynamical Synthesis (GDS) framework, a behavioral learning approach in which an autonomous neural dynamical system is trained from asymptotic state-transition samples associated with parameter-dependent bifurcation structures. Local one-step transition information is retained, while complete ordered trajectories, multistep supervision, and explicit governing equations are not required. After training, the learned feedforward neural map is recursively iterated as an autonomous discrete-time dynamical system. The framework is evaluated using the Logistic and Sine maps. In both cases, the learned models reproduce the principal in-domain bifurcation organization, supported by quantitative measures including bifurcation-set error, stability analysis, initialization sensitivity, and architecture sensitivity. Outside the training domain, the autonomous neural maps also generate structured nonlinear dynamics, including antimonotonicity and coexisting attractors in representative realizations. These behaviors are interpreted as properties of the learned neural models rather than guaranteed continuations of the reference systems. The results establish GDS as a proof-of-concept approach for behavior-oriented synthesis of parameter-dependent discrete-time dynamical systems.