Beyond invariance: Open challenges when dynamics and structure co-evolve

The success of nonlinear science has been largely built upon the existence of invariant structures. Pattern formation theories have traditionally assumed fixed domains and prescribed boundary conditions. Synchronization theory has relied on the existence of invariant synchronization manifolds. Complex network theory has mostly described systems through static graphs with pairwise interactions. These assumptions have enabled powerful analytical developments and have led to a deep understanding of a broad range of physical, biological, and technological phenomena. Many contemporary applications, however, challenge precisely such underlying assumptions. Domains evolve together with the dynamics they support. Coupling may alter the synchronized state itself. Interactions often occur among groups rather than pairs of units, while network architectures continuously reorganize across multiple functional scales. In this Perspective, I argue that the next generation of nonlinear science will be characterized by the progressive loss of the invariant structures that have historically organized its theoretical foundations. Through examples drawn from pattern formation, synchronization and control, and complex networks, I discuss a number of open problems that emerge from this transition and consider the role that machine learning and artificial intelligence may play in addressing them. The next frontier of nonlinear science may no longer lie in understanding dynamics on prescribed invariant structures, but in uncovering the principles governing the co-evolution of dynamics and structure across space, time, and organizational scales.