In my talk briefly, I will talk about nonlinear dynamics and complex networks, pattern formation in complex systems, chaos control mechanism, where we have controlled complex networks of chaotic systems to steady states and robustness of the network of bistable elements and multi-stable oscillators; connected in different network topologies using the concept of multi-node basin stability.
In detail, I will talk about the emergent dynamic stability of complex systems. As we know that the stable functionality of networked systems is a hallmark of their natural ability to coordinate between their multiple interacting components. Yet, strikingly, real-world networks seem random and highly irregular, apparently lacking any design for stability. What then are the naturally emerging organizing principles of complex-system stability? Encoded within the system's stability matrix, the Jacobian, the answer is obscured by the scale and diversity of the relevant systems, their broad parameter space, and their nonlinear interaction mechanisms. I will talk about the emergent patterns in the structure of the Jacobian, rooted in the interplay between the network topology and the system's intrinsic nonlinear dynamics. These patterns help us analytically identify the few relevant control parameters that determine a system's dynamic stability. We find that complex systems exhibit discrete stability classes, from asymptotically unstable, where stability is unattainable, to sensitive, in which stability abides within a bounded range of the system's parameters. Most crucially, alongside these two classes, we uncover a third class, asymptotically stable, in which a sufficiently large and heterogeneous network acquires guaranteed stability, independent of parameters, and therefore insensitive to an external perturbation.