Quadratic Dynamical Systems (Preliminary Version)

The paper promotes the study of computational aspects, primarily the convergence rate, of nonlinear dynamical systems from a combinatorial perspective. The authors identify the class of symmetric quadratic systems. Such systems have been widely used to model phenomena in the natural sciences, and also provide an appropriate framework for the study of genetic algorithms in combinatorial optimisation. They prove several fundamental general properties of these systems, notably that every trajectory converges to a fixed point. They go on to give a detailed analysis of a quadratic system defined in a natural way on probability distributions over the set of matchings in a graph. In particular, they prove that convergence to the limit requires only polynomial time when the graph is a tree. This result demonstrates that such systems, though nonlinear, are amenable to quantitative analysis. >

Authors

• Yuri Rabinovich
• Alistair Sinclair
• Avi Wigderson

Keywords

• Educational institutions
• Computer science
• Genetics
• State-space methods
• Nonlinear dynamical systems
• Convergence
• Polynomials
• Tree graphs
• Ear
• Extraterrestrial phenomena
• System Dynamics
• Markov Chain
• State Space
• Convergence Rate
• Normal Population
• Fixed Point
• Linear System
• Set Of Equations
• Nonlinear Systems
• Proof Of Theorem
• State Population
• Natural Phenomena
• Types Of Questions
• Aperiodic
• Vertices
• Tree Size
• Limit Point
• Obvious Way
• Pair Distribution