Modeling Networks of Four Elements

In this article, fourth-order systems of ordinary differential equations are studied. These systems are of a special form, which is used in modeling gene regulatory networks. The nonlinear part depends on the regulatory matrix W, which describes the interrelation between network elements. The behavior of solutions heavily depends on this matrix and other parameters. We research the evolution of trajectories. Two approaches are employed for this. The first approach combines a fourth-order system of two two-dimensional systems and then introduces specific perturbations. This results in a system with periodic attractors that may exhibit sensitive dependence on initial conditions. The second approach involves extending a previously identified system with chaotic solution behavior to a fourth-order system. By skillfully scanning multiple parameters, this method can produce four-dimensional chaotic systems.