New topological tool for multistable dynamical systems

We introduce a new method for investigation of dynamical systems which allows us to extract as much information as possible about potential system dynamics, based only on the form of equations describing it. The discussed tool of critical surfaces, defined by the zero velocity (and/or) acceleration field for particular variables of the system is related to the geometry of the attractors. Particularly, the developed method provides a new and simple procedure allowing to localize hidden oscillations. Our approach is based on the dimension reduction of the searched area in the phase space and has an advantage (in terms of complexity) over standard procedures for investigating full-dimensional space. The two approaches have been compared using typical examples of oscillators with hidden states. Our topological tool allows us not only to develop alternate ways of extracting information from the equations of motion of the dynamical system, but also provides a better understanding of attractors geometry and their capturing in complex cases, especially including multistable and hidden attractors. We believe that the introduced method can be widely used in the studies of dynamical systems and their applications in science and engineering.