Two-parameter analysis of nonlinear systems.
Atlas of charts of dynamical regimes

Equations of dynamical systems often contain parameters, the constant quantities determining character of observing regimes. In the case of two control parameters there exists a useful visual representation of the system behavior by means of charts of dynamical regimes -diagrams on the parameter plane where domains of qualitatively distinct regimes are indicated by colors. To depict such a chart one scans step by step an area of interest on the parameter plane. At each point associated with one pixel the differential equation for the system under study is solved numerically (or the respective map is iterated in the case of discrete-time systems). Then, the nature of a regime is analyzed after decay of transients and arrival to attractor, and the point is marked by an appropriate color. In those cases when multistability occurs (two or more attractors coexist at the same parameter values), the chart should be imagined as a collection of two or more overlapping sheets, each having its own coloring. On this site we present charts of dynamical regimes for several known dynamical systems.

One-dimensional maps

Two-dimensional maps

Driven nonlinear oscillators and self-oscillators

Artificial flow systems

Electronic self-oscillators

Coupled systems

Saratov group
of theoretical nonlinear
dynamics
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