Abstract
A new approach for demonstrating the global stability of ordinary differential equations is given. It is shown that if the curvature of solutions is bounded on some set, then any nonconstant orbits that remain in the set, must contain points that lie some minimum distance apart from each other. This is used to establish a negative-criterion for periodic orbits. This is extended to give a method of proving an equilibrium to be globally stable. The approach can also be used to rule out the sudden appearance of large-amplitude periodic orbits.
Highlights
A key issue in the analysis of a system of ordinary differential equations is to determine the long-term dynamics
It is not necessary to calculate solutions to the differential equations since the curvature calculation uses the derivative of a solution, and this is given by the vector field that defines the flow
If it is known that the curvature of a vector field is bounded on a set, it follows that the rate at which trajectories turn is bounded
Summary
It is shown that if the curvature of solutions is bounded on some set, any nonconstant orbits that remain in the set, must contain points that lie some minimum distance apart from each other. This is used to establish a negative-criterion for periodic orbits. This is extended to give a method of proving an equilibrium to be globally stable.
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