Exact linearization and stability of fixed points in synchronization models

Abstract A significant number of synchronization models can be formulated as a system of matrix equations <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mrow> <mml:mover> <mml:mi>g</mml:mi> <mml:mo>˙</mml:mo> </mml:mover> </mml:mrow> <mml:mi>i</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>M</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:msub> <mml:mi>g</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> </mml:math> for square matrices <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>g</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>M</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> </mml:math> , where g i are the variables and M i denotes the the mean-fields to which all variables couple. Examples include the Kuramoto model with the possible inclusion of phase lag angles, higher-order systems with three- or four-body interactions, and higher-dimensional generalizations to the unit sphere. The transformation to the matrix form is accomplished for Riccati systems by means of linear fractional transformations, and for models on the sphere with cubic nonlinearities via the unit vector map. Since M i is constant at fixed points, the matrix equations <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mrow> <mml:mover> <mml:mi>g</mml:mi> <mml:mo>˙</mml:mo> </mml:mover> </mml:mrow> <mml:mi>i</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>M</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:msub> <mml:mi>g</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> </mml:math> reduce to a linear system which is solved exactly in terms of the matrix exponential. Stability of the asymptotic trajectories is determined by the eigenvalues of M i , and the fixed point is related to the eigenvector corresponding to the largest real eigenvalue. This leads to relatively simple conditions which are satisfied by asymptotically stable fixed points. These properties are demonstrated by example for models on the unit circle such as periodic ring oscillators, and for models on the two-sphere with pairwise or three-body interactions.