Original Paper

Nonlinear Dynamics

, Volume 69, Issue 3, pp 1225-1235

First online:

Open Access This content is freely available online to anyone, anywhere at any time.

The largest transversal Lyapunov exponent and master stability function from the perturbation vector and its derivative dot product (TLEVDP)

  • Artur DabrowskiAffiliated withDivision of Dynamics, Technical University of Lodz Email author 


The behavior of systems of coupled nonlinear oscillators and, connected with it, the synchronization phenomena are of significant interest in many areas of science. One of the most important problems in this field is the stability of the synchronous state. The most often applied tool which allows one to quantify this stability is the largest Transversal Lyapunov Exponent (TLE) and, connected with it, Master Stability Function (MSF) theory (Pecora and Carroll in Phys. Rev. Lett. 80:2109, 1998). Thus there is still need to elaborate fast and simple methods of TLE calculation. The new method of the TLE estimation is presented in this paper. It applies the perturbation vector and its derivative Dot Product to calculate the largest Lyapunov exponent in the direction transversal to the synchronization manifold. The method (TLEVDP) of the TLE calculation is very simple in its background and numerical application. It does not require calculations of the Jacobi matrix eigenvalues and orthonormalization in each integration step. Thus it is fast and simple. Used methodology applies the method of Master Stability Function proposed by Pecora and Carroll (in Phys. Rev. Lett. 80:2109, 1998) which is undependable on the system type. Moreover on the base of the one MSF calculation one can predict complete synchronization of the sets of oscillators coupled in any coupling scheme. The theoretical improvement of the method is introduced. Results of the numerical simulations are shown and compared with other publications. Investigations of the system of Duffing oscillators coupled in ring scheme as well as the coupled Van der Pol oscillators made with use of the (TLEVDP) method are presented. Fast stabilization of the TLE value was shown.


Chaos synchronization Transversal Lyapunov exponent Master stability function Stability Nonlinear dynamics