FO_Lyapunov_q

Program to compute LEs as function of q, of autonomous systems of commensurate Caputo's Fractional Order

August 2022: plot modified to overcome the problems with plot function in the last matlab variants

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**slight changes**

The program, FO_Lyapunov_q, can be used either alone, to determined the LEs of a FO system for a fixed fractional order q (see e.g. LE_RF.m which contains the extended system), or can be used to obtain the variation of LEs as function of q, case when the code run_FO_LE_q must be used [1].

1. To obtain the LEs for a given q, for, e.g., RF system, one uses

LE=FO_Lyapunov_q(ne,ext_fcn,t_start,h_norm,t_end,x_start,h,q);

For example, for the RF system [1]

LE=FO_Lyapunov_q(3,@LE_RF,0,0.02,200,[0.1;0.1;0.1],0.02,0.998)

2. If one intends to obtain the evolution of LEs as function of q one uses

run_FO_LE_q(ne,ext_fcn,t_start,h_norm,t_end,x_start,h,q_min,q_max,n)

E.g., for the same system RF

run_FO_LE_q(3,@LE_RF,0,0.02,150,[0.1;0.1;0.1],0.02,0.9,1,800)

Note that FO_Lyapunov_q.m, LE_RF.m, run_FO_LE_q and FDE12.m (necessary to integrate the system) must be in the same folder.

As mentioned in [1], the relation between h_norm and h is essential. Here both are chosen equal (0.02), but multiple of h for h_norm should be tried

[1] Marius-F. Danca and N. Kuznetsov, Matlab code for Lyapunov exponents of fractional order systems, International Journal of Bifurcation and Chaos, 28(05)(2018), 1850067