Matlab code for Lyapunov exponents of fractional order

Program to compute LEs of autonomous continuous 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, determines the Lyapunov Exponents (LEs) of a commensurate Fractional Order (FO) autonomous continuous-time system modeled by Capto's derivative [1] (for non-commensurate order see [2])

Because there are continuous citations of the paper presenting this program for Lyapunov Exponents (LEs) and due to some (minor) errors existing in the codes, here there are the slightly improved variants of related codes

Thus, the names of the several used codes are synchronized so that the use of programs is easier now

For commensurate order, all programs uses FDE12.m to integrate the FDEs:

https://goo.gl/XScYmi

The function containing the extended system (see e.g. LE_RF.m or LE_RF_p.m), has to be in the same folder with FDE12.m and FO_Lyapunov.m

Note that there are the following possibilities:

1. The program, FO_Lyapunov.m, determines LEs of the (extended) system (see e.g. LE_RF.m) as function of time.

2.1 A modified variant of FO_Lyapunov (FO_Lyapunov_q.m) determines the LEs of a system (see e.g. LE_RF.m) for a given value of q.

2.2 FO_Lyapunov_q.m can be also used to obtain the variation of LEs as function of q. For this purpose it is used by run_FO_LE_q.m

3.1 Another modified variant FO_Lyapunov (FO_Lyapunov_p.m) determines the LEs of a system (see e.g. LE_RF_p.m) for a given value of p.

3.2 FO_Lyapunov_p.m can be used to obtain the variation of LEs as function of p. For this purpose it is used by run_LE_FO_p.m

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 (e.g. h=0.002 and h_norm=0.2, but to the detriment of computational time)

For non-commensurate order see FO_NC_Lyapunov.m which uses another program to integrate non-commensurate FDEs (see [2]) .

The proof of the memory principle of the algorithm can be found in [3]

[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

[2] Marius.-F. Danca, Matlab code for Lyapunov exponents of fractional-order systems, Part II: The non-commensurate case, International Journal of Bifurcation and Chaos, 31(12), 2150187, (2021)

[3] Marius-F. Danca, Michal Feckan, Memory Principle of the MATLAB Code for Lyapunov Exponents of Fractional-Order, International Journal of Bifurcation and Chaos, 2024, 2450156