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Nonlinear dynamics phenomena in fractional van der Pol-FitzHugh-Nagumo model

Yu. Luchko*, B. Datsko

**and V. Meleshko

**

*Technical University of Applied Sciences, Berlin, Germany**Institute of Applied Problems of Mechanics and Mathematics, NASU, Lviv, Ukraine

Summary. In this article, the stability domains and nonlinear phenomena of dissipative fractional dynamical systems with the fractionalderivatives of different orders are discussed. As an example, the van der Pol-FitzHugh-Nagumo model is considered. In particular, the stability

domain of this model is investigated for different parameters values. By means of computer simulations some new complex nonlinear dynamic

phenomena that cannot be found in dynamical systems of the integer order are identified.

Introduction and problem formulation

Both ordinary and partial differential equations of fractional order have been used within the last few decades for modelling

of many physical and chemical processes and in engineering, especially for modelling of the so called anomalous

phenomena in the complex systems (see [1,5] and the references therein). The main reason for the utility of the fractional

dynamical systems like the ones considered in this paper is that they can be used for representing of the long-memory and

non-local dependence of many anomalous processes. In this paper, stability domain and nonlinear dynamics in the van der

Pol-FitzHugh-Nagumo-like models of the fractional order are analysed.

We investigate a system of the coupled fractional differential equations with the nonlinear feedbacks in the form

( )AuFuD ,= (1)

where ( )TdtuddtudwD 2211 /,/ 21

= is the fractional differential operator, ( )2,0, 21 , ( ) ( )( ) ,, 21T

tutuu = )(),( 21 tutu

being the activator and inhibitor variables, respectively, ( ) ( )TAuufAuufAuF ),,(),,,(, 212211= is a nonlinear vector-function

depending on an external parameterA , and the vector ( )T21 , = represents the characteristic times of the system. The

fractional derivatives of order +R are understood in the Caputo sense:

( )( )

( )( )

( ).,1,

1:

0

1Nmmmd

t

x

mtx

dt

dt

m

m

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2/)( - (a), 21 = - (b), 21 < - (c). The

results of computer simulation for 0.2= and 1 =1.5, 2 =1.25 1(a) ; 1 =1.75, 2 =1.5 2(a); 1 =1.75,

2 =1.25 3(a); 1 =1.75, 2 =1.0 4(a); 1 = 2 =1.0 1(b); 1 = 2 =1.25 2(b); 1 = 2 = 1.5 3(b); 1 = 2 =

1.75 4(b); 1 =1.25, 2 =1.5 1(c) ; 1 =1.0, 2 =1.75 2(c); 1 =1.5, 2 =1.75 3(c); 1 =1.25, 2 =1.75 4(c);

(a) (b)

(c) (d)

Fig. 2: Examples of the time-domain oscillations (left) and the corresponding two-dimensional phase portraits (right)

for different values of fractional derivatives: Dynamics of variables 1n (black lines) and 2n (grey lines) for : 1 =1.5,

2 =0.75, 01.1=b , 0.0=A (a); 1 =0.8, 2 =0.4 , 0.2=b , 4.0=A (b); 1 =1.5, 2 =0.75, 01.1=b ,

3.0=A (c); 1 =1.75, 2 =1.75, 01.1=b , 4.0=A (d). The other parameters are 1.01 = , 0.12 =

The results of the computer simulations show that the instability domain and the system dynamics significantly depend on

the ratio between the orders of the fractional derivatives. Thus some new complex nonlinear dynamics phenomena that

cannot be found in dynamical systems of the integer order can be identified for the fractional van der Pol-FitzHugh-

Nagumomodel (see the Figs. 1, 2 for the evolution of the instability domains and typical oscillations, as well as the two-

dimensional phase portraits for different governing parameters and orders of the fractional derivatives).

References

[1] Agrawal, O., Machado, J. Tenreiro, Sabatier, J. (2007) Advances in Fractional Calculus: Theory and Applications in Physics and Engineering,

Springer, Dordrecht.

[2] Gafiychuk, V., Datsko, B. (2008) Spatiotemporal pattern formation in fractional reaction-diffusion systems.Phys. Rev. E77:066210-1-9.[3] Gafiychuk, V., Datsko, B., Meleshko, V. (2008) Math. modeling of time fractional react.-diffus. Systems.J. Comp. Appl. Math. 372 :215-225.

[4] Landa, P. (1996) Nonlinear oscillations and waves in dynamical systems. Kluwer, Dordrecht.

[5] Luchko, Yu., Rivero, M., Trujillo, J., Velasco M. (2010) Fractional models, non-locality and complex systems. Comp. Math. with Appl.59:1048-1056.