Page 490 - Mathematical Techniques of Fractional Order Systems

Dynamics, Synchronization and Fractional Order Form Chapter | 16 477

16.2 MODEL AND DYNAMICS OF THE SYSTEM WITH AN
INFINITE NUMBER OF EQUILIBRIUM POINTS

Recently, Gotthans et al. have proposed an approach for investigating new
chaotic systems with an infinte number of equilibrium (Gotthans and Petrˇ zela,
2015; Gotthans et al., 2016). By constructing general models with expected
features and applying a systematical search routine, authors have introduced
chaotic flows with circle and square equilibrium (Jafari et al., 2013).
Based on the effective approach of Gotthans et al., in this work we con-
sider a general form given by
8
_ x 52 z
>
<
2
_ y 5 xz 1 asgnðzÞ ð16:1Þ
>
:
_ z 5 f 1 ðx; yÞ 1 zf 2 ðx; y; zÞ
in which state variables are x, y, and z and a is a positive parameter. In the
general form, two nonlinear functions are denoted as f 1 ðx; yÞ and f 2 ðx; y; zÞ.It
is noted that the signum function sgnðzÞ has been used in known systems
because it can be conveniently implemented using an operational amplifier
(Piper and Sprott, 2010). The signum function is defined by:
2 1; x , 0
8
>
<
sgnðzÞ 5 0; x 5 0 ð16:2Þ
>
1; x . 0
:
It is trivial to find the equilibrium points of general form (16.1) by solv-
ing the three following equations:
2z 5 0 ð16:3Þ

2
xz 1 asgnðzÞ 5 0 ð16:4Þ
f 1 ðx; yÞ 1 zf 2 ðx; y; zÞ 5 0 ð16:5Þ
From Eq. (16.3) we have z 5 0. Thus, Eq. (16.4) is correct for all x.By
substituting Eq. (16.3) into Eq. (16.5), we get

f 1 ðx; yÞ 5 0: ð16:6Þ
It means that the equilibrium points of general form (16.1) are located on
the curve (16.6). In this chapter we select the nonlinear function f 1 ðx; yÞ as:

f 1 x; yð Þ 5 x 2 be y ð16:7Þ
where b is a positive parameter. The nonlinear function f 2 ðx; y; zÞ is chosen
as follows:

f 2 x; y; zð Þ 5 a 1 x 1 a 2 y 1 a 3 z 1 a 4 xy 1 a 5 xz 1 a 6 yz
2 2 2 ð16:8Þ
1a 7 x 1 a 8 y 1 a 9 z 1 a 10
in which 10 parameters are a i ði 5 1; ...; 10Þ).