In order to determine the direction and stability of the Pitchfork bifurcation, it is necessary to use the center manifold theory [2]. The center manifold theory demonstrates that the mapping behavior in the bifurcation is equivalent to the complex mapping below:
SINK ZONE
SOURCE ZONE
SADDLE ZONE SADDLE ZONE
SOURCE ZONE σ2
σ1 Pitchfork bifurcation
|W| = w11+w22–1
Fig. 2 The stability regions and the Pitchfork bifurcation line at the fixed point (0, 0)
10 Study of Pitchfork Bifurcation in Discrete Hopfield Neural Network 125
uðkþ1Þ ¼uðkÞ þcð0ÞuðkÞ3þoðuðkÞ4Þ (6) The parameterc(0) is [2]
cð0Þ ¼1
6hp;Cðq;q;qÞi (7)
whereCis the third derivative terms of the Taylor development, the notation
< . , . > represents the scalar product, and p, q are the eigenvector Jacobian matrix and its transpose, respectively. These vectors satisfy the normalization condition
p;q h i ¼1
The above coefficients are evaluated for the critical parameter of the system where the bifurcation takes place. Thec(0) sign determines the bifurcation direc-tion. Whenc(0) is negative, a stable fixed point becomes an unstable fixed point and two additional stable symmetrical fixed points appear. In the opposite case,c(0) positive, an unstable fixed point becomes a stable fixed point and two additional unstable symmetrical fixed points appear.
In the neural network mapping,pandqare q¼ d
eþd e w21X2;0;1
(8)
p¼ e
w12X1;0;1
(9)
where
d¼w11X1;01 e¼w22X2;01 X1;0¼1x21;0
X2;0¼1x22;0
x1,0andx2.0are the fixed point coordinates where the bifurcation appears.
126 R. Marichal et al.
The Taylor development term is
wheredijis the Kronecker delta.
In order to determine the parameterc(0), it is necessary to calculate the third derivate of the mapping (1) given by Eq. (10)
@fi
Taking into account the previous equations and theqautovector Eq. (8)
Cðq;q;qÞ ¼2
It can be shown in Eq. (3) that the zero is always a fixed point. Replacing the expressions forC(q,q,q),qandpgiven by Eqs. (8) (9) and (11), respectively, and evaluating them at the zero fixed point, the previousc(0) coefficient is
cð0Þ ¼1
6hp;Cðq;q;qÞi ¼ w212ðw221Þ þ ðw111Þ3 3ð1w11Þ2ð2w11w22Þ The previous expression is not defined for the following cases a. w11¼1
b. w11+w22¼2.
In this paper we only consider condition (a) since condition (b) shows the presence of another bifurcation, known as Neimark-Sacker and which is analyzed in another paper [16].
10 Study of Pitchfork Bifurcation in Discrete Hopfield Neural Network 127
Taking into account condition a) and the bifurcation parameter equation j j ¼W w11þw221
then
w12w21¼0
In this particular case, the eigenvalues match with the diagonal elements of the weight matrix
l1¼w11
l2¼w22
The newqandpeigenvectors are given by q¼f1;0g p¼ 1; w12
w221
Thec(0) coefficient is cð0Þ ¼1
6hp;Cðq;q;qÞi ¼ 1
3w311 ¼ 1 3:
Therefore, in this particular case, the coefficient of the normal formc (0) is negative, a stable fixed point becomes a saddle fixed point and two additional stable symmetrical fixed points appear.
5 Simulations
In order to examine the results obtained. The simulation shows the Pitchfork bifurcation (Fig.3). The Pitchfork bifurcation is produced by the diagonal element weight matrixw11. Figure3ashows the dynamic configuration before the bifurcation is produced, with only one stable fixed point. Subsequently, when the bifurcation is produced (Fig.3b), two additional stable fixed points appear and the zero fixed point changes its stability from stable to unstable (the normal form coefficientcis negative).
6 Conclusion
In this paper we considered the Hopfield discrete two-neuron network model. We discussed the number of fixed points and the type of stability. We showed the bifurcation Pitchfork direction and the dynamical behavior associated with the bifurcation.
The two-neuron networks discussed above are quite simple, but they are poten-tially useful since the complexity found in these simple cases might be carried
128 R. Marichal et al.
over to larger Hopfield discrete neural networks. There exists the possibility of generalizing some of these results to higher dimensions and of using them to design training algorithms that avoid the problems associated with the learning process.
–1 – 0.5 0 0.5 1
–1 – 0.8 – 0.6 – 0.4 – 0.2 0 0.2 0.4 0.6 0.8 1
X2 X2
X1 X1
–1 – 0.5 0 0.5 1
–1 – 0.8 – 0.6 – 0.4 – 0.2 0 0.2 0.4 0.6 0.8
a 1
b
Fig. 3 The dynamic behavior when the Pitchfork bifurcation is produced. + andDare the saddle and source fixed points, respectively. (a)w11¼0.9,w12¼0.1,w21¼1 andw22¼0.5; (b)w11¼ 1.1,w12¼0.1,w21¼1 andw22¼0.5
10 Study of Pitchfork Bifurcation in Discrete Hopfield Neural Network 129
References
1. R. Hush, B.G. Horne, Progress in supervised neural network. IEEE Signal Process. Mag. 8–39 (1993)
2. Y.A. Kuznetsov, Elements of Applied Bifurcation Theory, 3rd edn. (Springer-Verlag, New York, 2004)
3. K.T. Alligood, T.D. Sauer, J.A. Yorke, Chaos: An Introduction to Dynamical Systems (Springer Verlag, New York, 1998)
4. C.M. Marcus, R.M. Westervelt, Dynamics of iterated-map neural networks. Phys. Rev. A40, 501–504 (1989)
5. Cao Jine, On stability of delayed cellular neural networks. Phys. Lett. A261(5–6), 303–308 (1999)
6. X. Wang, Period-doublings to chaos in a simple neural network: an analytical proof. Complex Syst.5, 425–441 (1991)
7. X. Wang, Discrete-time dynamics of coupled quasi-periodic and chaotic neural network oscillators. International Joint Conference on Neural Networks (1992)
8. F. Pasemann, Complex dynamics and the structure of small neural networks. Netw. Comput.
Neural Syst.13(2), 195–216 (2002)
9. Tino et al., Attractive periodic sets in discrete-time recurrent networks (with emphasis on fixed-point stability and bifurcations in two-neuron networks). Neural Comput. 13(6), 1379–1414 (2001)
10. J. Hopfield, Neurons with graded response have collective computational properties like those of two-state neurons. Procs. Nat. Acad. Sci. USA81, 3088–3092 (1984)
11. D.W. Tank, J.J. Hopfield, Neural computation by concentrating information in time, Proc.
Nat. Acad. Sci. USA,84, 1896–1991 (1984)
12. X. Liao, K. Wong, Z. Wu, Bifurcation analysis on a two-neuron system with distributed delays. Physica D149, 123–141 (2001)
13. S. Guo, X. Tang, L. Huang, Hopf bifurcating periodic orbits in a ring of neurons with delays, Physica D,183, 19–44 (2003)
14. S. Guo, X. Tang, L. Huang, Stability and bifurcation in a discrete system of two neurons with delays. Nonlinear Anal. Real World. doi:10.1016/j.nonrwa.2007.03.002 (2007)
15. S. Guo, X. Tang, L. Huang, Bifurcation analysis in a discrete-time single-directional network with delays. Neurocomputing71, 1422–1435 (2008)
16. R. Marichal et al., Bifurcation analysis on hopfield discrete neural networks.WSEASTrans.
Syst.5, 119–124 (2006)
130 R. Marichal et al.