Bifurcation analysis in an associative memory model

We previously reported the chaos induced by the frustration of interaction in a nonmonotonic sequential associative memory model, and showed the chaotic behaviors at absolute zero. We have now analyzed bifurcation in a stochastic system, namely, a finite-temperature model of the nonmonotonic sequential associative memory model. We derived order-parameter equations from the stochastic microscopic equations. Two-parameter bifurcation diagrams obtained from those equations show the coexistence of attractors, which do not appear at absolute zero, and the disappearance of chaos due to the temperature effect.

Figure 1

(Color) Transition of overlap $m\left(t\right)$ and variance of crosstalk noise $\alpha R(t,t)$ for $\alpha =0.065,\theta =1.20,T=0.10$. Fixed points $P,P^{\prime },Q$ are from stationary state equations: (a) our theory, (b) simulation $(N=100000)$.

Figure 2

(Color) Two-parameter bifurcation diagram $(\theta ,\alpha )$ for (a) fixed point $Q$ and (b) fixed point $P$ at $T=0$. The blue region represents period-1 attractors, red period-2, green period-3, yellow period-4, purple period-5, sky blue period-6, and black for more than six periods, quasiperiodic or chaotic.

Figure 3

(Color online) Attractor $Q$ in (a) region $A(\alpha =0.30,\theta =1.20)$ and (b) region $A^{\prime }(\alpha =0.30,\theta =0.70)$ at $T=0$ by our theory. The area inside the semielliptic arc is the orientation-preserving area and the area outside is the orientation-reversing area.

Figure 4

(Color online) Attractors $P$ and $P_2$ in (a) region $B(\alpha =0.20,\theta =1.50)$ and (b) region $B^{\prime }(\alpha =0.20,\theta =0.20)$ at $T=0$ by our theory.

Figure 5

(Color online) Attractors around $P$ in region $C$: (a) quasiperiodic attractor $(\alpha =0.10,\theta =1.30)$, (b) period-6 attractor $(\alpha =0.01,\theta =1.20)$, and (c) chaotic attractor $(\alpha =0.10,\theta =1.15)$.

Figure 6

(Color online) Attractors around $P$ in region $D(\alpha =0.10,\theta =1.00)$ vanish due to the boundary crisis.

Figure 7

(Color) Two-parameter bifurcation diagrams on invariant line $m=0$ for (a) absolute zero $(T=0)$ and (b)–(h) finite temperature $(T=0.05-0.35)$. The abscissa denotes $\theta (0<\theta <1.7)$, and ordinate denotes $\alpha (0<\alpha <0.302)$ on a logarithmic scale. Colors denote the period of attractors as in Fig. 2.

Figure 8

(Color) Two-parameter bifurcation diagrams around fixed point $P$ for (a) absolute zero $(T=0)$ and (b)–(h) finite temperature $(T=0.05-0.40)$. The abscissa denotes $\theta (0<\theta <1.7)$ and the ordinate denotes $\alpha (0<\alpha <0.302)$ on a logarithmic scale. Colors denote period of attractors as in Fig. 2.

Figure 9

(Color) Curve of eigenvalue $\lambda =0$ and area of cusp point for (a) $T=0$ and (b) $T=0.05$. The cusp point is in the rectangular area.

Figure 10

Return map of variance of crosstalk noise $\alpha R(t,t)$ for $\alpha =0.001,\theta =1.0$. Solid and broken lines denote cases of $T=0$ and $T=0.3$, respectively.

Figure 11

Return map of $R(t,t)$ for $\alpha =0.01,\theta =0.8$. Solid and broken lines denote cases of $T=0$ and $T=0.1$, respectively.