This section describes the several numerical methods used herein, includ-ing the Poincar´e map, the FFT and the Lyapunov exponent.
The trajectory of the system (1.1) must first be generated. Numeri-cally, for a given set of parameters, a template A = [r, p, s] that satisfies (1.3), an amplitude b and period T , the system of differential equations is solved in FORTRAN 90 by calling a subroutine, RKF45, using the RUNGE-KUTTA-FEHLBERG (4,5) methods described in [Fehlberg, 1968], with step size=0.05, absolute error 1× 10−10 and relative error 1× 10−8.
Since the ω-limit set ω(b, T, A) is of greatest concern, 2× 106 steps are taken in the RKF45 integration. The first 1×106 steps were ignored, and the following numerical methods applied to the remaining data; the last 1× 106 points were taken as the ω-limit set ω(b, T, A).
The ω-limit set ˆω(b, T, A) of Poincar´a T -map is taken every T /stepsize points from ω(b, T, A). The relative error of the Poincar´e map can be easily computed. For example, in the ZN-case T = 4 with a step size 0.05, 80 steps must be integrated for each point on the Poincar`e map. Therefore, the relative error 1× 10−8× 80 = 8 × 10−7 is obtained for each successive point of the Poincar´e map.
The Lyapunov exponents are obtained by averaging eigenvalues of DF (ξ1, ξ2)
on each point in ˆωb. Here, a convergent condition is imposed that the relative error is less than 1× 10−4. Moreover, the first 1× 106 steps in the numerical integration are ignored to accelerate the convergence.
9 Conclusions
Zou & Nossek [1991] discovered a chaotic attractor in a two-cells CNN with template A = [1.2, 2,−1.2] and input b sin2πT with T = 4 and b ∼= 4.08.
This work investigates bifurcations and chaos for broad range of templates A = [r, p, s], input period T > 0 and input amplitude b > 0.
The bifurcations of (1.1) involves five parameters; r, p, s, T and b. The strategy used herein is to begin with b = 0 and A satisfying (1.3). Section 4 studied the existence and multiplicity problems of periodic cycle of (1.1).
The existence of the limit cycle Λ0(A) is proven when (1.3) holds. The existence and multiplicity of exterior periodic cycles are studied under further assumptions. The uniqueness problem is still open for general A that satisfies (1.3). The numerical evidence suggests that the limit cycle is unique.
The bifurcation problem is studied by examining how ”typical” trajec-tories vary with b, T and A. In particular, the trajectory Γ(b, T, A) and its ω-limit set ω(b, T, A) with initial conditions at the origin O = (0, 0), are considered. The system (1.1) is considered to be chaotic, if
(i) Γ(b, T, A) has a positive Lyapunov exponent,
(ii) The Poincar´e T -map ˆω(b, T, A) of ω(b, T, A) is fractal and (iii) FFT of Γ(b, T, A) has a broad-band.
Section 6 presented a approach to study the effects of the input T by examining the asymptotic limit cycle Λ∞(T, A) with period T of Γ(b, T, A) as b → ∞. When T ≤ T0∗, as defined in (6.10), then Λ∞ and −Λ∞ can be separated. Therefore, ω(b, T, A) and −ω(b, T, A) are separated for large b and may collide when b is small. If b becomes even smaller then a chaotic attractor may develop. The onset of chaos induced by crises ω(b, T, A) and
−ω(b, T, A) was observed for suitable T and b. These cases includes the ZN-case, A = [1.2, 2,−1.2] and T = 4. However, (6.10) alone does not cause chaos.
Section 3 presented a heuristic argument to determine the range of b over which maximum variation of Γ(b, T, A) may occur. b∗(T ) = c0T0(A)/T
is introduced in (3.9) where c0 = c0(A, T ) and c0 = 1 is a good guess for practical purposes. In most cases, the chaotic regions occur with c0 ∈ [0.8, 1].
After a range of interest of T and b are identified, the effect of b can be studied. The primary mean is to compare the relative strengths of sustained limit cycle Λ0(A) (without input) and the input bu(t).
The FFT of Γ(b, T, A) now is obtained and the ratioA(b) ≡ |aT(b)|/|a1(b)|, given in (1.10) of the largest amplitude a1(b) except for T -mode to the am-plitude aT(b) of the periodic T -mode is considered. Section 5 considered A(b) ≪ 1, i.e., Λ0(A) dominates. ω(b, T, A) is found to be either quasi-periodic or quasi-periodic. The Conjecture 5.1, which was partially proven, stated that periodic windows forms a devil’s staircase when b ∈ (b∗, b∗0) where 0 < b∗ < b∗0. The conjecture will be proven if some continuity and transver-sality conditions are met for the two dimensional Poincar´e map F (ξ1, ξ2) at periodic points. A well-known example of the devil’s staircase is the bifur-cations of one dimensional logistic map Fλ(x) = λx(1− x), λ ∈ [1, 4], where periodic windows forms a devil’s staircase.
Section 7 examined the chaos by studying the effects of b, T and A in three subsections. When A(b) ∼ 1, i.e., b ∼ b∗(T, A), the Lyapunov exponents of Γ(b, T, A) and ˆω(b, T, A) are computed. In many interesting cases, finitely many chaotic regions and window regions interweave with each other.
Subsection 7.1 studies in detail the ZN-case with varying b. Figure 7.4 plots typical chaotic attractors and basic periodic cycles. Different character-istics of ω(b, T, A) in windows are considered. In each window, a sequence of periodic-doubling is observed to the left of the basic periodic cycle. A quasi-periodic region is to the right of the basic quasi-periodic cycle. Several outstanding questions remain to be answered. For examples,
(i) Is any fine structures present in the quasi-periodic region of the win-dows?
(ii) Can any ergodic measure exist in chaotic regions? How do they change with regions?
(iii) Does any specific relationship exist between the chaotic regions and their neighboring window regions?
Subsection 7.2 considered the effect of input period T . The chaotic at-tractors, including especially the shape of ˆω(b, T, A), depend strongly on T . In the ZN-case, A = [1.2, 2,−1.2], ˆω(b, T, A) looks like a lady’s shoe for
T ∈ [3.5, 4.5]. However, for T is smaller, say T ≤ 3, and the shape change.
See Fig. 7.6. Apparently, the horseshoe or the partial horseshoe is visible in all cases. A detailed study of the effect of input period T is being conducted.
Subsection 7.3 considered the fundamental role of the temple A. Figures 7.8 and 7.9 present some preliminary and interesting results obtained by varying templates in the antisymmetric and asymmetric regions. Considering p = 2, the regions in (r,−s) parameter space were investigated to locate where chaos can occur. The chaotic parameters and non-chaotic parameters are spread out. Chaotic parameters cluster in some specific places which are easily identified, for example, along the antisymmetric line, i.e. r = −s. It is of great interests to study these chaotic bifurcations.
Acknowledgement
: The authors would like to thank Dr. Ban, Jung Chao for several interesting discussions during the preparation of this work.References
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