(a) Bifurcation diagram of x, and (b) LEs and Dky of the map (1) calculated for a ∈ [2.29, 2.73] and (b, c) = (1, 0.1) using the initial value (1, –3). In panel (b), the largest Lyapunov exponent (Le1), the smallest Lyapunov exponent (Le2), and Lyapunov (Kaplan–Yorke) dimension (Dky) are indicated by red, blue, and black lines, respectively. Additional windows demonstrate representative phase portraits of the map (1) calculated for (i) a = 2.300 (period-2 solution), (ii) a = 2.400 (period-4 solution), (iii) a = 2.522 (period-8 solution), (iv) a = 2.542 (period-16 solution), (v) a = 2.550 (five-piece chaotic solution), (vi) a = 2.625 (period-6 solution), (vii) a = 2.634 (period-12 solution), (viii) a = 2.636 (six-piece chaotic solution), (ix) a = 2.726 (one-piece chaotic solution).