6. Chaos

Discrete-time systems can display chaotic behaviour at low dimensions. One example is the well-known logistic model which describes the dynamics of a single species with nonoverlapping generations and intraspecific competition [6]. This system is known to undergo the so-called Feigenbaum (period-doubling) route to chaos [15]. In order to identify the chaotic regions in map (1), we compute the full spectrum of Lyapunov exponents using the algorithm described in Ref. [16], pp. 74–80. Figure 8(a) displays a bifurcation diagram obtained by iteration for increasing values of β . Notice that the fixed point P<sup>∗</sup> <sup>3</sup> becomes unstable and a

#### Figure 9.

(A) Enlarged view of the framed region in grey colour in Figure 8(c) displaying the first Lyapunov exponent, Λ1, in the parameter region (μ, β ∈ ½ � 2:5, 4 ). In the orange-red regions, the dynamics are chaotic with Λ<sup>1</sup> >0: (B) Second Lyapunov exponent, Λ2, within the range μ ∈½ � 2:8, 4 and β ∈ ½ � 3, 4 . The orange-red regions correspond to the hyperchaotic regimes since Λ<sup>1</sup>,<sup>2</sup> >0. Lower row of pictures: four plots of the set ∩∞ <sup>i</sup>¼0T<sup>i</sup> ð ÞS found in the regions labelled with the white numbers in panel (A), period-6 fixed point (a), using μ ¼ 3:25, β ¼ 3:25, and three examples of strange chaotic attractors, (b) μ ¼ 3:7, β ¼ 3:2, (c) μ ¼ 3:8, β ¼ 3:5, and (d) <sup>μ</sup> <sup>¼</sup> <sup>3</sup>:7, <sup>β</sup> <sup>¼</sup> <sup>3</sup>:95. In all of the phase portraits, we plot the fixed points P<sup>∗</sup> <sup>1</sup> (red), P<sup>∗</sup> <sup>2</sup> (blue), and P<sup>∗</sup> 3 (orange). See movie3.mp4 to visualise the dynamics of Eq. (1) for increasing parameter μ and setting β ¼ 3:9.

Neimark-Sacker bifurcation takes place. This bifurcation has been detected with the Lyapunov exponents shown in Figure 8(b), with Λ1,2 ¼ 0 at the bifurcation value. After this bifurcation the first Lyapunov exponent is 0 and the second one is negative. Then the dynamics are governed by attracting invariant curves; further increase of β involves the entry into the chaotic regime, where the first Lyapunov exponent, Λ<sup>1</sup> (in black), becomes positive. Notice the presence of hyperchaotic attractors, with Λ1,2 >0.

Enlarged views of the Lyapunov exponents in the parameter space ð Þ μ, β are represented in Figure 9, as well as four examples of the sets ∩<sup>∞</sup> <sup>i</sup>¼0T<sup>i</sup> ð ÞS found in the regions labelled with letters in Figure 9(A).
