Strange attractors in a predator–prey system with non-monotonic response function and periodic perturbation

Figure 1.  Two parameter bifurcation diagram ($ \beta-\alpha $) of system (1) with initial condition $ \delta = 1.1 $, $ \lambda_0 = 0.01 $, $ \mu = 0.1 $, $ \alpha = 0.002 $, $ \beta = 0.25 $, $ \omega = 1 $, $ x = 1.816 $, and $ y = 1.434 $, while $ \varepsilon = 0 $. The curve labelled by $ \textsf{F}_1 $, and $ \textsf{F}_2 $ are the fold curves. The curves $ \textsf{H}_1 $ and $ \textsf{H}_3 $ are the Hopf bifurcation curves. The curve $ \textsf{H}_1 $ and $ \textsf{H}_3 $ are joint together at the point $ \textsf{H}_2 $. The curve plotted with a dashed line (labelled by $ \textsf{Hom} $) is the curve of Homoclinic bifurcations while the $ \textsf{FLC} $ curve is the fold of limit cycle bifurcations curve. The $ \textsf{Hom} $ curve coincides with some part of the $ \textsf{F}_2 $-curve which is labelled by $ \textsf{F}'_2 $

Figure 2.  In this figure we have plotted the magnification of the region $ \textsf{B} $ of Figure 1. We have indicated five regions on that diagram, i.e. $ \textsf{B}_1 $, $ \textsf{B}_2 $, $ \textsf{B}_3 $, $ \textsf{B}_4 $, and $ \textsf{B}_5 $. The diagrams on the second and third rows are the phase portraits in each of these regions. The four phase portraits for $ (\alpha, \beta) $ in $ \textsf{B}_1 $, $ \textsf{B}_2 $, $ \textsf{B}_5 $, and $ \textsf{B}_3 $, correspond to the Bogdanov-Takens bifurcations. The transition from phase portrait when $ (\beta, \alpha) \in \textsf{B}_3 $ to $ (\beta, \alpha) \in \textsf{B}_4 $, or when $ (\beta, \alpha) \in \textsf{B}_5 $ to $ (\beta, \alpha) \in \textsf{B}_4 $ corresponds to fold bifurcation of equilibrium of system (1) for $ \varepsilon = 0 $; the latter with the creation of orbit homoclinic to the degenerate equilibrium

Figure 3.  We have plotted the magnification of the region $ \textsf{A} $ of Figure 1. We have indicated seven regions on that diagram, i.e. $ \textsf{A}_j $, $ j = 1, 2, \ldots, 7 $. These regions are separated from each other by the bifurcation curves: $ \textsf{F}_2 $ — where a fold bifurcation occurs—, $ \textsf{H}_{1,3} $ — where a Hopf bifurcation occurs—, $ \textsf{Hom} $ — where a homoclinic bifurcation occurs —, and $ \textsf{FLC} $ — where a fold bifurcation of limit cycles occurs

Figure 4.  We have plotted seven diagrams which correspond to the phase portraits of system (1), for parameter value of $ (\beta, \alpha) $ in region: $ \textsf{A}_j $, $ j = 1,2, \ldots, 7 $ and $ \varepsilon = 0 $. The topological changes of these phase portraits when the parameter moves from $ \textsf{A}_1 $, to $ \textsf{A}_2 $, to $ \textsf{A}_3 $, and back, are in agreement with the scenario of Bautin (or degenerate Hopf) bifurcation

Figure 5.  A plot of a negative time attractor in the strobocospic map (4) (also known as the Poincaré section) showing evidence of the existence of a strange repeller (a strange negative-time attractor). The value of $ \varepsilon = 0.07 $, $ \delta = 1.1 $, $ \lambda_0 = 0.01 $, $ \mu = 0.1 $, $ \omega = 1 $, $ \alpha = 0.007 $, and $ \beta = 0.08 $. On the left diagram we have plotted the cross section of the strange repeller while on the right we have plotted the magnification of a part of the cross section, indicated by the box $ \textsf{K} $

Figure 6.  A comparison between the phase portrait of the stroboscopic map of System (2) for $ \varepsilon = 0 $ (diagrams in the first row) and $ \varepsilon = 0.07 $ (diagrams in the second column). The value of $ (\beta,\alpha) = (0.07,0.005) $ (for the diagrams in the most left), $ (\beta,\alpha) = (0.08,0.005) $ (for the diagrams in middle column), and $ (\beta,\alpha) = (0.09,0.005) $ (for the diagrams in the most right column)

Figure 7.  The chaotic transient behaviour in the neighborhood of the fold of limit cycles bifurcation