How general is the above route to chaos? To address this question, we now give three more numerical examples.
(i) Coupling forest trees with limited pollen availability [1-2]. The dynamics on its synchronous manifold is described by fd,β : [−1 + d, 1] → [−1 + d, 1] of the form
fd,β(x) =
x + 1 =: f1(x), if x ≤ 0,
−dxβ+1+ 1 =: f2,d,β(x) if x > 0.
(3)
Here d > 0 is a depletion coefficient and β ≥ 0 is the coupling strength. Since f2,d,β(x) is concave down for x > 0, it behaves like a quadratic map. Like gµ(x), fd,β(x) consists of two part, one part, which is concave, has a tendency of producing period-doubling cascade to chaos while another part, which is non-concave, produces only simple dynamics. It is expected that fd,β(x) should have a finite period-doubling cascade to chaos. This is supported by our numerical computation, see Table 3. In Table 3, β is fixed to be 1, we let d increase from d0≈ 0.75 to d7≈ 1.1628237. The corresponding map undertakes a finite period-doubling. As d moves past d7, a chaotic attractor occurs. We remark that such finite period-doubling route to chaos holds true for any arbitrary fixed β > 0.
0 < d < d0
d > d7 chaotic attractor
Table 3: Bifurcation values for fd,1. At d > d7, the bifurcation comes a sudden end and a chaotic attractor appears.
(ii) An impact oscillator. In [27], a method for deriving the global form of the stroboscopic map for the impact oscillator which considers the linear dynam-ics on either side of the grazing bifurcation was presented. The corresponding regularized discontinuous map has the following form [27]:
f (x; λ1, λ2, µ, ) =
The graphs of f , f2 and f4 for λ1 = 0.7, λ2 = −0.9, = 1 and µ = 1.2 are shown in Fig 6.
Figure 6: The graphs of f , f2 and f4 for λ1 = 0.7, λ2 = −0.9, µ = 1.2 and
= 1.
One part of the graph y = f1(x) is a line segment with slope 0.7 which is nonconvex and yields only simple dynamics. The remaining part of the graph, particularly near the turning point x = 0, is described by a square root map, which is convex and capable of generating a chaos set without transition. Such map has a stable period two orbit and no chaotic set. As µ decreases, say, to µ = 0.795, the stable period two orbit is still preserved. However, a chaotic set, a cantor set of measure zero, is created, see Fig 7.
Figure 7: The graphs of (a) f , f2 and (b) f4, f8 for λ1 = 0.7, λ2 = −0.9, µ = 0.795 and = 1.
Note that in Fig 7(b), f8 extends out of the box. Therefore, an invariant cantor set whose dynamics is conjugate to the shift map of two symbols is generated. In real applications, we are more interested in finding an attractor.
As µ keeps decreasing, the stable period two and the chaotic set of measure zero remain coexisted until µ reaches around 0.772891. By then, the period two orbit becomes unstable and f4 extends out of the box B2 and still stays in the box B1, see Fig 8.
Figure 8: The graphs of f , f2 and f4 for λ1 = 0.7, λ2 = −0.9, µ = 0.772 and
= 1.
Consequently, a chaotic attractor is born. This completes a finite period doubling route to chaos, which is summarized in Table 4.
0.80523 < µ < 1.2 stable period 2 0.772891 < µ < 0.805232 stable period 2 + chaotic set
0.7 < µ < 0.772891 chaotic attractor
Table 4: Bifurcation values for the piecewise linear map f ( · ; 0.7, −0.9, µ, 1).
For smaller , the corresponding f also exhibits a similar route to chaos.
The numerical computation of f (x; 0.7, −0.9, µ, λ) as µ varies from 0.2 to 0.155 is summarized in Table 5.
µ0< µ < 0.2 (µ0≈ 0.1613646)
stable period 4 point + chaotic set
µ1< µ < µ0 (µ1≈ 0.1580433)
stable period 8 point + chaotic set
0.155 < µ < µ1 chaotic attractor
Table 5: Bifurcation values for f ( · ; 0.7, −0.9, µ, 0.07). As µ decreases from 0.2 to 0.155, the bifurcation comes a sudden end and a chaotic attractors appears.
(iii) Friction oscillator and DC-DC buck converter (see [44]) Even simpler than square-root maps are those that are completely linear in each of two halves of their domain. Maps of this form can be used to explain the dynamics observed in the friction oscillator and DC-DC converter case studies (see e.g., [44]). Those maps, without loss of generality, can be written in the form
f (x) =
f1(x) = αx + µ if x ≤ 0, f2(x) = βx + µ, if x > 0.
(5)
The most interesting dynamics occurs for α > 0 and β < 0. Indeed, we let µ = 1, α = 0.4 and let β vary from −6 to −6.4. The system undergoes the finite period-doubling. See Table 6 and Fig 9.
Figure 9: The graph of f6for β = −6.1, β = −6.25 and β = −6.4, respectively.
Fig 9(a) is the box where f6|[−0.5,0]with β = −6.1 stays. Stable period three point of f is situated at the lower left corner. As β decreases to −6.25, a portion of the graph of f6 is coincide with the diagonal. Consequently, a stable period 6 is born. When β decreases past −6.25, the slope of both pieces of segments of f6 inside the box have absolute values greater than 1. Hence, like tent map, the chaotic dynamic instantly begins. If one further decreases the value of β, then the period-adding bifurcations also occur.
−6.25 < β < −6 stable period 3 + chaotic set β = −6.25 stable period 6 + chaotic set
−6.4 < β < −6.25 chaotic attractor
Table 6: Bifurcation values for the piecewise linear map f .
5 Conclusion
Satake and Iwasa proved that the generalized budget resource model is chaotic when d > 1 by computing the Lyapunov exponent. In [3], the model was shown to have Devaney’s chaos on an invariant set by proving its topological entropy is positive for d > 1.00026. In this thesis, we clearly point out that the generalized resource budget model is chaotic in the sense of Devaney as the depletion coefficient d > 1 on an invariant set.
The second part of thesis, we present a finite period-doubling route to chaos for a class of nonsmooth maps. Those maps are piecewise smooth functions, which consists of nonconvex and nonconcave parts. Each part may generate a certain type of dynamics as a parameter of the system changes so that when combined together a finite period doubling route to chaos is created. For in-stance, it could have that one piece of the function, as the system parameter varies, tends to chaos through period-doubling cascades while the other piece produces chaos without transition. The maps defined in (1) and (2) fit into the combination described above. Both maps (4) and (5) has a nonconcave
piece, which is capable of generating chaos without transition, and a noncon-vex part yielding only simple dynamics. The third possibility comes from map (3), for which its nonconvex part’s route to chaos is through period-doubling.
Note that its nonconcave piece produces a simple dynamics. The competition between these two pieces seems to be the mechanism for producing a finite period-doubling route to chaos. No one seems to win out. The numerical com-putation seems also suggest that the finite period doubling route to chaos for nonsmooth maps is generic.
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