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Chaotic, Forced, and Coupled Oscillators
Published in LM Pismen, Working with Dynamical Systems, 2020
The parametric evolution of the attractor (for trajectories originating just above x = 1/2) at γ = 0.08 and increasing values of μ is shown in Fig. 4.16b. This particular line fy = const passes through the areas with 1, 2, and 0 fixed points as μ increases. In the area 1, the trajectories are attracted to the nearest fixed point. As the border of the area 2 is crossed, the change in dynamics is not straightforward. The entering unstable fixed points serve in the subarea 2c as domain boundaries between a chaotic attractor and a stable fixed point lying in an alternative half-interval, similar to the behavior of the Lorenz system following the homoclinic connection. As μ increases further, the extent of the chaotic attractor grows before it collapses and the fixed point in the alternative half-interval becomes the only attractor in the subarea 2a. Finally, after the locus of the saddle-node bifurcation is crossed, the large-amplitude chaotic attractor prevails in the area 0.
Nonlinear Dynamics
Published in Bogdan M. Wilamowski, J. David Irwin, Control and Mechatronics, 2018
When a pair of conjugate complex eigenvalues of a FP leaves the unit circle, the so-called Neimark-Sacker bifurcation occurs, which is a kind of local bifurcation. In that case, a closed curve looking like a limit cycle develops in the xd–xq plane corresponding to a quasi-periodic “orbit” in the state space. Looking at Figure 1.41b, a pair of conjugate complex eigenvalues of FP XII+ leaving the unit circle at the critical value μ3 can be detected. Indeed, a “limit cycle” develops, as can be seen from the global phase portrait shown in Figure 1.41f. The bifurcation diagram supports this finding but seemingly there is no difference in the bifurcation diagram before and after μ3. The explanation is that at the critical value μ2, a global bifurcation, the so-called homoclinic bifurcation of the saddle XII– takes place. In Figure 1.41c, the global structure of manifolds belonging to XII– is presented at the critical value μ2. It can be seen that one branch of unstable manifolds connects to the stable manifold of XII– creating a closed so-called “big” homoclinic connection denoted as Γμ2. The other unstable and stable branches of manifolds of XII– are directed to the stable XII+ and come from the unstable fixed point XI, respectively. Either increasing or decreasing the bifurcation parameter in respect to μ2, the homoclinic connection disappears It is a topologically unstable structure of the state space. At both sides of the homoclinic bifurcation, global phase portraits are calculated (see in Figure 1.41d and e, respectively.) Decreasing μ from μ2, the unstable manifold comes to the inner side of the stable manifold, and finally is directed to the stable FP XII+. However, when μ is increased over μ2, the unstable manifold runs to the other side of the stable manifold of XII– and connects to a stable “limit cycle” Lμ that surrounds all three FPs. So, in the region [μ2; μ3], two attractors, the stable “limit cycle” Lμ and the spiral node XII+ exist. When μ3 is reached, the spiral node XII+ becomes a spiral saddle, the only attractor remained is the “limit cycle” Lμ (Figure 1.41f).
Robustly shadowable chain transitive sets and hyperbolicity
Published in Dynamical Systems, 2018
Mohammad Reza Bagherzad, Keonhee Lee
Recently, Gan et al. [4] showed that if M is robustly shadowable for , then there is no singularity σ ∈ Sing(X) exhibiting homoclinic connection. Here the homoclinic connection is the closure of a orbit of a regular point which is contained in both the stable and the unstable manifolds of σ.