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Singular Perturbation and Chaos
Published in Wilfrid Perruquetti, Jean-Pierre Barbot, Chaos in Automatic Control, 2018
Another important concept of the theory of dynamical systems is the occurrence of bifurcations. Many dynamical systems have parameters appearing in their equations. When these parameter values (bifurcation values) are changed, one may observe modifications (bifurcations) of the qualitative structure of their solution flows. The local bifurcations of a system are studied by analyzing the vector field near an equilibrium point [9] (also see Chapter 2). Global bifurcations are related to the description of the global changes of flows when local analysis is not useful. In the global approach, particular trajectories are studied like homoclinic orbits (orbits connecting fixed points to themselves) or heteroclinic orbits (orbits connecting distinct fixed points).
Nonlinear Dynamics
Published in Bogdan M. Wilamowski, J. David Irwin, Control and Mechatronics, 2018
To obtain complete understanding of the global dynamics of nonlinear systems, the knowledge of invariant manifolds is absolutely crucial. The invariant manifolds or briefly the manifolds are borders in state space separating regions. A trajectory born in one region must remain in the same region as time evolves. The manifolds organize the state space. There are stable and unstable manifolds. They originate from saddle points. If the initial condition is on the manifold or subspace, the trajectory stays on the manifold. Homoclinic orbit is established when stable and unstable manifolds of a saddle point intersect. Heteroclinic orbit is established when stable and unstable manifolds from different saddle points intersect.
Bifurcation and Chaos
Published in Wai-Kai Chen, Feedback, Nonlinear, and Distributed Circuits, 2018
Michael Peter Kennedy, Vandenberghe Lieven
Chaos is associated with two characteristic connections of the stable and unstable manifolds. A homoclinic orbit (see Figure 14.9a) joins an isolated equilibrium point XQ to itself along its stable and unstable manifolds. A heteroclinic orbit (Figure 14.9b) joins two distinct equilibrium points, XQ1 and XQ2, along the unstable manifold of one and the stable manifold of the other.
Imaging phase plane models
Published in International Journal of Mathematical Education in Science and Technology, 2023
Richard F. Melka, Hashim A. Yousif
Case 1: When a < 0, there is a saddle point at x = a, y = a. We choose a = 1 to exemplify case. Also note that (0, 0) is an unstable node and there is a heteroclinic orbit connecting (0, 0) and (1, 1). A heteroclinic orbit is a path connecting two critical points. The easiest way to realise such an orbit is to zoom in the plane portrait. Figure 2 shows a zoom in phase-plane plot for the heteroclinic orbit; in addition, Figure 3 shows a zoomed-out plot (a global picture) of this case. The proof that such an orbit exists is outside of the scope of this paper, however, the interested reader should consult (Ranu, 2016) for the determination of a heteroclinic orbit in a second-order nonlinear differential equation. Below is a portion of the program’s output for this case.
Dynamic modeling and extended bifurcation analysis of flexible-link manipulator
Published in Mechanics Based Design of Structures and Machines, 2020
Omid Mehrjooee, Siavash Fathollahi Dehkordi, Moharam Habibnejad Korayem
The Melnikov analysis approach is one of the analytical methods for studying homoclinic and heteroclinic bifurcations. In this approach, the bifurcation conditions and the possibility of chaos are presented by considering the intersection between the stable and unstable system manifolds in Poincaré map. In homoclinic/heteroclinic bifurcations, changes in the system control parameter generate stable and unstable manifolds of homoclinic/heteroclinic orbits close to each other that eventually intersect one another. With this intersection, the horseshoe phenomenon occurs in the Poincaré map which represents a chaotic behavior. Melnikov analysis can be used as an analytical approach for predicting the chaotic behavior in nonlinear systems. When the perturbation terms are added to the system Hamiltonian equations, homoclinic/heteroclinic orbits of system are broken. Therefore, the possibility of intersection of manifolds exists that may appear as chaos. Melnikov analysis defines the condition of intersection between manifolds in the parametric space. According to the Poincaré-Bendixon theorem, if the Melnikov function has a simple zero, the corresponding system is expected to exhibit chaotic behavior (Wiggins 1988).