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Stochastic rotational stability of tower cranes under gusty winds
Published in Alphose Zingoni, Insights and Innovations in Structural Engineering, Mechanics and Computation, 2016
The unperturbed system describes a closed trajectory of constant energy H called a homoclinic orbit. This motion presents a period () T=2∫q1q2dqq˙=2∫−2H2H12H−q2dq=2π.
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.
Chaos in Systems with Time Variable Parameters
Published in L. Cveticanin, Dynamics of Machines with Variable Mass, 2022
It is known that chaos appears if the homoclinic orbit exists after perturbation. It is shown by Wiggins [1], that the manifold of the unperturbed system persists under general sufficiently differentiable perturbations. For the appearance of chaos it is necessary that the stable and unstable manifolds intersect transversely (5.1.10). Melnikov introduced the distance measurement along the direction ∂H0/∂pθ and ∂H0/∂pρ as follows () d(p,μ)=[∂H0∂pθ(pθμu−pθμs)+∂H∂pρ(pρμu−pρμs)]/‖∂H0∂pθ,∂H0∂pρ‖,
Criteria on existence of horseshoes near homoclinic tangencies of arbitrary orders
Published in Dynamical Systems, 2018
Sergey Gonchenko, Ming-Chia Li, Mikhail Malkin
Homoclinic orbit, or Poincaré homoclinic orbit, is an orbit that is bi-asymptotic to a saddle periodic trajectory. By its definition, any homoclinic orbit belongs to the intersection Ws∩Wu of the stable and unstable manifolds of the corresponding periodic orbit. Depending on whether this intersection is transverse or non-transverse, the homoclinic orbit is called transverse or non-transverse, respectively. The latter case is called also the homoclinic tangency.