Newhouse Laminations of polynomials on ℂ2

It has been recently discovered that in smooth unfoldings of maps with a rank-one homoclinic tangency there are codimension two laminations of maps with infinitely many sinks. Indeed, these laminations, called Newhouse laminations, occur also in the holomorphic context. In the space of polynomials of [Formula: see text], with bounded degree, there are Newhouse laminations.

The onset of transient turbulence in minimal plane Couette flow

The onset of transient turbulence in minimal plane Couette flow has been identified theoretically as homoclinic tangency with respect to a simple edge state for the Navier–Stokes equation, i.e., the gentle periodic orbit (the lower branch of a saddle-node pair) found by Kawahara & Kida (J. Fluid Mech., vol. 449, 2001, pp. 291–300). The first tangency of a pair of distinct homoclinic orbits to this periodic edge state has been discovered at Reynolds number $Re\equiv Uh/\unicode[STIX]{x1D708}=Re_{T}\approx 240.88$ ($U$, $h$, and $\unicode[STIX]{x1D708}$ being half the difference of the two wall velocities, half the wall separation, and the kinematic viscosity of fluid, respectively). At $Re>Re_{T}$ a Smale horseshoe appears on the Poincaré section through transversal homoclinic points to generate a transient chaos that eventually relaminarises. In numerical experiments a sustaining chaos, which is a consequence of period-doubling cascade stemming from the upper branch of another saddle-node pair of periodic orbits, is observed in a narrow range of the Reynolds number, $Re\approx 240.40$–240.46. At the upper edge of this $Re$ range it is found that the chaotic set touches the lower branch of this pair, i.e., another edge state. The corresponding chaotic attractor is replaced by a chaotic saddle at $Re\approx 240.46$, and subsequently this saddle touches the gentle periodic edge state on the boundary of the laminar basin at the tangency Reynolds number $Re=Re_{T}$. After this crisis on the boundary of the laminar basin, for $Re>Re_{T}$, chaotic transients that eventually relaminarise can be observed.

Hamiltonian suspension of perturbed Poincaré sections and an application

We construct a Hamiltonian suspension for a given symplectomorphism which is the perturbation of a Poincaré map. This is especially useful for the conversion of perturbative results between symplectomorphisms and Hamiltonian flows in any dimension 2d. As an application, using known properties of area-preserving maps, we prove that for any Hamiltonian defined on a symplectic 4-manifold M and any point p ∈ M, there exists a C2-close Hamiltonian whose regular energy surface through p is either Anosov or contains a homoclinic tangency.

Homoclinic Tangencies for a Rotor-Active Magnetic Bearings System

The homoclinic tangency for a rotor-active magnetic bearings (AMB) system with the time-varying stiffness are considered in this paper. The zeros of Melnikov equation are paid more attentions and a 3-order zero was gained and some numerical results under the parameter perturbations were shown.

LOCAL AND GLOBAL BIFURCATIONS IN THREE-DIMENSIONAL, CONTINUOUS, PIECEWISE SMOOTH MAPS

In this work, we study the dynamics of a three-dimensional, continuous, piecewise smooth map. Much of the nontrivial dynamics of this map occur when its fixed point or periodic orbit hits the switching manifold resulting in the so-called border collision bifurcation. We study the local and global bifurcation phenomena resulting from such borderline collisions. The conditions for the occurrence of nonsmooth period-doubling, saddle-node, and Neimark–Sacker bifurcations are derived. We show that dangerous border collision bifurcation can also occur in this map. Global bifurcations arise in connection with the occurrence of nonsmooth Neimark–Sacker bifurcation by which a spiral attractor turns into a saddle focus. The global dynamics are systematically explored through the computation of resonance tongues and numerical continuation of mode-locked invariant circles. We demonstrate the transition to chaos through the breakdown of mode-locked torus by degenerate period-doubling bifurcation, homoclinic tangency, etc. We show that in this map a mode-locked torus can be transformed into a quasiperiodic torus if there is no global bifurcation.