Y. Charles Li

Chaotic dynamics of spin-valve oscillators

Recent experimental and theoretical studies on the magnetization dynamics driven by an electric current have uncovered a number of unprecedented rich dynamic phenomena. We predict an intrinsic chaotic dynamics that has not been previously anticipated. We explicitly show that the transition to chaotic dynamics occurs through a series of period doubling bifurcations. In chaotic regime, two dramatically different power spectra, one with a well-defined peak and the other with a broadly distributed noise, are identified and explained.

Homoclinic tubes and chaos in perturbed sine-Gordon equation

Abstract Sine-Gordon equation under a quasi-periodic perturbation or a chaotic perturbation is studied. Existence of a homoclinic tube is proved. Established are chaos associated with the homoclinic tube, and “chaos cascade” referring to the embeddings of smaller scale chaos in larger scale chaos.

A Resolution of the Sommerfeld Paradox

Sommerfeld paradox roughly says that mathematically Couette linear shear is linearly stable for all Reynolds number, but experimentally arbitrarily small perturbations can induce the transition from the linear shear to turbulence when the Reynolds number is large enough. The main idea of our resolution of this paradox is to show that there is a sequence of linearly unstable shears which approaches the linear shear in the kinetic energy norm but not in the enstrophy (vorticity) norm. These oscillatory shears are single Fourier modes in the Fourier series of all the shears. In experiments, such linear instabilities will manifest themselves as transient nonlinear growth leading to the transition from the linear shear to turbulence no matter how small the intitial perturbations to the linear shear are. Under the Euler dynamics, these oscillatory shears are steady, and cats eye structures bifurcate from them as travelling waves. The 3D shears

CHAOS IN WAVE INTERACTIONS

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Chaotic spin dynamics of a long nanomagnet driven by a current

in a neighborhood of these oscillatory shears are linearly unstable too. Under the Navier-Stokes dynamics, these oscillatory shears are not steady rather drifting slowly. When these oscillatory shears are viewed as frozen, the corresponding Orr-Sommerfeld operator has unstable eigenvalues which approach the corresponding inviscid eigenvalues when the Reynolds number tends to infinity. All the linear instabilities mentioned above offer a resolution to the Sommerfeld paradox, and an initiator for the transition from the linear shear to turbulence.

Isospectral theory of Euler equations

Existence of chaos is proved in finite-dimensional invariant subspaces for both two- and three-wave interactions. For a simple Galerkin truncation of the 2D Navier–Stokes equation, existence of chaos is also proved.

Stability criteria and turbulence paradox problem for type II 3D shears

We study the spin dynamics of a long nanomagnet driven by an electrical current. In the case of only DC current, the spin dynamics has a sophisticated bifurcation diagram of attractors. One type of attractors is a weak chaos. On the other hand, in the case of only AC current, the spin dynamics has a rather simple bifurcation diagram of attractors. That is, for small Gilbert damping, when the AC current is below a critical value, the attractor is a limit cycle; above the critical value, the attractor is chaotic (turbulent). For normal Gilbert damping, the attractor is always a limit cycle in the physically interesting range of the AC current. We also developed a Melnikov integral theory for a theoretical prediction on the occurrence of chaos. Our Melnikov prediction seems performing quite well in the DC case. In the AC case, our Melnikov prediction seems predicting transient chaos. The sustained chaotic attractor seems to have extra support from parametric resonance leading to a turbulent state.

ON THE CHAOTIC FLUX DYNAMICS IN A LONG JOSEPHSON JUNCTION

Isospectral problem of both 2D and 3D Euler equations of inviscid fluids, is investigated. Connections with the Clay problem are described. Spectral theorem of the Lax pair is studied.

A mathematical model of demand-supply dynamics with collectability and saturation factors

There are two types of 3D shears in channel flows: (

Recurrence in 2D inviscid channel flow

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