SRB Measures for A Class of Partially Hyperbolic Attractors in Hilbert spaces

In this paper, we study the existence of SRB measures and their properties for infinite dimensional dynamical systems in a Hilbert space. We show several results including (i) if the system has a partially hyperbolic attractor with nontrivial finite dimensional unstable directions, then it has at least one SRB measure; (ii) if the attractor is uniformly hyperbolic and the system is topological mixing and the splitting is H\"older continuous, then there exists a unique SRB measure which is mixing; (iii) if the attractor is uniformly hyperbolic and the system is non-wondering and and the splitting is H\"older continuous, then there exists at most finitely many SRB measures; (iv) for a given hyperbolic measure, there exist at most countably many ergodic components whose basin contains an observable set

Existence of SRB Measures for A Class of Partially Hyperbolic Attractors in Banach spaces

In this paper, we study the existence of SRB measures for infinite dimensional dynamical systems in a Banach space. We show that if the system has a partially hyperbolic attractor with nontrivial finite dimensional unstable directions, then it has an SRB measure.Comment: arXiv admin note: text overlap with arXiv:1508.0330

Symmetry restoration and quantumness reestablishment

A realistic quantum many-body system, characterized by a generic microscopic Hamiltonian, is accessible only through approximation methods. The mean field theories, as the simplest practices of approximation methods, commonly serve as a powerful tool, but unfortunately often violate the symmetry of the Hamiltonian. The conventional BCS theory, as an excellent mean field approach, violates the particle number conservation and completely erases quantumness characterized by concurrence and quantum discord between different modes. We restore the symmetry by using the projected BCS theory and the exact numerical solution and find that the lost quantumness is synchronously reestablished. We show that while entanglement remains unchanged with the particle numbers, quantum discord behaves as an extensive quantity with respect to the system size. Surprisingly, discord is hardly dependent on the interaction strengths. The new feature of discord offers promising applications in modern quantum technologies.Comment: 17 pages and 3 figure

Absolute continuity of stable foliations for systems on Banach spaces

AbstractWe prove the absolute continuity of stable foliations for C1,α maps of Banach spaces satisfying a globally defined infinitesimal invariant cones condition. Proofs of regularity for center and stable manifolds needed for the main theorem are included. Our results are applicable to dynamical systems generated by ordinary, partial, or functional differential equations, including non-autonomous differential equations that are periodic in time

Experimental and simulation study on nonlinear pitch control of Seagull underwater glider

1008-1015The Seagull underwater glider, developed by the Shanghai Jiao Tong University, is designed as a test-bed glider for the development and validation of various algorithms to enhance the glider’s long-term autonomy. In this paper, an adaptive backstepping control (ABC) method is proposed for the nonlinear pitch control of the underwater glider gliding in the vertical plane. The linear quadratic regulator (LQR) control and proportional-integral-derivative (PID) control are applied and evaluated with the ABC method to control a glider in saw-tooth motion. Simulation results demonstrate inherent effectiveness and superiority of the LQR or PID based method. According to Lyapunov stability theory, the ABC control scheme is derived to ensure the tracking errors asymptotically converge to zero. The ABC controller has been implemented on Seagull underwater glider, and verified in field experiments in the Qiandao Lake, Zhejiang