Polynomial and linearized normal forms for almost periodic differential systems

Agraïments: The first author is partially supported by NSFC key program of China (no. 11231001). The MINECO/FEDER grant UNAB13-4E-1604. And the third is supported by NSFC for Young Scientists of China (no. 11001047) and NSF of Jiangsu, China (no. BK20131285).For almost periodic differential systems ˙x = εf(x, t, ε) with x ∈ Cn, t ∈ R and ε > 0 small enough, we get a polynomial normal form in a neighborhood of a hyperbolic singular point of the system ˙x = ε limT→∞1T∫ T0f(x,t, 0) dt, if its eigenvalues are in the Poincaré domain. The normal form linearizes if the real part of the eigenvalues are non-resonant

Polynomial systems : a lower bound for the weakened 16th Hilbert problem

In this paper we provide the greatest lower bound about the number of (non-infinitesimal) limit cycles surrounding a unique singular point for a planar polynomial differential system of arbitrary degree

Reduction of periodic difference systems to linear or autonomous ones

Agraïments: The first author is partially supported by NSF grants no. 10531010 and NNSF of China (no. 10525104).We extend Floquet theory for reducing nonlinear periodic difference systems to autonomous ones (actually linear) by using normal form theor

Reduction of periodic difference systems to linear or autonomous ones

We extend Floquet theory for reducing nonlinear periodic difference systems to autonomous ones (actually linear) by using normal form theory

Poincaré type theorems for non-autonomous systems

AbstractIn this paper we establish analytic equivalence theorems of Poincaré and Poincaré–Dulac type for analytic non-autonomous differential systems based on the dichotomy spectrum of their linear part. As applications of the theorem, normal forms linearize for two illustrative examples

Structural design and manufacturing of three-dimensional monolithic porous superstructures for flexible self-powered electronics, energy conversion and storage devices, and water purification systems

Three dimensional (3D) porous superstructures with designed hierarchy have received intensive research interest due to the grand potencies in applications of energy devices, environment remediation, and flexible electronics. Three dimensional metallic foams or polymer sponges have been utilized as catalytic substrates or templates for the growth of a variety of functional 3D materials with replicated morphologies, including 3D foams made of the unique class of 2D materials (graphene, molybdenum disulfide), metal and metal oxides/hydroxides, as well as polymers. Here, we developed a rational approach to create innovative nickel foams with hierarchical porosity to substantially enhance the specific surface area by up to 3 folds compared to that of commercial nickel foam. The resulted 3D graphite foams (GF) provide enhanced electrochemical performances when applied as supercapacitor electrode supports. The as-fabricated flexible supercapacitor can be readily integrated with our wearable GF/polymer strain sensors or nanomotor manipulation systems as self-powered devices, which can detect both large and small motion induced strains, or compel nanomotors to trace letters, such as “U” and “T”, respectively. After sulfurizing and electrochemically activate the surface of the unique nickel foams with dendritic microstructures, we obtained ternary metal oxyhydroxides that are utilized as high-performance and stable oxygen evolution reaction catalysts for water splitting. We also exploited electrostatic assembly to achieve 3D composite foams with hydrothermally synthesized MoS₂/C microbeads attached on polyurethane sponges for high-rate solar-steaming water treatment with synergistic mercury removal. Finally, we achieved one of the best solar-steaming performances in both energy-conversion efficiency and evaporation rate via strategical origami folding of photothermal thin films into 3D-structured roses that exhibit optimized performances in light absorption and water evaporation. Moreover, a portable low-pressure water purification-collection uni-system was developed that can remarkably enhance the efficiency of water collection, which is the first to the best of our knowledge. This dissertation work, exploring innovative structural design and manufacturing paradigms of 3D monolithic porous superstructures has made an important forward step in the interdisciplinary field of nanomanufacturing, 3D porous materials, flexible energy storage systems, self-powered devices, and solar water treatment.Mechanical Engineerin

Isochronous properties in fractal analysis of some planar vector fields

AbstractWe extend the classical Belitskii normal form theorem uniformly for planar nilpotent foci and limit cycles. Then by using fractal analysis, isochronous properties near such invariant sets are well investigated

Polynomial and linearized normal forms for almost periodic differential systems

Agraïments: The first author is partially supported by NSFC key program of China (no. 11231001). The MINECO/FEDER grant UNAB13-4E-1604. And the third is supported by NSFC for Young Scientists of China (no. 11001047) and NSF of Jiangsu, China (no. BK20131285).For almost periodic differential systems ˙x = εf(x, t, ε) with x ∈ Cn, t ∈ R and ε > 0 small enough, we get a polynomial normal form in a neighborhood of a hyperbolic singular point of the system ˙x = ε limT→∞1T∫ T0f(x,t, 0) dt, if its eigenvalues are in the Poincaré domain. The normal form linearizes if the real part of the eigenvalues are non-resonant