Low-dimensional Galerkin approximations of nonlinear delay differential equations
Mickaël D. Chekroun Michael Ghil Honghu Liu Shouhong Wang
Discrete & Continuous Dynamical Systems - A 2016, 36(8): 4133-4177 doi: 10.3934/dcds.2016.36.4133
This article revisits the approximation problem of systems of nonlinear delay differential equations (DDEs) by a set of ordinary differential equations (ODEs). We work in Hilbert spaces endowed with a natural inner product including a point mass, and introduce polynomials orthogonal with respect to such an inner product that live in the domain of the linear operator associated with the underlying DDE. These polynomials are then used to design a general Galerkin scheme for which we derive rigorous convergence results and show that it can be numerically implemented via simple analytic formulas. The scheme so obtained is applied to three nonlinear DDEs, two autonomous and one forced: (i) a simple DDE with distributed delays whose solutions recall Brownian motion; (ii) a DDE with a discrete delay that exhibits bimodal and chaotic dynamics; and (iii) a periodically forced DDE with two discrete delays arising in climate dynamics. In all three cases, the Galerkin scheme introduced in this article provides a good approximation by low-dimensional ODE systems of the DDE's strange attractor, as well as of the statistical features that characterize its nonlinear dynamics.
keywords: Koornwinder polynomials El Niño--Southern Oscillation (ENSO). inner product with a point mass orthogonal polynomials ``nearly-Brownian'' chaotic dynamics distributed delays Galerkin approximation
Global structure of 2-D incompressible flows
Tian Ma Shouhong Wang
Discrete & Continuous Dynamical Systems - A 2001, 7(2): 431-445 doi: 10.3934/dcds.2001.7.431
The main objective of this article is to classify the structure of divergence-free vector fields on general two-dimensional compact manifold with or without boundaries. First we prove a Limit Set Theorem, Theorem 2.1, a generalized version of the Poincaré-Bendixson to divergence-free vector fields on 2-manifolds of nonzero genus. Namely, the $\omega$ (or $\alpha$) limit set of a regular point of a regular divergence-free vector field is either a saddle point, or a closed orbit, or a closed domain with boundaries consisting of saddle connections. We call the closed domain ergodic set. Then the ergodic set is fully characterized in Theorem 4.1 and Theorem 5.1. Finally, we obtain a global structural classification theorem (Theorem 3.1), which amounts to saying that the phase structure of a regular divergence-free vector field consists of finite union of circle cells, circle bands, ergodic sets and saddle connections.
keywords: Compact manifold global structural classification theorem. Limit Set Theorem
Tropical atmospheric circulations: Dynamic stability and transitions
Tian Ma Shouhong Wang
Discrete & Continuous Dynamical Systems - A 2010, 26(4): 1399-1417 doi: 10.3934/dcds.2010.26.1399
In this article, we present a mathematical theory of the Walker circulation of the large-scale atmosphere over the tropics. This study leads to a new metastable state oscillation theory for the El Niño Southern Oscillation (ENSO), a typical inter-annual climate low frequency oscillation. The mathematical analysis is based on 1) the dynamic transition theory, 2) the geometric theory of incompressible flows, and 3) the scaling law for proper effect of the turbulent friction terms, developed recently by the authors.
keywords: tropical atmospheric circulation dynamic transition Walker circulation metastable states. new mechanism of ENSO El Niño Southern Oscillation (ENSO)
Thermodynamical potentials of classical and quantum systems
Ruikuan Liu Tian Ma Shouhong Wang Jiayan Yang
Discrete & Continuous Dynamical Systems - B 2017, 22(11): 1-38 doi: 10.3934/dcdsb.2018214

The aim of the paper is to systematically introduce thermodynamic potentials for thermodynamic systems and Hamiltonian energies for quantum systems of condensates. The study is based on the rich previous work done by pioneers in the related fields. The main ingredients of the study consist of 1) SO(3) symmetry of thermodynamical potentials, 2) theory of fundamental interaction of particles, 3) the statistical theory of heat developed recently [23], 4) quantum rules for condensates that we postulate in Quantum Rule 4.1, and 5) the dynamical transition theory developed by Ma and Wang [20]. The statistical and quantum systems we study in this paper include conventional thermodynamic systems, thermodynamic systems of condensates, as well as quantum condensate systems. The potentials and Hamiltonian energies that we derive are based on first principles, and no mean-field theoretic expansions are used.

keywords: Potential-descending principle order parameters control parameters SO(3) symmetry spinor representation PVT system N-component system magnetic system dielectric system Bose-Einstein condensates (BEC) superfluid superconductor liquid helium-4 liquid helium-3
Unified field equations coupling four forces and principle of interaction dynamics
Tian Ma Shouhong Wang
Discrete & Continuous Dynamical Systems - A 2015, 35(3): 1103-1138 doi: 10.3934/dcds.2015.35.1103
The main objective of this article is to postulate a principle of interaction dynamics (PID) and to derive field equations coupling the four fundamental interactions based on first principles. PID is a least action principle subject to div$_A$-free constraints for the variational element with $A$ being gauge potentials. The Lagrangian action is uniquely determined by 1) the principle of general relativity, 2) the $U(1)$, $SU(2)$ and $SU(3)$ gauge invariances, 3) the Lorentz invariance, and 4) the principle of representation invariance (PRI), introduced in [11]. The unified field equations are then derived using PID. The field model spontaneously breaks the gauge symmetries, and gives rise to a new mass generation mechanism. The unified field model introduces a natural duality between the mediators and their dual mediators, and can be easily decoupled to study each individual interaction when other interactions are negligible. The unified field model, together with PRI and PID applied to individual interactions, provides clear explanations and solutions to a number of outstanding challenges in physics and cosmology, including e.g. the dark energy and dark matter phenomena, the quark confinement, asymptotic freedom, short-range nature of both strong and weak interactions, decay mechanism of sub-atomic particles, baryon asymmetry, and the solar neutrino problem.
keywords: unified field equations dual particle fields solar neutrino problem Higgs mechanism dark energy short-range nature of strong and weak interactions baryon asymmetry. Principle of interaction dynamics (PID) fundamental interactions asymptotic freedom quark confinement principle of representation invariance gauge symmetry breaking dark matter
Structure of 2D incompressible flows with the Dirichlet boundary conditions
Tian Ma Shouhong Wang
Discrete & Continuous Dynamical Systems - B 2001, 1(1): 29-41 doi: 10.3934/dcdsb.2001.1.29
We study in this article the structure and its stability of 2-D divergence-free vector fields with the Dirichlet boundary conditions. First we classify boundary points into two new categories: $\partial$−singular points and $\partial$−regular points, and establish an explicit formulation of divergence-free vector fields near the boundary. Second, local orbit structure near the boundary is classified. Then a structural stability theorem for divergence-free vector fields with the Dirichlet boundary conditions is obtained, providing necessary and sufficient conditions of a divergence-free vector fields. These structurally stability conditions are extremely easy to verify, and examples on stability of typical flow patterns are given.
The main motivation of this article is to provide an important step for a forthcoming paper, where, for the first time, we are able to establish precise rigorous criteria on boundary layer separations of incompressible fluid flows, a long standing problem in fluid mechanics.
keywords: structural stability Divergence-free vector fields Dirichlet boundary conditions.
Asymptotic structure for solutions of the Navier--Stokes equations
Tian Ma Shouhong Wang
Discrete & Continuous Dynamical Systems - A 2004, 11(1): 189-204 doi: 10.3934/dcds.2004.11.189
We study in this article the large time asymptotic structural stability and structural evolution in the physical space for the solutions of the 2-D Navier-Stokes equations with the periodic boundary conditions. Both the Hamiltonian and block structural stabilities and structural evolutions are considered, and connections to the Lyapunov stability are also given.
keywords: divergence-free vector fields Navier-Stokes equations Lyapunov stability. periodic boundary conditions structural stability block stability Hamiltonian stability
Boundary layer separation and structural bifurcation for 2-D incompressible fluid flows
Tian Ma Shouhong Wang
Discrete & Continuous Dynamical Systems - A 2004, 10(1&2): 459-472 doi: 10.3934/dcds.2004.10.459
The main objective of this article and the previous articles [2, 3, 7] is to provide a rigorous characterization of the boundary layer separation of 2-D incompressible viscous fluids. First we establish a simple equation linking the separation location and time with the Reynolds number, the external forcing the boundary curvature, and the initial velocity field. Second, we show that external forcing with reverse orientation to the initial velocity field leads to structural bifurcation at a degenerate singular point with integer index of the velocity field at the critical bifurcation time. Necessary and sufficient kinematic conditions are given to identify the case for boundary layer separation.
keywords: location and time of separation kinematic conditions. Navier-Stokes equations structural bifurcation boundary layer separation
Cahn-Hilliard equations and phase transition dynamics for binary systems
Tian Ma Shouhong Wang
Discrete & Continuous Dynamical Systems - B 2009, 11(3): 741-784 doi: 10.3934/dcdsb.2009.11.741
The process of phase separation of binary systems is described by the Cahn-Hilliard equation. The main objective of this article is to give a classification on the dynamic phase transitions for binary systems using either the classical Cahn-Hilliard equation or the Cahn-Hilliard equation coupled with entropy, leading to some interesting physical predictions. The analysis is based on dynamic transition theory for nonlinear systems and new classification scheme for dynamic transitions, developed recently by the authors.
keywords: dynamic transition theory unified time-dependent Ginzburg-Landau equation. binary system Cahn-Hilliard equation phase transition dynamics dynamic classification scheme of phase transitions
Gravitational Field Equations and Theory of Dark Matter and Dark Energy
Tian Ma Shouhong Wang
Discrete & Continuous Dynamical Systems - A 2014, 34(2): 335-366 doi: 10.3934/dcds.2014.34.335
The main objective of this article is to derive new gravitational field equations and to establish a unified theory for dark energy and dark matter. The gravitational field equations with a scalar potential $\varphi$ function are derived using the Einstein-Hilbert functional, and the scalar potential $\varphi$ is a natural outcome of the divergence-free constraint of the variational elements. Gravitation is now described by the Riemannian metric $g_{\mu\nu}$, the scalar potential $\varphi$ and their interactions, unified by the new field equations. From quantum field theoretic point of view, the vector field $\Phi_\mu=D_\mu \varphi$, the gradient of the scalar function $\varphi$, is a spin-1 massless bosonic particle field. The field equations induce a natural duality between the graviton (spin-2 massless bosonic particle) and this spin-1 massless bosonic particle. Both particles can be considered as gravitational force carriers, and as they are massless, the induced forces are long-range forces. The (nonlinear) interaction between these bosonic particle fields leads to a unified theory for dark energy and dark matter. Also, associated with the scalar potential $\varphi$ is the scalar potential energy density $\frac{c^4}{8\pi G} \Phi=\frac{c^4}{8\pi G} g^{\mu\nu}D_\mu D_\nu \varphi$, which represents a new type of energy caused by the non-uniform distribution of matter in the universe. The negative part of this potential energy density produces attraction, and the positive part produces repelling force. This potential energy density is conserved with mean zero: $\int_M \Phi dM=0$. The sum of this potential energy density $\frac{c^4}{8\pi G} \Phi$ and the coupling energy between the energy-momentum tensor $T_{\mu\nu}$ and the scalar potential field $\varphi$ gives rise to a unified theory for dark matter and dark energy: The negative part of this sum represents the dark matter, which produces attraction, and the positive part represents the dark energy, which drives the acceleration of expanding galaxies. In addition, the scalar curvature of space-time obeys $R=\frac{8\pi G}{c^4} T + \Phi$. Furthermore, the proposed field equations resolve a few difficulties encountered by the classical Einstein field equations.
keywords: gravitational eld equations spin-1 massless particle spin-2 massless graviton gravitational force formula. scalar potential energy dark matter Dark energy

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