Jacobian matrix: a bridge between linear and nonlinear polynomial-only problems

By using the Hadamard matrix product concept, this paper introduces two generalized matrix formulation forms of numerical analogue of nonlinear differential operators. The SJT matrix-vector product approach is found to be a simple, efficient and accurate technique in the calculation of the Jacobian matrix of the nonlinear discretization by finite difference, finite volume, collocation, dual reciprocity BEM or radial functions based numerical methods. We also present and prove simple underlying relationship (theorem (3.1)) between general nonlinear analogue polynomials and their corresponding Jacobian matrices, which forms the basis of this paper. By means of theorem 3.1, stability analysis of numerical solutions of nonlinear initial value problems can be easily handled based on the well-known results for linear problems. Theorem 3.1 also leads naturally to the straightforward extension of various linear iterative algorithms such as the SOR, Gauss-Seidel and Jacobi methods to nonlinear algebraic equations. Since an exact alternative of the quasi-Newton equation is established via theorem 3.1, we derive a modified BFGS quasi-Newton method. A simple formula is also given to examine the deviation between the approximate and exact Jacobian matrices. Furthermore, in order to avoid the evaluation of the Jacobian matrix and its inverse, the pseudo-Jacobian matrix is introduced with a general applicability of any nonlinear systems of equations. It should be pointed out that a large class of real-world nonlinear problems can be modeled or numerically discretized polynomial-only algebraic system of equations. The results presented here are in general applicable for all these problems. This paper can be considered as a starting point in the research of nonlinear computation and analysis from an innovative viewpoint.Comment: Six chapters, 28 pages, Original MS. Word format, interested readers can contact me in [email protected] or [email protected]

Pseudo almost periodic solutions for neutral type high-order Hopfield neural networks with mixed time-varying delays and leakage delays on time scales

In this paper, a class of neutral type high-order Hopfield neural networks with mixed time-varying delays and leakage delays on time scales is proposed. Based on the exponential dichotomy of linear dynamic equations on time scales, Banach's fixed point theorem and the theory of calculus on time scales, some sufficient conditions are obtained for the existence and global exponential stability of pseudo almost periodic solutions for this class of neural networks. Our results are completely new. Finally, we present an example to illustrate our results are effective. Our example also shows that the continuous-time neural network and its discrete-time analogue have the same dynamical behaviors for the pseudo almost periodicity.Comment: 24 page

Weighted pseudo-almost periodic functions on time scales with applications to cellular neural networks with discrete delays

In this paper, we first propose a concept of weighted pseudo-almost periodic functions on time scales and study some basic properties of weighted pseudo-almost periodic functions on time scales. Then, we establish some results about the existence of weighted pseudo-almost periodic solutions to linear dynamic equations on time scales. Finally, as an application of our results, we study the existence and global exponential stability of weighted pseudo-almost periodic solutions for a class of cellular neural networks with discrete delays on time scales. The results of this paper are completely new.Comment: 25 page

Structural Compactness and Stability of Pseudo-Monotone Flows

Fitzpatrick's variational representation of maximal monotone operators is here extended to a class of pseudo-monotone operators in Banach spaces. On this basis, the initial-value problem associated with the first-order flow of such an operator is here reformulated as a minimization principle, extending a method that was pioneered by Brezis, Ekeland and Nayroles for gradient flows. This formulation is used to prove that the problem is stable w.r.t.\ arbitrary perturbations not only of data but also of operators. This is achieved by using the notion of evolutionary $\Gamma$-convergence w.r.t.\ a nonlinear topology of weak type. These results are applied to the Cauchy problem for quasilinear parabolic PDEs. This provides the structural compactness and stability of the model of several physical phenomena: nonlinear diffusion, incompressible viscous flow, phase transitions, and so on.Comment: arXiv admin note: text overlap with arXiv:1509.0381

Geometry and nonlinear analysis

Nonlinear analysis has played a prominent role in the recent developments in geometry and topology. The study of the Yang-Mills equation and its cousins gave rise to the Donaldson invariants and more recently, the Seiberg-Witten invariants. Those invariants have enabled us to prove a number of striking results for low dimensional manifolds, particularly, 4-manifolds. The theory of Gromov-Witten invariants was established by using solutions of the Cauchy-Riemann equation. These solutions are often refered as pseudo-holomorphic maps which are special minimal surfaces studied long in geometry. It is certainly not the end of applications of nonlinear partial differential equations to geometry. In this talk, we will discuss some recent progress on nonlinear partial differential equations in geometry. We will be selective, partly because of my own interest and partly because of recent applications of nonlinear equations. There are also talks in this ICM to cover some other topics of geometric analysis by R. Bartnik, B. Andrew, P. Li and X.X. Chen, etc

Stability and Stabilization of Fractional-order Systems with Different Derivative Orders: An LMI Approach

Stability and stabilization analysis of fractional-order linear time-invariant (FO-LTI) systems with different derivative orders is studied in this paper. First, by using an appropriate linear matrix function, a single-order equivalent system for the given different-order system is introduced by which a new stability condition is obtained that is easier to check in practice than the conditions known up to now. Then the stabilization problem of fractional-order linear systems with different fractional orders via a dynamic output feedback controller with a predetermined order is investigated, utilizing the proposed stability criterion. The linear matrix inequality based procedure of developing stabilizing output feedback control is preserved in spite of the complexity of assuming the most complete linear controller model, with direct feedthrough parameter. The proposed stability and stabilization theorems are applicable to FO-LTI systems with different fractional orders in one or both of and intervals. Eventually, some numerical examples are presented to confirm the obtained analytical results

Stability and instability of expanding solutions to the Lorentzian constant-positive-mean-curvature flow

We study constant mean curvature Lorentzian hypersurfaces of $\mathbb{R}^{1,d+1}$ from the point of view of its Cauchy problem. We completely classify the spherically symmetric solutions, which include among them a manifold isometric to the de Sitter space of general relativity. We show that the spherically symmetric solutions exhibit one of three (future) asymptotic behaviours: (i) finite time collapse (ii) convergence to a time-like cylinder isometric to some $\mathbb{R}\times\mathbb{S}^d$ and (iii) infinite expansion to the future converging asymptotically to a time translation of the de Sitter solution. For class (iii) we examine the future stability properties of the solutions under arbitrary (not necessarily spherically symmetric) perturbations. We show that the usual notions of asymptotic stability and modulational stability cannot apply, and connect this to the presence of cosmological horizons in these class (iii) solutions. We can nevertheless show the global existence and future stability for small perturbations of class (iii) solutions under a notion of stability that naturally takes into account the presence of cosmological horizons. The proof is based on the vector field method, but requires additional geometric insight. In particular we introduce two new tools: an inverse-Gauss-map gauge to deal with the problem of cosmological horizon and a quasilinear generalisation of Brendle's Bel-Robinson tensor to obtain natural energy quantities.Comment: Version 2: 60 pages, 1 figure. Changes mostly to fix typographical errors, with the exception of Remark 1.2 and Section 9.1 which are new and which explain the extrinsic geometry of the embedding in more detail in terms of the stability result. Version 3: updated reference

Existence of Doubly-Weighted Pseudo Almost Periodic Solutions to Some Classes of Nonautonomous Differential Equations

The main objective of this paper is twofold. We first show that if the doubly-weighted Bohr spectrum of an almost periodic function exists, then it is either empty or coincides with the Bohr spectrum of that function. Next, we investigate the problem which consists of the existence of doubly-weighted pseudo-almost periodic solutions to some nonautonomous abstract differential equations

Dynamic robust stabilization of fractional-order linear systems with nonlinear uncertain parameters: An LMI approach

This paper considers the problem of robust stability and stabilization for linear fractional-order system with nonlinear uncertain parameters, with fractional order 0<a<2. A dynamic output feedback controller, with predetermined order, for asymptotically stabilizing such uncertain fractional-order systems is designed. The derived stabilization conditions are in LMI form. Simulation results of two numerical examples illustrate that the proposed sufficient theoretical results are applicable and effective for tackling robust stabilization problems. Keywords: Fractional-order system, nonlinear uncertain parameters, linear matrix inequality (LMI), robust stabilization, dynamic output feedback.Comment: arXiv admin note: substantial text overlap with arXiv:1807.1082

Stability of fractional-order nonlinear systems by Lyapunov direct method

In this paper, by using a characterization of functions having fractional derivative, we propose a rigorous fractional Lyapunov function candidate method to analyze stability of fractional-order nonlinear systems. First, we prove an inequality concerning the fractional derivatives of convex Lyapunov functions without the assumption on the existence of derivative of pseudo-states. Second, we establish fractional Lyapunov functions to fractional-order systems without the assumption on the global existence of solutions. Our theorems fill the gaps and strengthen results in some existing papers