Search Results (20)

We study the quantitative structure of the reciprocity gap method for inclusions with spatially varying conductivity. Motivated by the variable-coefficient reciprocity gap identity, we investigate the discrete approximation mechanism underlying the bounded/blow-up dichotomy for sampling points inside and outside the inclusion. The reciprocity gap functional is discretized by harmonic test functions, and the resulting ill-conditioned linear system is regularized by a Tikhonov term consistent with the $L^2(\partial D)$ trace norm appearing in the weighted formulation. The regularization parameter is selected by the L-curve criterion. For constant, radially varying, and angularly oscillating contrasts, the numerical results show that exterior sampling points exhibit an approximately exponential growth of $\parallel v_N{\left(z\right)\parallel }_{L^2(\partial D)}$ with respect to the harmonic order N, whereas interior points remain bounded. This behavior is quantified through fitted growth rates and contrast indicators, and its dependence on geometry and model parameters is examined. The results provide a quantitative description of the reciprocity gap approximation mechanism in heterogeneous media and show that the bounded/blow-up dichotomy remains numerically detectable beyond the constant-coefficient setting. Full article

We study the long-time behavior of nonlinear stochastic evolution equations in a separable Hilbert space driven by a Q-Wiener process. The linear part of the equation is generated by a strongly continuous semigroup with an exponential dichotomy, which provides fixed rates of decay and growth. The nonlinear drift and diffusion terms are globally Lipschitz and become small as time tends to infinity. Our main result shows that under these conditions, the mean-square Lyapunov exponents of the nonlinear system coincide with those of the linear part. In other words, nonlinear stochastic perturbations that decay in time do not change the main growth or decay rates of solutions in the mean-square sense. This result provides simple and verifiable criteria ensuring that the long-time Lyapunov behavior of the nonlinear stochastic equation is fully determined by the linear semigroup, even in the presence of time-dependent stochastic perturbations. Full article

This paper develops a structured analytical framework and a robust numerical methodology for the spectral stability of time-varying bilateral quaternion differential equations of the form $\dot{q}=A\left(t\right)q+qB\left(t\right)$. By systematically extending classical real matrix theory to non-commutative dynamical systems via exact isometric real representations, this study utilizes the Kronecker product of real adjoint matrices to rigorously elucidate the underlying tensor structure of the bilateral evolution operator. This tensor-based reformulation proves that the Floquet multipliers of the bilaterally coupled system can be strictly decoupled into the product of the spectra corresponding to the left and right unilateral subsystems. Second, a “Scalar-Vector Stability Separation Principle” based on logarithmic norms is proposed, demonstrating that the transient energy evolution of the system is governed exclusively by the Hermitian real parts of the coefficient matrices, remaining entirely independent of the anti-Hermitian imaginary parts (rotation terms). Furthermore, for constant-coefficient and slowly varying systems, the Riesz projection from holomorphic functional calculus is introduced to establish algebraic criteria for exponential dichotomies, thereby revealing a cubic scaling law that relates the robustness threshold to the spectral gap (${\epsilon }_0\propto {\beta }^3$). Numerically, a Quaternion Chebyshev Spectral Collocation Method (Q-CSCM) is embedded within this exact vectorization framework to ensure that the algebraic symmetries of the bilateral system are strictly preserved through the isomorphic mapping. By explicitly constructing the fully discrete Kronecker product matrix via the exact real vectorization isomorphism, discrete energy estimates are utilized to rigorously prove that the numerical scheme successfully inherits the intrinsic spectral accuracy of the Chebyshev approximation. Comprehensive numerical experiments demonstrate that, within the low-dimensional regime, this methodology exhibits substantial temporal approximation efficiency advantages and superior numerical robustness compared to an alternative Legendre spectral baseline, as well as traditional explicit and state-of-the-art implicit symplectic Runge–Kutta methods, particularly when solving stiff and critically stable problems such as nonlinear Riccati oscillators. Full article

This paper studies the existence and uniqueness of periodic solutions to a class of nonlinear neutral matrix difference systems with multiple delays. The analysis is based on the construction of a suitable Green operator combined with fixed-point methods under exponential dichotomy assumptions. The existence of periodic solutions is established using Krasnoselskii’s fixed-point theorem, while uniqueness is demonstrated under a natural contraction condition via Banach’s principle. The results extend previous contributions on neutral difference systems and provide discrete analogues of related differential models. Examples are included to illustrate the applicability of the theory. Full article

The paper utilizes the continuous finite element method to solve stiff ordinary differential equations and proves that the linear finite element method and the quadratic finite element method have A-stability in solving autonomous ordinary differential equations, and exponential dichotomy in solving non-autonomous ordinary differential equations. In the numerical experiments of nonlinear autonomous and non-autonomous strongly and moderately stiff ordinary differential equations, a relatively large step size of $h=0.1$ was adopted over a longer period of time, with the numerical solution accuracy reaching ${10}^{-4}$. The superconvergence order maintained the theoretical order. A new approach is provided for solving stiff ordinary differential equations. Full article

In this paper, we investigate differential equations with generalized piecewise constant delay, DEGPCD in short, and establish the existence and stability of a unique almost periodic solution that is exponentially stable. Our results are derived by utilizing the properties of the $({\mu }_1,{\mu }_2)$-exponential dichotomy, Cauchy and Green matrices, a Gronwall-type inequality for DEGPCD, and the Banach fixed point theorem. We apply these findings to derive new criteria for the existence, uniqueness, and convergence dynamics of almost periodic solutions in both the linear inhomogeneous and quasilinear DEGPCD systems through the $({\mu }_1,{\mu }_2)$-exponential dichotomy for difference equations. These results are novel and serve to recover, extend, and improve upon recent research. Full article

The present paper deals with two of the most significant behaviors in the theory of dynamical systems: the uniform exponential dichotomy and the uniform polynomial dichotomy for evolution operators in Banach spaces. Assuming that the evolution operator has uniform exponential growth, respectively uniform polynomial growth, we give some characterizations for the uniform exponential dichotomy, respectively for the uniform polynomial dichotomy. The proof techniques that we use for the polynomial case are new. In addition, connections between the concepts approached are established. Full article

Nonlinearities, exponential trends, and Euler equations are three key features of standard dynamic volatility models of speculation, economic growth, or macroeconomic fluctuations with occasionally binding constraints and endogenous state-dependent volatility. A natural way to estimate a model with all such three features could be to use the observed nonstationary data in a single step without preliminary linearization, log-linearization, or preliminary detrending. Adoption of this natural strategy confronts a serious challenge that has been neither articulated nor solved: a dichotomy in the empirical model implied by the Euler equation. This leads to a discontinuity in the regression in the limit, rendering the approaches employed in available proofs of consistency inapplicable. We characterize the problem and develop a novel method of proof of consistency and asymptotic normality. Our methodological contribution establishes a foundation for consistent estimation and hypothesis testing of nonstationary models without resorting to preliminary detrending, an a priori assumption that any trend is exactly zero, linearization, or other restrictions on the model. Full article

In the present paper, we consider the problem of dichotomic behaviors of dynamical systems described by discrete-time skew evolution cocycles in Banach spaces. We study two concepts of uniform dichotomy: uniform exponential dichotomy and uniform polynomial dichotomy. Some characterizations of these notions and connections between these concepts are given. Full article

Cellular neural networks with D operator and time-varying delays are found to be effective in demonstrating complex dynamic behaviors. The stability analysis of the pseudo-almost periodic solution for a novel neural network of this kind is considered in this work. A generalized class neural networks model, combining cellular neural networks and the shunting inhibitory neural networks with D operator and time-varying delays is constructed. Based on the fixed-point theory and the exponential dichotomy of linear equations, the existence and uniqueness of pseudo-almost periodic solutions are investigated. Through a suitable variable transformation, the globally exponentially stable sufficient condition of the cellular neural network is examined. Compared with previous studies on the stability of periodic solutions, the global exponential stability analysis for this work avoids constructing the complex Lyapunov functional. Therefore, the stability criteria of the pseudo-almost periodic solution for cellular neural networks in this paper are more precise and less conservative. Finally, an example is presented to illustrate the feasibility and effectiveness of our obtained theoretical results. Full article

In this article, we construct a $C^1$ stable invariant manifold for the delay differential equation $x^{\prime }=Ax\left(t\right)+L{x}_t+f(t,x_t)$ assuming the $\rho$-nonuniform exponential dichotomy for the corresponding solution operator. We also assume that the $C^1$ perturbation, $f(t,x_t)$, and its derivative are sufficiently small and satisfy smoothness conditions. To obtain the invariant manifold, we follow the method developed by Lyapunov and Perron. We also show the dependence of invariant manifold on the perturbation $f(t,x_t)$. Full article

We investigate a control problem leading to the optimal payment of dividends in a Cramér-Lundberg-type insurance model in which capital injections incur proportional cost, and may be used or not, the latter resulting in bankruptcy. For general claims, we provide verification results, using the absolute continuity of super-solutions of a convenient Hamilton-Jacobi variational inequality. As a by-product, for exponential claims, we prove the optimality of bounded buffer capital injections $(-a,0,b)$ policies. These policies consist in stopping at the first time when the size of the overshoot below 0 exceeds a certain limit a, and only pay dividends when the reserve reaches an upper barrier b. An exhaustive and explicit characterization of optimal couples buffer/barrier is given via comprehensive structure equations. The optimal buffer is shown never to be of de Finetti ($a=0$) or Shreve-Lehoczy-Gaver ($a=\infty$) type. The study results in a dichotomy between cheap and expensive equity, based on the cost-of-borrowing parameter, thus providing a non-trivial generalization of the Lokka-Zervos phase-transition Løkka-Zervos (2008). In the first case, companies start paying dividends at the barrier $b^{*}=0$, while in the second they must wait for reserves to build up to some (fully determined) $b^{*}>0$ before paying dividends. Full article

Fractal groups (also called self-similar groups) is the class of groups discovered by the first author in the 1980s with the purpose of solving some famous problems in mathematics, including the question of raising to von Neumann about non-elementary amenability (in the association with studies around the Banach-Tarski Paradox) and John Milnor’s question on the existence of groups of intermediate growth between polynomial and exponential. Fractal groups arise in various fields of mathematics, including the theory of random walks, holomorphic dynamics, automata theory, operator algebras, etc. They have relations to the theory of chaos, quasi-crystals, fractals, and random Schrödinger operators. One important development is the relation of fractal groups to multi-dimensional dynamics, the theory of joint spectrum of pencil of operators, and the spectral theory of Laplace operator on graphs. This paper gives a quick access to these topics, provides calculation and analysis of multi-dimensional rational maps arising via the Schur complement in some important examples, including the first group of intermediate growth and its overgroup, contains a discussion of the dichotomy “integrable-chaotic” in the considered model, and suggests a possible probabilistic approach to studying the discussed problems. Full article

An exponential dichotomy is studied for linear differential equations. A constructive method is presented to derive a roughness result for perturbations giving exponents of the dichotomy as well as an estimate of the norm of the difference between the corresponding two dichotomy projections. This roughness result is crucial in developing a Melnikov bifurcation method for either discontinuous or implicit perturbed nonlinear differential equations. Full article