Springer Science and Business Media LLC

This paper proposes a dynamical model of influence volume of small-world-network with memory to investigate the effects of multiple delays on network dynamics. We calculate the influence volume covered by the spreading quantity, discuss the effect of finite size on the network dynamics, and then give the saturate time. The dynamical control is also investigated by introducing the delayed state feedback to simulate the adaptivity of network. With properly chosen delay and gain in feedback path, the controlled model may have stable equilibrium, periodic solution, quasi-periodic solution, or a complex chaotic attractor from a sequence of period-doubling bifurcations. It shows delayed feedback control may find important applications in the management and dynamical control of complex networks.

Mặc dù các kỹ sư tin rằng dao động tự kích thích và sự hỗn loạn là có hại, nhưng nghiên cứu gần đây đã tiết lộ rằng dao động chu kỳ và dao động hỗn loạn tự kích thích có thể được sử dụng một cách hiệu quả trong nhiều quá trình và thiết bị kỹ thuật mang lại lợi ích đáng kể. Do đó, một bộ điều khiển phản hồi phi tuyến đơn giản được đề xuất để tạo ra dao động chu kỳ và dao động hỗn loạn tự kích thích trong một bộ dao động cơ học. Các biểu thức phân tích liên quan đến biên độ của chu kỳ giới hạn và các tham số điều khiển được xác định bằng phương pháp nhiều thang thời gian. Kết quả phân tích được xác thực thông qua các mô phỏng số được thực hiện trong MATLAB–SIMULINK. Sự tồn tại của các dao động hỗn loạn được xác nhận qua các mô phỏng số. Một nghiên cứu tham số sâu rộng được thực hiện để làm sáng tỏ bản chất của các dao động hỗn loạn và sự phụ thuộc của chúng vào các tham số điều khiển. Các kết quả lý thuyết cuối cùng được xác thực qua các thí nghiệm. Thông thường, có thể quan sát rằng cùng một bộ điều khiển có thể được sử dụng để kích thích cả dao động chu kỳ và dao động hỗn loạn chỉ bằng cách đảo ngược pha của bộ điều khiển. Đây có thể là ví dụ đầu tiên cho thấy cùng một bộ điều khiển có thể được ứng dụng để tạo ra dao động chu kỳ và dao động hỗn loạn trong một hệ thống cơ học. Các phát hiện của bài báo được cho là có thể ứng dụng trong các hệ thống cơ học vĩ mô và vi mô.

In this paper, the robust global exponential estimating problem is investigated for Markovian jumping reaction-diffusion delayed neural networks with polytopic uncertainties under Dirichlet boundary conditions. The information on transition rates of the Markov process is assumed to be partially known. By introducing a new inequality, some diffusion-dependent exponential stability criteria are derived in terms of relaxed linear matrix inequalities. Those criteria depend on decay rate, which may be freely selected in a range according to practical situations, rather than required to satisfy a transcendental equation. Estimates of the decay rate and the decay coefficient are presented by solving these established linear matrix inequalities. Numerical examples are provided to demonstrate the advantage and effectiveness of the proposed method.

The two vectors of applied and constraint forces form a plane, which is not directly related to the Frenet geometry. Nonetheless, the resultant of the applied and constraint forces lies in a plane defined by the Frenet geometry when the motion trajectory (MT) of the mass center of a rigid body is considered. This, however, is not the case for non-centroidal-point trajectories. For both centroidal and non-centroidal points, an instantaneous motion plane defined by the Frenet-frame osculating plane can be determined. This plane contains the absolute velocity and acceleration vectors, inertia forces including centrifugal force, and resultant of the applied and constraint forces acting at the body center of mass. It is shown that the component of the resultant of the applied and constraint forces along the Frenet bi-normal vector is zero when mass-center trajectories are considered, and therefore, the Frenet bi-normal vector is an instantaneous zero-force axis. The MT analysis is generalized to the case of non-centroidal points in which the bi-normal vector is not orthogonal to the plane formed by the two vectors of applied and constraint forces only. Complexities that arise in case of points different from the mass center are highlighted. At zero-curvature points, singularities that can lead to software crashing can be avoided by proper definition of the vector normal to the space curve. Consequently, the spatial Newton equations can always be transformed to instantaneous planar equations. Developing real-time onboard-computer MT algorithms for autonomous vehicles and positive-train control can contribute to avoiding linearization and simplifications of the equations of motion that may lead to wrong results, particularly in extreme dynamics that characterizes accidents.

Under investigation in this paper is a discrete modified exponential Toda lattice equation which may describe vibration of particles in a lattice. Firstly, we construct an integrable lattice hierarchy associated with this equation from a $$2\times 2$$ matrix spectral problem, and some related integrable properties such as Hamiltonian structures, Liouville integrability and conservation laws of this hierarchy are discussed. Secondly, we present the discrete generalized $$(m, N-m)$$ -fold Darboux transformation of the modified exponential Toda lattice equation on the basis of its known Lax representation. As applications of the obtained discrete generalized Darboux transformation, multi-kink-soliton solutions on an exponential surface background and an inclined plane background are obtained when $$m=2N$$ , the discrete rational and semi-rational solutions are derived when $$m=1$$ , and the mixed solutions of usual soliton solutions and rational solutions are given when $$m=2$$ . Based on the asymptotic and graphic analysis, soliton elastic interaction phenomena and limit states related to rational solutions are discussed and analyzed. Furthermore, some mathematical features of rational solutions are summarized. Finally, numerical simulations are used to explore the dynamical behaviors of such soliton solutions which show the soliton evolutions are robust against a small noise. These results and properties given in this paper might be useful for understanding nonlinear lattice dynamics.

This paper is concerned with nonlinear modeling and analysis of the COVID-19 pandemic. We are especially interested in two current topics: effect of vaccination and the universally observed oscillations in infections. We use a nonlinear Susceptible, Infected, & Immune model incorporating a dynamic transmission rate and vaccination policy. The US data provides a starting point for analyzing stability, bifurcations and dynamics in general. Further parametric analysis reveals a saddle-node bifurcation under imperfect vaccination leading to the occurrence of sustained epidemic equilibria. This work points to the tremendous value of systematic nonlinear dynamic analysis in pandemic modeling and demonstrates the dramatic influence of vaccination, and frequency, phase, and amplitude of transmission rate on the persistent dynamic behavior of the disease.

The time fractional diffusion equation is obtained from the standarddiffusion equation by replacing the first-order time derivative with afractional derivative of order β ∋ (0, 1). From a physicalview-point this generalized diffusion equation is obtained from afractional Fick law which describes transport processes with longmemory. The fundamental solution for the Cauchy problem is interpretedas a probability density of a self-similar non-Markovian stochasticprocess related to a phenomenon of slow anomalous diffusion. By adoptinga suitable finite-difference scheme of solution, we generate discretemodels of random walk suitable for simulating random variables whosespatial probability density evolves in time according to this fractionaldiffusion equation.