Analysis of codimension 2 bifurcations for high-dimensional discrete systems using symbolic computation methods

A fast symbolic SNR computation method and its Verilog-A implementation for Sigma-Delta modulator design optimization

The authors show how to use symbolic computing methods to obtain an analytic representation of the solution of

By using symbolic computation methods, we systematically study a generalized (3+1)-dimensional Kadomtsev–Petviashvili–Boussinesq (KP-Boussinesq) equation, which can be used to explain some interesting physical phenomena in the fields of fluids. We first obtain bilinear representation of the equation by employing the properties of Bell's polynomials. Based on that, its a series of exact solutions, such as one kink soliton solutions, two kink soliton solutions and lump wave solutions, are constructed in detail. It is worth mentioning that we also study the interaction solutions between lump waves and kink waves. Moreover, in order to describe the dynamic behaviours of those solutions, some graphical analyses are given by selecting different parameter values in such solutions. Those analyses are helpful for enriching the physical phenomena in KP-type equations.

This paper investigates duopoly competition under both constant and decreasing returns to scale in a market characterized by an isoelastic demand function, where firms adjust their strategies using a gradient adjustment mechanism. To establish the stability conditions of the model, we adopt different analytical approaches depending on the type of returns to scale. Under constant returns to scale, we employ a traditional approach by deriving the closed-form solution of the Nash equilibrium and analyzing the Jacobian matrix to verify whether the moduli of all eigenvalues are less than one. In contrast, under decreasing returns to scale, we analyze the local stability of the Nash equilibrium using symbolic computation methods without deriving a closed-form solution. The results show that when firms have heterogeneous costs, the model can exhibit both period-doubling and Neimark–Sacker bifurcations under both types of returns to scale. However, when costs are homogeneous, only period-doubling bifurcations occur. Numerical simulations support these analytical results and further demonstrate the emergence of complex dynamics, including chaotic behavior.

Schwarz’s alternating method in a matrix form and its applications to composites

Stewart platform is a parallel linkage, which is widely used in the area of parallel robotics. Generalized Stewart Platform (GSP) consists of two rigid bodies connected with six constraints between six pairs of geometric primitives in the base and the moving platform respectively. Direct kinematics to GSP is to determine the pose of moving platform according to six constraints between the base and the moving platform. Strategies on direct kinematics to GSP were presented, which were based on graph theory and symbolic computation method. A GSP can be decomposed into smaller ones if possible, and the corresponding parameters conditions of real solutions to direct kinematics to the GSP can be obtained according to these strategies.

Discrete chaotic system, as a pseudo-random signal source, plays a very important role in realizing secure communication. However, many low-dimensional chaotic systems are prone to chaos degradation. Therefore, many scholars have studied the construction of high-dimensional chaotic systems. However, many existing algorithms for constructing high-dimensional chaotic systems have relatively high time complexity and relatively complex structures. To solve this problem, this paper explores an <i>n</i>-dimensional discrete hyperchaotic system with a simple structure. Firstly, the <i>n</i>-dimensional discrete hyperchaotic system is constructed by using sine function and power function and simple operations. Then, it is theoretically analyzed based on Jacobian matrix method that the system can have the positive Lyapunov exponents. Next, the algorithm time complexity, sample entropy, correlation dimension and other indexes are compared with those of the existing methods. The experimental results show that our system has a simple structure, high complexity and good algorithm time complexity. Therewith, a six-dimensional chaotic system is chosen as an example, and the phase diagram, bifurcation diagram, Lyapunov expnonents, complexity and other characteristics of the system are analyzed. The results show that the proposed system has good chaotic characteristics. Moreover, to show the application of the proposed system, we apply it to audio encryption. According to this system, we combine it with the XOR operation and true random numbers to explore a novel method of one-cipher audio encryption. Through experimental simulation, compared with some existing audio encryption algorithms, this algorithm can satisfy various statistical tests and resist various common attacks. It is also validated that the proposed system can be effectively applied to the field of audio encryption.

This textbook provides a broad introduction to continuous and discrete dynamical systems. With its hands-on approach, the text leads the reader from basic theory to recently published research material in nonlinear ordinary differential equations, nonlinear optics, multifractals, neural networks, and binary oscillator computing. Dynamical Systems with Applications Using Python takes advantage of Python’s extensive visualization, simulation, and algorithmic tools to study those topics in nonlinear dynamical systems through numerical algorithms and generated diagrams. After a tutorial introduction to Python, the first part of the book deals with continuous systems using differential equations, including both ordinary and delay differential equations. The second part of the book deals with discrete dynamical systems and progresses to the study of both continuous and discrete systems in contexts like chaos control and synchronization, neural networks, and binary oscillator computing. These later sections are useful reference material for undergraduate student projects. The book is rounded off with example coursework to challenge students’ programming abilities and Python-based exam questions. This book will appeal to advanced undergraduate and graduate students, applied mathematicians, engineers, and researchers in a range of disciplines, such as biology, chemistry, computing, economics, and physics. Since it provides a survey of dynamical systems, a familiarity with linear algebra, real and complex analysis, calculus, and ordinary differential equations is necessary, and knowledge of a programming language like C or Java is beneficial but not essential.

We describe the destabilizing (in the sense of a decrease in the reserve of mean-square asymptotic stability) effect of random parametric perturbations of the white-noise type in quasilinear continuous and discrete dynamical systems (Lur’e-Postnikov systems of automatic control with nonlinear feedback). We use stochastic Lyapunov functions in the form of linear combinations of the types “a quadratic form of phase coordinates plus the integral of a nonlinearity” (continuous systems) and “a quadratic form of phase coordinates plus the integral sum for a nonlinearity” (discrete systems) and the matrix algebraic Sylvester equations associated with stochastic Lyapunov functions of this form.

On the fluidization of Petri nets and marking homothecy

Sensitivity of non-autonomous discrete dynamical systems

With the rapid development of computer science, there are more and more kinds of discrete dynamic systems. Computer integrated system CIMS, network communication database administrator system, and human behavior analysis system are all discrete dynamic systems. At present, many researchers have studied by adding human behavior data to discrete dynamic systems. This paper aims to study the behavior data of English learners by using the discrete dynamic modeling technology of complex systems and the discrete dynamic system modeling method of Petri nets. By adding the behavior data of learners to the discrete dynamic system of fuzzy Petri nets, the system is diagnosed and optimized. The experimental results show that the complex discrete dynamic system in this paper has achieved good experimental results according to the performance indicators selected in theory. Based on the combination of the above technologies and systems, the fuzzy Petri net discrete dynamic system studied in this paper improves the processing speed of English learners' behavior data.

We study the qualitative behavior of localized synchronous oscillations organized by synaptic inhibition in two types of spatially extended neuronal network models driven by a time-independent, localized excitatory input. Each network is formulated as a one-dimensional network of conductance-based models constituting a high-dimensional dynamical system of nonlocal differential equations. Although such equations readily generate highly complex dynamic behavior, in the case of strong inhibitory coupling the response of the network to a localized Gaussian input is a solution in which a single, continuous band of cells fire nearly synchronous action potentials, in an approximately periodic fashion in time. Tracking the cycle-to-cycle evolution of the width of the band of synchronous action potentials reveals the characteristic behavior of low-dimensional, discrete dynamical systems. Based upon a continuum formulation of the conductance-based model, we heuristically develop and analyze one- and two-dimensional implicit discrete maps for both a purely inhibitory and an excitatory-inhibitory network of neurons. Although the discrete maps do not predict the band widths precisely, they generally reflect the qualitative behavior of the conductance-based model. The most salient features of the bifurcations of fixed points to period 2 orbits and resonances indicate that in some cases these high-dimensional continuous dynamical systems exhibit behavior which can be captured in related low-dimensional discrete maps. Finally, we describe a global bifurcation in the discrete map for the excitatory-inhibitory network in which a strong (1:2) resonance bifurcation occurs on a period 2 orbit, giving rise to a pair of double homoclinic tangles that generate nontrivial dynamics.

The paper considers the issues of control and stabilization of nonlinear dynamic systems described by a set of differential and difference equations, the latter of which contain a control vector. The states of these systems have both continuous and discrete components, so such systems are called continuous-discrete or hybrid. Necessary and sufficient features of controllability of nonlinear hybrid systems with a constant discretization step are established, which imply a transition from these systems to equivalent, in the natural sense, nonlinear discrete dynamic systems. A transformation is presented that allows reducing a linear discrete system to the canonical Brunovsky form and constructing a stabilizing control on its basis for the corresponding continuous-discrete system with scalar control. An algorithm for reducing a first approximation system of a nonlinear discrete system with scalar control to the canonical Brunovsky form and an algorithm for constructing a stabilizing control for nonlinear hybrid systems with scalar control are developed and illustrated with examples. Sufficient signs of stabilization of nonlinear hybrid systems are presented both without and with the feedback controller.

The traditional models of distributed computing systems employ the mathematical formalisms of discrete event dynamical systems along with Petri Nets. However, such models are complex to analyze and computationally expensive. Interestingly, the evolving distributed computing systems closely resemble the discrete dynamical systems. This paper constructs and analyzes the model of distributed computing systems by applying mathematical formalisms of the discrete dynamical systems. The proposed model embeds the partial ordering of states under happened-before relation in the set of globally observable states. The stability of the proposed model is analyzed using the first order linear approximation.