Autonomous Differential Equations Research Articles

Bifurcation analysis for a two-dimensional peristaltic driven flow of power-law fluid in asymmetric channel

In the present article, the bifurcations of equilibrium points and their streamlined patterns for the peristaltic transport of shear-thinning and shear-thickening fluids through an asymmetric channel are studied by incorporating a power-law model. An exact solution in the wave frame of reference is obtained under the vanishing Reynolds number and long wavelength approximations. A system of non-linear autonomous differential equations is developed to locate the equilibrium points in the flow. The qualitative nature of equilibrium points and their bifurcations are investigated through the dynamical system method. There exist three distinct flow conditions (backward flow, trapping, and augmented flow). It is observed that the shifting of these flow phenomena corresponds to bifurcations where non-hyperbolic degenerate points are conceived. The impacts of various embedded parameters on flow phenomena and their bifurcations are demonstrated through graphical representations. It is found that the trapping phenomenon manifests at a high flow rate for shear-thinning fluids. That is, the backward flow region shrinks for large values of the power-law index. Trapping in mechanical devices can be diminished by enlarging the phase difference of channel walls, while an opposite trend is observed for amplitude ratios. The obtained results are concluded through global bifurcation diagrams. At the end, findings of this analysis are verified by making a comparison with the existing literature.

Extreme Multistability and Complex Dynamics of a Memristor-Based Chaotic System

A memristor with coexisting pinched hysteresis loops and twin local activity domains is presented and analyzed, with an emulator being designed and applied to the classic Chua’s circuit to replace the diode. The memristive system is modeled with four coupled first-order autonomous differential equations, which has three equilibria determined by three static equilibria of the memristor but not controlled by the system parameters. The complex dynamics of the system are analyzed by using compound coexisting bifurcation diagrams, Lyapunov exponent spectra and phase portraits, including point attractors, limit cycles, symmetrical chaotic attractors and their blasting, extreme multistability, state-switching without parameter, and transient chaos. Of particular surprise is that the extreme multistability of the system is hidden and symmetrically distributed. It is found that the existence of transient chaos in the specified parameter domain is determined by using bifurcation diagrams within different time durations and Lyapunov exponents with chaotic sequences. Finally, the symmetrical chaotic attractor and the system blasting are verified by digital signal processing experiments, which are consistent with the numerical analysis.

Existence and convergence of Puiseux series solutions for autonomous first order differential equations

Given an autonomous first order algebraic ordinary differential equation F(y,y′)=0, we prove that every formal Puiseux series solution of F(y,y′)=0, expanded around any finite point or at infinity, is convergent. The proof is constructive and we provide an algorithm to describe all such Puiseux series solutions. Moreover, we show that for any point in the complex plane there exists a solution of the differential equation which defines an analytic curve passing through this point.

Mixture distributions in a stochastic gene expression model with delayed feedback: a WKB approximation approach

Noise in gene expression can be substantively affected by the presence of production delay. Here we consider a mathematical model with bursty production of protein, a one-step production delay (the passage of which activates the protein), and feedback in the frequency of bursts. We specifically focus on examining the steady-state behaviour of the model in the slow-activation (i.e. large-delay) regime. Using a formal asymptotic approach, we derive an autonomous ordinary differential equation for the inactive protein that applies in the slow-activation regime. If the differential equation is monostable, the steady-state distribution of the inactive (active) protein is approximated by a single Gaussian (Poisson) mode located at the globally stable fixed point of the differential equation. If the differential equation is bistable (due to cooperative positive feedback), the steady-state distribution of the inactive (active) protein is approximated by a mixture of Gaussian (Poisson) modes located at the stable fixed points; the weights of the modes are determined from a WKB approximation to the stationary distribution. The asymptotic results are compared to numerical solutions of the chemical master equation.

On the nonlinear dynamics of constant stiffness coefficients 16-pole rotor active magnetic bearings system

The present study is devoted to investigate the oscillatory behaviors of the 16-pole rotor active magnetic bearing system. A controllable electromagnetic force generated via a conventional proportional-derivative controller is utilized to stabilize the system lateral oscillations that excited by the rotating disk eccentricity when the spinning speed (Ω) is close to or equal the system linear natural frequency (ω). The nonlinear dynamical equations governing the controlled system lateral vibrations at constant stiffness coefficients are derived in this article for the first time. Then, four nonlinear autonomous first-order differential equations to describe the considered system oscillation amplitudes and the corresponding phase angles are obtained applying the asymptotic analysis. Bifurcation behavior of the system periodic motions under varying the different control parameters is explored. The main acquired results confirm that the 16-pole rotor-AMB system at constant stiffness coefficients can exhibit one of three oscillatory motions that are periodic, quasiperiodic, or chaotic motions depending on the derivative gain coefficient. Moreover, the system may respond with one-stable solution, bi-stable solutions, tri-stable solutions, or quadri-stable solutions depending on the proportional gain coefficient. Numerical simulations for different system motions are validated via the system time response, Poincare map, orbit plot, and frequency spectrum that are showed an excellent agreement with the obtained analytical results.

Algebraic, Rational and Puiseux Series Solutions of Systems of Autonomous Algebraic ODEs of Dimension One

In this paper, we study the algebraic, rational and formal Puiseux series solutions of certain type of systems of autonomous ordinary differential equations. More precisely, we deal with systems which associated algebraic set is of dimension one. We establish a relationship between the solutions of the system and the solutions of an associated first order autonomous ordinary differential equation, that we call the reduced differential equation. Using results on such equations, we prove the convergence of the formal Puiseux series solutions of the system, expanded around a finite point or at infinity, and we present an algorithm to describe them. In addition, we bound the degree of the possible algebraic and rational solutions, and we provide an algorithm to decide their existence and to compute such solutions if they exist. Moreover, if the reduced differential equation is non trivial, for every given point (x_0,y_0) in mathbb {C}^2, we prove the existence of a convergent Puiseux series solution y(x) of the original system such that y(x_0)=y_0.

Robustness of Exponential Dichotomy in a Class of Generalised Almost Periodic Linear Differential Equations in Infinite Dimensional Banach Spaces

In this paper we study the robustness of the exponential dichotomy in nonautonomous linear ordinary differential equations under integrally small perturbations in infinite dimensional Banach spaces. Some applications are obtained to the case of rapidly oscillating perturbations, with arbitrarily small periods, showing that even in this case the stability is robust. These results extend to infinite dimensions some results given in Coppel (Dichotomies in stability theory. Lecture notes in mathematics, Springer, Berlin, 1970). Based in Rodrigues (Invariância para sistemas de equacoes diferenciais com retardamento e aplicacoes, Tese de Mestrado, Universidade de Sao Paulo, Sao Carlos, 1970) and in Kloeden and Rodrigues (Nonlinear Anal 74:2695–2719, 2011), Rodrigues et al. (Stability problems in non autonomous linear differential equations in infinite dimensions. arXiv:1906.04642, 2019) we use the class of functions that we call Generalized Almost Periodic Functions that extend the usual class of almost periodic functions and are suitable to model these oscillating perturbations. We also present an infinite dimensional example of the previous results.

Nonstandard finite difference method revisited and application to the Ebola virus disease transmission dynamics

We provide effective and practical guidelines on the choice of the complex denominator function of the discrete derivative as well as on the choice of the nonlocal approximation of nonlinear terms in the construction of nonstandard finite difference (NSFD) schemes. Firstly, we construct nonstandard one-stage and two-stage theta methods for a general dynamical system defined by a system of autonomous ordinary differential equations. We provide a sharp condition, which captures the dynamics of the continuous model. We discuss at length how this condition is pivotal in the construction of the complex denominator function. We show that the nonstandard theta methods are elementary stable in the sense that they have exactly the same fixed-points as the continuous model and they preserve their stability, irrespective of the value of the step size. For more complex dynamical systems that are dissipative, we identify a class of nonstandard theta methods that replicate this property. We apply the first part by considering a dynamical system that models the Ebola Virus Disease (EVD). The formulation of the model involves both the fast/direct and slow/indirect transmission routes. Using the specific structure of the EVD model, we show that, apart from the guidelines in the first part, the nonlocal approximation of nonlinear terms is guided by the productive-destructive structure of the model, whereas the choice of the denominator function is based on the conservation laws and the sub-equations that are associated with the model. We construct a NSFD scheme that is dynamically consistent with respect to the properties of the continuous model such as: positivity and boundedness of solutions; local and/or global asymptotic stability of disease-free and endemic equilibrium points; dependence of the severity of the infection on self-protection measures. Throughout the paper, we provide numerical simulations that support the theory.

One kind motion of controllable constrained Birkhoffian system: the absence of constraints

This paper is devoted to discuss the motion of controllable constrained Birkhoffian system along with its absence of constraints. The first step is to establish the autonomous and non-autonomous differential equations of motion of the system, based on Pfaff–Birkhoff principle. Secondly, the existence of constraint multipliers are exhaustively discussed. Thirdly, the definition of one kind motion of the system, called free motion, is given, which is described and analyzed by the absence of constraints that are determined by constraint multipliers. Lemma 2 illustrates that one system can realize its free motion by selecting proper control parameters. In particular, theorem 2 provides that one system can naturally realize free motion when we consider the integral of the unconstrained Birkhoffian system as the constraints of constrained Birkhoffian system. Finally, the results obtained are illustrated by several examples.

Application of Second Order Real Matrix to Plane Autonomous Linear Ordinary Differential Equation

According to the matrix standard form of the second order real matrix, we study the qualitative issue of the planar autonomous linear ordinary differential equation, construct the qualitative optimization system which is admissible by the equation, and carry out symmetry reduction. At the same time, the solution method of second order real square matrix for the plane autonomous linear ordinary differential equation is given in the Bergman space. The approximate solutionis obtained by truncating the exact solutionin the Bergman space. In the end, the standard type of the plane autonomous linear ordinary differential equations is summed up and the exact solutions of the plane autonomous linear ordinary differential equations are given in combination with an example.

Applications of Conley index theory on difference equations with non-resonance

Inspired by Chang (1981) and Long (1990), we consider the existence of periodic solutions for second-order difference equations with asymptotically linear nonlinearity via Conley index theory. Moreover, when we study the following one-dimensional autonomous difference equations with form Δ2xn−1+f(xn)=0,n∈Z,xn∈R,we can derive at least three periodic solutions under suitable assumptions.

On the Omega-Limit Sets of Planar Nonautonomous Differential Equations with Nonpositive Lyapunov Exponents

The well-known Poincare-Bendixson theorem tells us that the structure of the omega-limit sets of planar autonomous differential equations can be described by fixed points, limit cycles, or finite number of fixed points together with homoclinic and heteroclinic orbits connecting them. However, this is very different for planar nonautonomous differential equations. In this paper, we study the omega-limit sets of three classes of planar nonautonomous differential equations, and their corresponding dynamical behavior. (i) The first type is linear, the omega-limit set is an annulus, the Lyapunov exponents are zero, there is no periodic orbit except for one fixed point, and the orbits are transitive in the omega-limit sets, implying the existence of infinitely many transitive components depending on initial conditions. (ii) The second type has butterfly-like omega-limit sets, two zero Lyapunov exponents, and neither periodic orbits nor sensitive dependence on initial conditions. (iii) The last type has a unique omega-limit set (a global attractor) independent on initial conditions, and has two negative Lyapunov exponents, where the omega-limit set is an annulus or a subset (homeomorphic to a disk) of an annulus for different parameter regions.

METHODOLOGY FOR MODELING MACROECONOMIC LIMITATIONS OF SCIENTIFIC AND TECHNOLOGICAL PROGRESS

The subject of this article is the methodology of macroeconomic modeling of scientific and technological progress from the point of view, which is an alternative to the modern paradigm of unlimited potential for economic efficiency.The research is based on the general scientific method of transition from simple to complex. As a starting point, the general laws of scientific and technical progress are selected, independent of individual economic factors. The analysis of long-term economic dynamics is based on the model of limited scientific and technological progress (LSTP-model), first proposed in this article. During the simulation, it is shown that even with infinite resources directed to the R&D sector, the rate of STP has its limit. In this connection, the conclusion is formulated that not only the volume and quality of resources in the R&D sector, but also the duration of their use are important for technological development.The methodology used in the article implements the general scientific principle of correspondence. It is shown that: 1) the classical models of unlimited STP are the limiting case of one of the proposed equations; 2) traditional production functions proved to be the limiting case of the proposed production function with restrictions on maximum product efficiency and minimum efficiency of production factors.A feature of the study is a multivariate approach - consideration and comparative analysis of possible macroeconomic functions and equations of dynamics. Based on this approach, functions and equations were determined that describe the unevenness of economic development, fluctuations in the rate of growth of efficiency, and endogenous waves of STP. It is shown that the new indicators (the potential for increasing efficiency, the sufficiency index of the STP factor) and the functions made up of them allow us to adequately analyze the increasingly complex aspects of scientific and technological progress. A function of exogenous dynamics reflecting the exogenous effect on STP is proposed. An analysis of this function made it possible to adequately describe the historical dynamics of various economic systems that differ in the balance of positive and negative exogenous factors.The general laws of STP are modeled using complex (including implicit) functions. Depending on the specifics of the process under study, complex functions of higher orders can be applied. From a mathematical point of view, the presented equations of dynamics are classified as ordinary autonomous first-order differential equations. Such a mathematical apparatus made it possible to demonstrate the need to observe the general scientific principle of the homogeneity of time. It is shown that limited scientific and technological progress can be described using the utility function.

Asymptotic analysis of a multiferroic model

We consider a system of nonlinear autonomous differential equations that describe the orientation of the antiferromagnetic vector in a multiferroic film and find the conditions for the existence of spatially modulated structures of the cycloid type. We investigate the stability of such structures under spatial perturbations.

A continuation principle for Fredholm maps II: application to homoclinic solutions

AbstractWhen studying the behavior of autonomous ordinary differential equations under time‐dependent perturbations vanishing for , their equilibria generically persist locally as homoclinic solutions. Using an abstract and flexible continuation theorem, we find even global branches of such homoclinic solutions for parametrized nonautonomous ordinary differential equations. Our approach is based on degree‐theoretical arguments. In particular, Landesman–Lazer conditions are proposed to obtain the existence of homoclinic solutions by means of a nonzero degree.

Complex Dynamics of a Novel Chaotic System Based on an Active Memristor

On the basis of the bistable bi-local active memristor (BBAM), an active memristor (AM) and its emulator were designed, and the characteristic fingerprints of the memristor were found under the applied periodic voltage. A memristor-based chaotic circuit was constructed, whose corresponding dynamics system was described by the 4-D autonomous differential equations. Complex dynamics behaviors, including chaos, transient chaos, heterogeneous coexisting attractors, and state-switches of the system were analyzed and explored by using Lyapunov exponents, bifurcation diagrams, phase diagrams, and Poincaré mapping, among others. In particular, a novel exotic chaotic attractor of the system was observed, as well as the singular state-switching between point attractors and chaotic attractors. The results of the theoretical analysis were verified by both circuit experiments and digital signal processing (DSP) technology.

You can create your own bifurcation

Textbooks on Dynamical Systems Theory usually present only three types of codimension-one bifurcations occurring in autonomous first-order nonlinear differential equations. These three types are called saddle-node, transcritical, and pitchfork. However, the number of bifurcations is virtually infinite. In this didactic work, the bifurcation diagrams of three other types of codimension-one bifurcations are analytically derived.

On stabilization of unstable steady states of autonomous ordinary differential equations via delayed feedback controls

The paper discusses stabilizing effects of some time-delayed feedback controls applied to unstable steady states of an autonomous system of ordinary differential equations. First, we derive explicit delay-dependent stability conditions that are applicable to a family of time-delayed systems with simultaneously triangularizable system matrices. Then, using this criterion and other argumentation, we employ diagonal delayed feedback controls of conventional and Pyragas type to stabilize unstable steady states of the studied autonomous system. More precisely, we formulate explicit, non-improvable and immediately applicable conditions on time delay and feedback strength that enable such a stabilization. As an illustration, we stabilize the unstable steady states of the Rössler dynamical system considered under the standard choice of entry parameters when the uncontrolled system displays a chaotic behavior. Also, we consider a non-diagonal feedback control (whose rotational gain matrix, involving a feedback strength and phase, commutes with the Jacobi matrix of the uncontrolled system) and show its larger stabilization potential with respect to the appropriate diagonal control. The obtained results are tested by numerical experiments and confronted with the existing results. As a supplement, we provide MATLAB codes supporting theoretical conclusions.

Contact-aided invariant extended Kalman filtering for robot state estimation

Legged robots require knowledge of pose and velocity in order to maintain stability and execute walking paths. Current solutions either rely on vision data, which is susceptible to environmental and lighting conditions, or fusion of kinematic and contact data with measurements from an inertial measurement unit (IMU). In this work, we develop a contact-aided invariant extended Kalman filter (InEKF) using the theory of Lie groups and invariant observer design. This filter combines contact-inertial dynamics with forward kinematic corrections to estimate pose and velocity along with all current contact points. We show that the error dynamics follows a log-linear autonomous differential equation with several important consequences: (a) the observable state variables can be rendered convergent with a domain of attraction that is independent of the system’s trajectory; (b) unlike the standard EKF, neither the linearized error dynamics nor the linearized observation model depend on the current state estimate, which (c) leads to improved convergence properties and (d) a local observability matrix that is consistent with the underlying nonlinear system. Furthermore, we demonstrate how to include IMU biases, add/remove contacts, and formulate both world-centric and robo-centric versions. We compare the convergence of the proposed InEKF with the commonly used quaternion-based extended Kalman filter (EKF) through both simulations and experiments on a Cassie-series bipedal robot. Filter accuracy is analyzed using motion capture, while a LiDAR mapping experiment provides a practical use case. Overall, the developed contact-aided InEKF provides better performance in comparison with the quaternion-based EKF as a result of exploiting symmetries present in system.

Existence results for a clamped beam equation with integral boundary conditions

In this paper we investigate the existence of positive solutions of fourthorder non autonomous differential equations with integral boundary conditions, the nonlinearity is a continuous function that depends on the spatial variable and its the second-order derivative. The approach relies an extension of Krasnoselskii’s fixed point theorem in a cone. Some examples are given to illustrate our results.