Autonomous Differential Equations Research Articles

Simple Chaotic Flow with Circle and Square Equilibrium

Simple systems of third-order autonomous nonlinear differential equations can exhibit chaotic behavior. In this paper, we present a new class of chaotic flow with a square-shaped equilibrium. This unique property has apparently not yet been described. Such a system belongs to a newly introduced category of chaotic systems with hidden attractors that are interesting and important in engineering applications. The mathematical model is accompanied by an electrical circuit implementation, demonstrating structural stability of the strange attractor. The circuit is simulated with PSpice, constructed, and analyzed (measured).

The mathematical modelling of the impact of electric (Seebeck) effects on end-milling

It is a well-known fact that thermoelectric currents, reaching even the scale of ampere, develop during chip formation in the machine-workpiece-tool-chip. The impact of these currents on tool wear in end-milling was examined with a qualitative thermoelectric model, in which wear is described by an autonomous non-linear differential equation. Cutting temperature was measured by a so-called natural thermoelement with C60 and P35 carbide pair, which were submerged into heated Sn bath at validation. The mathematical model can be solved by a numerical method, where the inverse of the differential equation of wear was used. The wear curves determined by calculation fitted well with the measurement results. The results were achieved by cutting experiments using P35 carbide conducted on the C45 quality steel workpiece. It was found that the optimal solution with respect to the wear of the tool behaving as a natural thermal element might be achieved if the current is compensated by an external power source. Based on the model, anomalies might occur in some cases in the thermoelectric system. Further research is necessary to decide if this is only a special characteristic of the model or the model is interpreting really the actual processes.

Heteroclinics for non autonomous third order differential equations

We study the existence of heteroclinics connecting the two equilibria $\pm1$ of the third order differential equation $$u'''=f(u)+p(t)u'$$ where $f$ is a continuous function such that $f(u)(u^2-1)> 0$ if $u\neq\pm1$ and $p$ is a bounded non negative function. Uniqueness is also addressed.

Long-Term Dynamics of Autonomous Fractional Differential Equations

This paper aims to investigate long-term dynamic behaviors of autonomous fractional differential equations with effective numerical method. The long-term dynamic behaviors predict where systems are heading after long-term evolution. We make some modification and transplant cell mapping methods to autonomous fractional differential equations. The mapping time duration of cell mapping is enlarged to deal with the long memory effect. Three illustrative examples, i.e. fractional Lotka–Volterra equation, fractional van der Pol oscillator and fractional Duffing equation, are studied with our revised generalized cell mapping method. We obtain long-term dynamics, such as attractors, basins of attraction, and saddles. Compared with some existing stability and numerical results, the validity of our method is verified. Furthermore, we find that the fractional order has its effect on the long-term dynamics of autonomous fractional differential equations.

Prolongations, Suspensions and Telescopes

Autonomous differential equations induced by continuous vector fields usually appear in non-smooth mechanics and other scientific contexts. For these type of equations, given an initial condition, one has existence theorems but, in general, the uniqueness of the solution can not be ensured. For continuous vector fields, the equation solutions do not generally present a continuous flow structure; one particular but interesting case, occurs when under some initial conditions one can ensure existence of solutions and uniqueness in forward time obtaining in this case continuous semi-flows. The discretization and return Poincare techniques induce the corresponding discrete flows and semi-flows and some inverse methods as the suspension can construct a flow from a discrete flow or semi-flow. The objective of this work is to give categorical models for the diverse phase spaces of continuous and discrete semi-flows and flows and for the relations between these different phase spaces. We also introduce some new constructions such as the prolongation of continuous and discrete semi-flows and the telescopic functors. We consider small Top-categories (weakly enriched over the category Top of topological spaces) and we take as categorical models of the solutions of these differential equations some categories of continuous functors from a small Top-category to the category of topological spaces. Moreover, the processes of discretizations, suspensions, prolongations, et cetera are described in terms of adjoint functors. The main contributions of this paper are the construction of a tensor product associated to a functor between small Top-categories and the interpretation of prolongations, suspensions and telescopes as particular cases of this general tensor product. In general, the paper is focused on the establishment of links between category theory and dynamical systems more than on the study of differential equations using some categorical terminology.

Impact-and-Friction Interaction of Elastically Coupled Bodies with a Rough Plane

Design and mechanical-mathematical models of a system of three solids are constructed. Translating synchronously, these bodies fall on a rough plane, slide for a short time, and may rebound from it. A system of three autonomous differential equations is derived and analytically solved. It models three stages of the plane motion. The exact solutions of the system are obtained. Results of this numerical modeling are presented

Linear and weakly nonlinear magnetoconvection in a porous medium with a thermal nonequilibrium model

Two-dimensional magnetoconvection in a layer of Brinkman porous medium with local thermal nonequilibrium (LTNE) model is investigated by performing both linear and weakly nonlinear stability analyses. Condition for the occurrence of stationary and oscillatory convection is obtained in the case of linear stability analysis. It is observed that the presence of magnetic field is to introduce oscillatory convection once the Chandrasekhar number exceeds a threshold value if the ratio of the magnetic diffusivity to the thermal diffusivity is sufficiently small. Besides, asymptotic solutions for both small and large values of the inter-phase heat transfer coefficient are presented for the steady case. A weakly nonlinear stability analysis is performed by constructing a system of nonlinear autonomous ordinary differential equations. It is observed that subcritical steady convection is possible for certain choices of physical parameters. Heat transport is calculated in terms of Nusselt number. Increasing the value of Chandrasekhar number, inter-phase heat transfer coefficient and the inverse Darcy number is to decrease the heat transport, while increasing the ratio of the magnetic diffusivity to the thermal diffusivity and the porosity modified conductivity ratio shows an opposite kind of behavior on the heat transfer.

Multidimensional quasilinear first-order equations and multivalued solutions of the elliptic and hyperbolic equations

We discuss an extension of the theory of multidimensional second-order equations of the elliptic and hyperbolic types related to multidimensional quasilinear autonomous first-order partial differential equations. Calculating the general integrals of these equations allows constructing exact solutions in the form of implicit functions. We establish a connection with hydrodynamic equations. We calculate the number of free functional parameters of the constructed solutions. We especially construct and analyze implicit solutions of the Laplace and d’Alembert equations in a coordinate space of arbitrary finite dimension. In particular, we construct generalized Penrose–Rindler solutions of the d’Alembert equation in 3+1 dimensions.

Observance of period-doubling bifurcation and chaos in an autonomous ODE model for malaria with vector demography

We illustrate that an autonomous ordinary differential equation model for malaria transmission can exhibit period-doubling bifurcations leading to chaos when ecological aspects of malaria transmission are incorporated into the model. In particular, when demography, feeding, and reproductive patterns of the mosquitoes that transmit the malaria-causing parasite are explicitly accounted for, the resulting model exhibits subcritical bifurcations, period-doubling bifurcations, and chaos. Vectorial and disease reproduction numbers that regulate the size of the vector population at equilibrium and the endemicity of the malaria disease, respectively, are identified and used to simulate the model to show the different bifurcations and chaotic dynamics. A subcritical bifurcation is observed when the disease reproduction number is less than unity. This highlights the fact that malaria control efforts need to be long lasting and sustained to drive the infectious populations to levels below the associated saddle-node bifurcation point at which control is feasible. As the disease reproduction number increases beyond unity, period-doubling cascades that develop into chaos closely followed by period-halving sequences are observed. The appearance of chaos suggests that characterization of the physiological status of disease vectors can provide a pathway toward understanding the complex phenomena that are known to characterize the dynamics of malaria and other indirectly transmitted infections of humans. To the best of our knowledge, there is no known unforced continuous time deterministic host-vector transmission malaria model that has been shown to exhibit chaotic dynamics. Our results suggest that malaria data may need to be critically examined for complex dynamics.

A Progression and Retrogression Mathematical Model for the Motor Learning Process

We present a novel mathematical model based on the concepts of progression and retrogression. The goal of the model is to provide a more general description of the motor learning process. The model structure is given as a set of two autonomous linear differential equations with two coefficients that characterize the learning process. The coefficients are estimated using data obtained from the learning process. This paper examines how well the data complied with the model and whether the model reflects the human learning process with three biofeedback learning methods. To evaluate them, trainees used our prepared motor learning program for the motion of “walking without vertical acceleration of the head.” 82.2% of the learning process data complied with the assumptions of the P-R model. To evaluate the impact of feedback, trainees used three motor learning programs. Audio feedback was perceived as “clearer,” the somatosensory feedback as “clear,” and the visual feedback as “fuzzy,” for learning the motion, and the results were evaluated based on the progression speed normalized by retrogression speed calculated by the model, which was 0.25 for the visual feedback learning program, 0.28 for the somatosensory feedback, and 0.69 for the audio feedback. These values had statistically significant differences. Thus, the descriptive model can express successive repeated learning processes under progression and retrogression components.

A solution method for autonomous first-order algebraic partial differential equations

In this paper we present a procedure for solving first-order autonomous algebraic partial differential equations in an arbitrary number of variables. The method uses rational parametrizations of algebraic (hyper)surfaces and generalizes a similar procedure for first-order autonomous ordinary differential equations. In particular we are interested in rational solutions and present certain classes of equations having rational solutions. However, the method can also be used for finding non-rational solutions.

On the nonautonomous Hopf bifurcation problem

Under well-known conditions, a one-parameter family of two-dimensional, autonomous ordinary differential equations admits a supercritical\break Andronov-Hopf bifurcation. Let such a family be perturbed by a non-autonomous term. We analyze the sense in which and some conditions under which the Andronov-Hopf pattern persists under such a perturbation.

Stability analysis of a delayed SIS epidemiological model

In this paper, we investigate global stability of the endemic steady state of the SIS epidemic model studied recently in terms of fluctuations. The epidemiological model has an exponential demographic structure, disease-related deaths and a delay corresponding to the infectious period. The disease spread is governed by a scalar delay differential equation. Our study is mainly based on the monotone dynamical systems theory. We begin by simplifying the classical framework of stability analysis for non-quasi-monotone scalar autonomous delay differential equations by using recent results on essentially strongly order-preserving semiflows. Under certain conditions, it is proved that the endemic equilibrium is globally asymptotically stable on a closed subset of the phase space C. To our knowledge, this is the first study proving the endemic steady state's global stability for this model. Numerical simulations which illustrate the results are carried out.

Differential Equations of Electrodiffusion: Constant Field Solutions, Uniqueness, and New Formulas of Goldman--Hodgkin--Katz Type

The equations governing one-dimensional, steady-state electrodiffusion are considered when there are arbitrarily many mobile ionic species present, in any number of valence classes, possibly also with a uniform distribution of fixed charges. Exact constant field solutions and new formulas of Goldman-Hodgkin-Katz type are found. All of these formulas are exact, unlike the usual approximate ones. Corresponding boundary conditions on the ionic concentrations are identified. The question of uniqueness of constant field solutions with such boundary conditions is considered, and is re-posed in terms of an autonomous ordinary differential equation of order $n+1$ for the electric field, where $n$ is the number of valence classes. When there are no fixed charges, the equation can be integrated once to give the non-autonomous equation of order $n$ considered previously in the literature including, in the case $n=2$, the form of Painlev\'e's second equation considered first in the context of electrodiffusion by one of us. When $n=1$, the new equation is a form of Li\'enard's equation. Uniqueness of the constant field solution is established in this case.

Boundedness and stability in third order nonlinear differential equations with multiple deviating arguments

In this paper, we establish some new sufficient conditions which guarantee the stability and boundedness of solutions of certain nonlinear and non autonomous differential equations of third order with delay. By defining appropriate Lyapunov function, we obtain some new results on the subject. By this work, we extend and improve some stability and boundedness results in the literature.

On differential equations with state-dependent delay: The principles of linearized stability and instability revisited

This paper deals with a dynamical systems approach for studying nonlinear autonomous differential equations with bounded state-dependent delay. Starting with the semiflow generated by solutions of such an equation, we revisit the principles of linearized stability and instability enabling the local stability analysis of equilibria via linearization. In particular, we prove both principles in an elementary way by using only a quantitative version of continuous dependence of the semiflow on initial data together with basic properties of the discrete semi-dynamical system induced by iterations of some time-t-map.

PERIODIC SOLUTIONS OF SYSTEMS OF AUTONOMOUS DIFFERENTIAL EQUATIONS WITH VARIABLE STRUCTURE AND IMPULSES

Autonomous systems of differential equations with variable struc- ture and impulsive effects are the main objects of our study. The structure changes and the impulsive effects realize at the switching moments, which are specific to each different solution of the system. In these moments, the trajec- tory of the corresponding initial value problem meets successively the switching sets. For this class of equations, sufficient conditions for t existence of peri- odic solutions are found. The results are based on the Brouwer's fixed point theorem. We investigate the question of existence of periodic solutions of a generalized model of predator-prey community.

Exploring branched Hamiltonians for a class of nonlinear systems

One of the less well-understood ambiguities of quantization is emphasized to result from the presence of higher-order time derivatives in the Lagrangians resulting in multiple-valued Hamiltonians. We explore certain classes of branched Hamiltonians in the context of nonlinear autonomous differential equation of Liénard type. Two eligible elementary nonlinear models that emerge are shown to admit a feasible quantization along these lines.

On the Response of Autonomous Sweeping Processes to Periodic Perturbations

If x 0 is an equilibrium of an autonomous differential equation $\dot x=f(x)$ and det∥f ′(x 0)∥≠0, then x 0 persists under autonomous perturbations and x 0 transforms into a T-periodic solution under non-autonomous T-periodic perturbations. In this paper we discover a similar structural stability for Moreau sweeping processes of the form $-\dot u\in N_{B}(u)+f_{0}(u)$ , $u\in \mathbb {R}^{2}, $ i. e. we consider the simplest case where the derivative is taken with respect to the Lebesgue measure and where the convex set B of the reduced system is a non-moving unit ball of $\mathbb {R}^{2}$ . We show that an equilibrium ∥u 0∥=1 persists under periodic perturbations, if the projection $\overline {f}:\partial B\to \mathbb {R}^{2}$ of f 0 on the tangent to the boundary ∂ B is nonsingular at u 0.

Periodic bifurcation problems for fully nonlinear neutral functional differential equations via an integral operator approach: the multidimensional degeneration case

We consider a $T$-periodically perturbed autonomous functional differential equation of neutral type. We assume the existence of a $T$-periodic limit cycle $x_0$ for the unperturbed autonomous system. We also assume that the linearized unperturbed equation around the limit cycle has the characteristic multiplier $1$ of geometric multiplicity $1$ and algebraic multiplicity greater than~$1$. The paper deals with the existence of a branch of $T$-periodic solutions emanating from the limit cycle. The problem of finding such a branch is converted into the problem of finding a branch of zeros of a~suitably defined bifurcation equation \hbox{$P(x,\varepsilon) +\varepsilon Q(x, \varepsilon)=0$.} The main task of the paper is to define a novel equivalent integral operator having the property that the $T$-periodic adjoint Floquet solutions of the unperturbed linearized operator correspond to those of the equation $P'(x_0(\theta),0)=0$, $\theta\in[0,T]$. Once this is done it is possible to express the condition for the existence of a branch of zeros for the bifurcation equation in terms of a multidimensional Malkin bifurcation function.