System Of Autonomous Differential Equations Research Articles

Orbital Stability of Small Periodic Solutions of an Autonomous System of Differential Equations

Orbital Stability of Small Periodic Solutions of an Autonomous System of Differential Equations

On limit cycles of autonomous systems

We consider the problem of the existence of limit cycles for autonomous systems of differential equations. We present quite elementary considerations that can be useful in discussing qualitative issues that arise in the course of ordinary differential equations. We establish that any simple closed curve defined by the equation \(F(x,y)=1\) with a sufficiently general function \(F\) is a limit cycle for the corresponding autonomous system on the plane (and even for an infinite number of systems depending on the real parameter). These systems are written out explicitly. We analyze in detail several specific examples. Graphic illustrations are provided.

New insights from GW170817 in the dynamical system analysis of Einstein Gauss–Bonnet gravity

In this work, we explore the dynamical system phase space of Einstein–Gauss–Bonnet theory in the cosmological minisuperspace in the light of GW170817. This approach binds the main features of the theory through a system of autonomous differential equations, in the context of a flat Friedmann–Lemaître–Robertson–Walker spacetime. We analyze the critical points that feature in this system to assess their stability criteria. The phase space of this form of scalar-tensor gravity is very rich due to the fourth-order contributions of the Gauss–Bonnet invariant together with the second order contribution of the scalar field together with their coupling dynamics. We find additional critical points as compared with previous works in the literature which may be important for understanding the larger evolution of standard background cosmology within this class of gravitational models.

Center conditions to find certain degenerate centers with characteristic directions

We consider the two-dimensional autonomous systems of differential equations where the origin is a monodromic degenerate singular point, i.e., with null linear part. In this work we give two heuristic procedures to obtain some center conditions (perhaps not necessary) for certain degenerate centers at the origin although they have characteristic directions.

Qualitative stability analysis of cosmological parameters in f(T, B) gravity

We analyze the cosmological solutions of f(T, B) gravity using dynamical system analysis where T is the torsion scalar and B be the boundary term scalar. In our work, we assume three specific cosmological models. For first model, we consider f(T,B)=f_{0}(B^{k}+T^{m}), where k and m are constants. For second model, we consider f(T,B)=f_{0}T B, for third model, we consider f(T,B)=alpha T^{2}. We generate an autonomous system of differential equations for each models by introducing new dimensionless variables. To solve this system of equations, we use dynamical system analysis. We also investigate the critical points and their natures, stability conditions and their behaviors of Universe expansion. For first and second models, we get two stable critical points, while for third model we get one stable critical point. The phase plots of this system are analyzed in detail and study their geometrical interpretations also. For these three models, we evaluated density parameters such as Omega _{r}, Omega _{m}, Omega _{Lambda } and omega _{eff} and deceleration parameter (q) and find their suitable range of the parameter lambda for stability. For first model, we get omega _{eff}=-0.833,-0.166 and for second model, we get omega _{eff}=-frac{1}{3}. This shows that both the models are in quintessence phase. For third model we get accelerated expansion of the Universe. Further, we compare the values of EoS parameter and deceleration parameter with the observational values.

Dynamical Systems Analysis of f(Q) Gravity

Modified gravity theories can be used for the description of homogeneous and isotropic cosmological models through the corresponding field equations. These can be cast into systems of autonomous differential equations because of their sole dependence on a well-chosen time variable, be it the cosmological time, or an alternative. For that reason, a dynamical systems approach offers a reliable route to study those equations. Through a model-independent set of variables, we are able to study all f(Q) modified gravity models. The drawback of the procedure is a more complicated constraint equation. However, it allows the dynamical system to be formulated in fewer dimensions than using other approaches. We focus on a recent model of interest, the power-exponential model, and generalize the fluid content of the model.

A study on the bifurcation of the peristaltic driven flow of Ellis fluid

In this work, a bifurcation analysis is presented for the transport of Ellis fluid model due to peristaltic activity. The position, qualitative nature and bifurcations of stagnation-points are deliberated through the dynamical system approach. An exact expression for stream function is found by using the lubrication approach. Based on the stream function, a system of autonomous differential equations is established to analyze the equilibrium-points. Three kinds of flows are observed, namely backward flow, trapping and augmented flow. The transition of one flow into other represents a bifurcation. In the first bifurcation, a saddle point bifurcates into two centers in the normal direction while the second bifurcation emerges where nearby saddle nodes join under the wave trough and then split into two branches in the transverse direction. Ramifications of different embedded parameters on the bifurcations and nature of stagnation-points are explored through graphical representations. It is found that bifurcations transpire earlier for viscous fluids as compared to shear-thinning fluids. Further, bifurcations appear at low flow rate for small values of the material parameter . Moreover, in Newtonian fluids, as opposed to shear-thinning fluids, the trapping starts earlier and the backward region expands for larger values of the material parameter .

FINITE APPROXIMATIONS OF THE PROBLEM OF OPTIMAL IMPULSE STABILIZATION FOR A SYSTEM WITH AFTEREFFECT

A method for constructing finite-dimensional approximations for the optimal impulse stabilizing control of a linear autonomous system of differential equations with aftereffect is proposed. Auxiliary finite-dimensional problems of optimal stabilization of linear autonomous systems of ordinary differential equations with non-impulse controls are used. A procedure for reducing the dimension of a system of algebraic matrix Riccati equations is described.

Centralizers of Jacobian derivations

Let K be an algebraically closed field of characte-ristic zero, K[x,y] the polynomial ring in variables x, y and let W2(K) be the Lie algebra of all K-derivations on K[x,y]. A derivation D∈W2(K) is called a Jacobian derivation if there exists f∈K[x,y] such that D(h)=det J(f,h) for any h∈K[x,y] (hereJ(f,h) is the Jacobian matrix for f and h). Such a derivation is denoted by Df. The kernel of Df in K[x,y] is a subalgebra K[p] where p=p(x,y) is a polynomial of smallest degree such that f(x,y)=φ(p(x,y) for some φ(t)∈K[t]. Let C=CW2(K)(Df) be the centralizer of Df in W2(K). We prove that C is the free K[p]-module of rank 1 or 2 over K[p] and point out a criterion of being a module of rank 2. These results are used to obtain a classof integrable autonomous systems of differential equations.

Adding a reaction-restoration type transmission rate dynamic-law to the basic SEIR COVID-19 model.

The classical SEIR model, being an autonomous system of differential equations, has important limitations when representing a pandemic situation. Particularly, the geometric unimodal shape of the epidemic curve is not what is generally observed. This work introduces the βSEIR model, which adds to the classical SEIR model a differential law to model the variation in the transmission rate. It considers two opposite thrives generally found in a population: first, reaction to disease presence that may be linked to mitigation strategies, which tends to decrease transmission, and second, the urge to return to normal conditions that pulls to restore the initial value of the transmission rate. Our results open a wide spectrum of dynamic variabilities in the curve of new infected, which are justified by reaction and restoration thrives that affect disease transmission over time. Some of these dynamics have been observed in the existing COVID-19 disease data. In particular and to further exemplify the potential of the model proposed in this article, we show its capability of capturing the evolution of the number of new confirmed cases of Chile and Italy for several months after epidemic onset, while incorporating a reaction to disease presence with decreasing adherence to mitigation strategies, as well as a seasonal effect on the restoration of the initial transmissibility conditions.

Dynamic stability of [formula omitted]-varying non-zero torsion cosmology

This paper aims to study the stability analysis of Friedmann-like spacetime in the presence of torsion through dynamical system techniques. For this purpose, we consider equations of motion which describe the evolution in an isotropic and homogeneous cosmological background with non-vanishing torsion and convert them into the first order autonomous system of differential equations. In order to identify system’s equilibrium points, we implement an ansatz for the torsion term ϕ(t)=λH. Furthermore, within this framework, the cosmological scenario of the varying vacuum is being considered where dark matter interacts with vacuum. We take into account various forms of interaction term to study the effects of linear and non-linear interaction and examine the stability in each case. It is found that these cosmological solutions can describe different evolutionary phases of the universe and behave as stable attractor for the specific values of the parameters. For each case of interaction model, the phase space portraits exhibit the attractive behavior of the universe.

Bifurcations of Limit Cycles in Rotating Shafts Mounted On Partial Arc and Lemon Bore Journal Bearings in Elastic Pedestals

Abstract The paper investigates the bifurcations encountered in a simple rotor dynamic system interacting with nonlinear impedance forces, generated by the supporting journal bearings of realistic profile geometry. Bearing configurations of finite arc length and of finite width, as implemented in standard design of turbomachinery have been selected, namely the cylindrical partial arc and the elliptical (lemon) bore profile. The way in which the key design parameters influence the stability of elastic or rigid Jeffcott rotor is discussed. In the scope of this study, the following bearing design parameters are considered: arc length, length to diameter ratio, geometric preload and offset, and properties of the supporting pedestal by codimension-two studies. The bearing model is coupled to a six degree of freedom shaft-disc-pedestal model with nonlinear forces calculated from the journal kinematics, bearing design and operating conditions by numerical evaluation of the Reynolds equation for laminar, isothermal flow on a two dimensional mesh. An autonomous system of differential equations is implemented. Stability of fixed points and of limit cycles for this system is evaluated applying numerical continuation. The results confirm that minor variations in journal bearing design and pedestal properties have the potential to render substantial changes in the quality of stability and the bifurcation set of the rotor dynamic system. Specific bearing profiles render significant increment of instability threshold speed while at the same time supercritical Hopf bifurcations can be shifted to subcritical with resulting instability envelopes to be generated at speeds lower than the threshold speed.

Dynamical analysis of arbitrary dark energy and coincidence problem

This paper reviews a systematic dynamical analysis on a general form of scalar field as Dark Energy (DE) with dark matter (DM) to sort out the “cosmic coincidence” problem. Here the autonomous system of differential equations is two-dimensional (2D) as well as nonlinear. So we have utilized nonlinear dynamical theory to explain various cosmological implications of this model. Nowadays, we have noted that some works are undertaking this nonlinear systems theory. Although we have seen that most of the works are simplifying the underlying nonlinear dynamical systems similar to a linear one, that can lead to flawed conclusions about the evolution of the universe. Since an important theorem, Poincare–Bendixson theorem asserts linearization of the nonlinear system and does not give “global” stability, unlike the linear one if the dimension is more than two. Anyway, our work is different from others in this regard. Here the dimension of the system is two, and we have obtained some interesting stuffs also. We have applied the above theorem of nonlinear dynamical systems and others to find the “global” stability. This theorem offers completely different stable solutions, contrary to the prediction of linear analysis. As a result, we have obtained two fixed points; one of them is a stable “attractor” (it is attracting “node” actually), and thereafter, we have analyzed the stability. To investigate the dynamical system behavior, we have drawn different figures. These figures include vector field and a new plotting strategy (explained later). These investigations suggest a way out of the coincidence problem (or, precisely speaking, what should be the mathematical form of the term “[Formula: see text]”, which indicates interaction between DE and DM to reduce coincidence). In this scenario, if the equation of state (EoS) of DE and DM obeys [Formula: see text], then coincidence problem may be avoided.

Bifurcation analysis for a flow of viscoelastic fluid due to peristaltic activity

In this article, bifurcation analysis is performed to study the qualitative nature of stagnation points and various flow regions for a peristaltic transport of viscoelastic fluid through an axisymmetric tube. The rheological behavior of viscoelastic fluid is characterized by the simplified Phan–Than–Tanner fluid model. An analytic solution in a wave frame is obtained subject to the low Reynolds number and long wavelength approximations. The stagnation points and their bifurcations (critical conditions) are explored by developing a system of autonomous differential equations. The dynamical system theory is employed to examine the nature and bifurcations of obtained stagnation points. The ranges of various flow phenomena and their bifurcations are scrutinized graphically through global bifurcation diagrams. This analysis reveals that the bifurcation in the flow is manifested at large flow rate for high extensional parameter and Weissenberg number. Backward flow phenomenon enhances and trapping diminishes with an increase in the Weissenberg number. At the end, the results of present analysis are verified by making a comparison with the existing literature.

Analytical Model of the Two-Mass Above Resonance System of the Eccentric-Pendulum Type Vibration Table

The article deals with atwo-mass above resonant oscillatory system of an eccentric-pendulum type vibrating table. Based on the model of a vibrating oscillatory system with three masses, the system of differential equations of motion of oscillating masses with five degrees of freedom is compiled using generalized Lagrange equations of the second kind. For given values of mechanical parameters of the oscillatory system and initial conditions, the autonomous system of differential equations of motion of oscillating masses is solved by the numerical Rosenbrock method. The results of analytical modelling are verified by experimental studies. The two-mass vibration system with eccentric-pendulum drive in resonant oscillation mode is characterized by an instantaneous start and stop of the drive without prolonged transient modes. Parasitic oscillations of the working body, as a body with distributed mass, are minimal at the frequency of forced oscillations.

Analysis of Zika virus dynamics with sexual transmission route using multiple optimal controls

In this paper, we formulate and analyse a nonlinear optimal control problem for a Zika virus (ZIKV) infection model with sexual transmission route. An existing deterministic 11-dimensional autonomous system of differential equations is extended to include five time-dependent control functions, namely personal protection, condom use, vaccination, treatment and spraying of insecticide. The necessary conditions for the existence of optimal control quintuple are shown, and we determine the control strategies to minimize the spread dynamics of the ZIKV in the population at the minimum cost of control implementation. The derived optimality system is solved numerically to demonstrate the effectiveness of the different combinations of the five optimal controls in curtailing the transmission and spread of the disease. More importantly, we conduct a cost-effectiveness analysis on the combinations of at least four optimal controls to ascertain the most cost-effective strategy that can be used to hamper the ZIKV spread in the population.

Universe with quadratic equation of state: A dynamical systems perspective

This paper deals with the dynamical systems analysis of Friedmann–Robertson–Walker (FRW) model of the Universe in the framework of general relativity with quadratic equation of state and bulk viscosity. The evolution equations are transformed into an autonomous system of differential equations using suitable variables transformation. Stability analysis of cosmological models with quadratic equation of state parameter are discussed in detail in two different scenarios viz, first Universe filled with barotropic fluid and second filled with bulk viscous fluid. The nature of critical points is analyzed for both cases accordance with respective eigenvalues. We have also analyzed the stable attractor for both cases and examined their properties from the point of cosmological view.

Bifurcations of stagnation points in a micropolar fluent media under the influence of an asymmetric peristaltic movement

In the present analysis, the effects of an asymmetric peristaltic movement on the bifurcations of stagnation points have been investigated. An exact analytic solution for a flow of an incompressible micropolar fluid has been established under long wavelength and vanishing Reynolds number assumptions in a moving frame of reference. The stagnation points are located through a system of autonomous differential equations. The behavior and bifurcations of these stagnation points and corresponding streamline patterns have been epitomized through dynamical system methods. Different flow situations manifesting in the flow are characterized as follows: backward flow and trapping and augmented flow. Two possible bifurcations encountered in the flow because of the transitions between these flow regions, where nonhyperbolic degenerate points appear and heteroclinic connections between saddles are conceived. The micropolar parameter, coupling number, amplitude ratios, and phase difference have significant impacts on the bifurcations of the stagnation points and the ranges of the flow rate, which are explored graphically by local bifurcation diagrams. The backward flow region is observed to shrink by increasing the micropolar parameter up to an optimal value, and later an opposite trend is found. Furthermore, the increment in the coupling number causes the trapping region to expand. A reduction in the trapping phenomenon is encountered by enlarging the phase difference, while the augmented flow region becomes smaller for large amplitudes of peristaltic waves propagating along the walls of the channel. At the end, global bifurcation diagrams are used to summarize the obtained results.

First integrals and asymptotic trajectories

We discuss the relationship between the singular points of an autonomous system of differential equations and the critical points of its first integrals. Applying the well-known Splitting Lemma, we introduce local coordinates in which the first integral takes a “canonical” form. These coordinates make it possible to introduce a quasihomogeneous structure in some neighbourhood of any singular point and so to prove general theorems on the existence of asymptotic trajectories which go into or out of that singular point. We consider quasihomogeneous truncations of the original system of differential equations and show that if the singular point is isolated, the quasihomogeneous system is Hamiltonian. For a general mechanical system with two degrees of freedom, we prove a theorem on the instability of an equilibrium when it is neither a local minimum nor a local maximum of the potential energy. Bibliography: 21 titles.

On stochastic properties of dynamic chaos in systems of autonomous differential equations, such as the Lorentz system

On stochastic properties of dynamic chaos in systems of autonomous differential equations, such as the Lorentz system