Search Results (10)

After linear differential systems in the plane, the easiest systems are quadratic polynomial differential systems in the plane. Due to their nonlinearity and their many applications, these systems have been studied by many authors. Such quadratic polynomial differential systems have been divided into ten families. Here, for two of these families, we classify all topologically distinct phase portraits in the Poincaré disc. These two families have already been studied previously, but several mistakes made there are repaired here thanks to the use of a more powerful technique. This new technique uses the invariant theory developed by the Sibirskii School, applied to differential systems, which allows to determine all the algebraic bifurcations in a relatively easy way. Even though the goal of obtaining all the phase portraits of quadratic systems for each of the ten families is not achievable using only this method, the coordination of different approaches may help us reach this goal. Full article

In this paper, we describe the global dynamics of the Painlevé–Gambier equations numbered XVIII: $x^{″}-{\left(x^{\prime }\right)}^2/\left(2x\right)-4x^2=0$, XXI: $x^{″}-3{\left(x^{\prime }\right)}^2/\left(4x\right)-3x^2$, and XXII: $x^{″}-3{\left(x^{\prime }\right)}^2/\left(4x\right)+1=0$. We obtain three rational functions as their first integrals and classify their phase portraits in the Poincaré disc. The main reason for considering these three Painlevé–Gambier equations is due to the paper of Guha, P., et al., where the authors studied these three differential equations in order to illustrate a method to generate nonlocal constants of motion for a special class of nonlinear differential equations. Here, we want to complete their studies describing all of the dynamics of these equations. This demonstrates that the phase portraits of equations XVIII and XXI in the Poincaré disc are topologically equivalent. Full article

Planar differential systems whose angular velocity is constant are called rigid or uniform differential systems. The first rigid system goes back to the pendulum clock of Christiaan Huygens in 1656; since then, the interest for the rigid systems has been growing. Thus, at this moment, in MathSciNet there are 108 articles with the words rigid systems or uniform systems in their titles. Here, we study the dynamics of the planar rigid polynomial differential systems with homogeneous nonlinearities of arbitrary degree. More precisely, we characterize the existence and non-existence of limit cycles in this class of rigid systems, and we determine the local phase portraits of their finite and infinite equilibrium points in the Poincaré disc. Finally, we classify the global phase portraits in the Poincaré disc of the rigid polynomial differential systems of degree two, and of one class of rigid polynomial differential systems with cubic homogeneous nonlinearities that can exhibit one limit cycle. Full article

The following differential quadratic polynomial differential system  $\frac{dx}{dt}=y-x, \frac{dy}{dt}=2y-\frac{y}{\gamma -1}\left(2-\gamma y-\frac{5\gamma -4}{\gamma -1}x\right),$ when the parameter $\gamma \in (1,2]$ models the structure equations of an isotropic star having a linear barotropic equation of state, being $x=m\left(r\right)/r$ where $m\left(r\right)\ge 0$ is the mass inside the sphere of radius r of the star, $y=4\pi r^2\rho$ where $\rho$ is the density of the star, and $t=ln(r/R)$ where R is the radius of the star. First, we classify all the topologically non-equivalent phase portraits in the Poincaré disc of these quadratic polynomial differential systems for all values of the parameter $\gamma \in \mathbb{R}\setminus \left\{1\right\}$. Second, using the information of the different phase portraits obtained we classify the possible limit values of $m\left(r\right)/r$ and $4\pi r^2\rho$ of an isotropic star when r decreases. Full article

The quadratic polynomial differential systems in a plane are the easiest nonlinear differential systems. They have been studied intensively due to their nonlinearity and the large number of applications. These systems can be classified into ten classes. Here, we provide all topologically different phase portraits in the Poincaré disc of two of these classes. Full article

Roughly speaking, the Poincaré disc ${\mathbb{D}}^2$ is the closed disc centered at the origin of the coordinates of ${\mathbb{R}}^2$, where the whole of ${\mathbb{R}}^2$ is identified with the interior of ${\mathbb{D}}^2$ and the circle of the boundary of ${\mathbb{D}}^2$ is identified with the infinity of ${\mathbb{R}}^2$, because in the plane ${\mathbb{R}}^2$, we can go to infinity in as many directions as points have the circle. The phase portraits of the quadratic Hamiltonian systems in the Poincaré disc were classified in 1994. Since then, no new interesting classes of Hamiltonian systems have been classified on the Poincaré disc. In this paper, we determine the phase portraits in the Poincaré disc of five classes of homogeneous Hamiltonian polynomial differential systems of degrees 1, 2, 3, 4, and 5 with finitely many equilibria. Moreover, all these phase portraits are symmetric with respect to the origin of coordinates. We showed that these polynomial differential systems exhibit precisely 2, 2, 3, 3, and 4 topologically distinct phase portraits in the Poincaré disc. Of course, the new results are for the homogeneous Hamiltonian polynomial differential systems of degrees 3, 4, and 5. The tools used here for obtaining these phase portraits also work for obtaining any phase portrait of a homogeneous Hamiltonian polynomial differential system of arbitrary degree. Full article

We apply the theory of quadratic differentials, to present a classification of orthogonal pairs of foliations of the hyperbolic plane by hyperbolic conics. Light rays are represented by trajectories of meromorphic differentials, and mirrors are represented by trajectories of the quadratic differential that represents the geometric mean of two such differentials. Using the notion of a hyperbolic conic as a mirror, we classify the types of orthogonal pairs of foliations of the hyperbolic plane by confocal conics. Up to diffeomorphism, there are nine types: three of these types admit one parameter up to isometry; the remaining six types are unique up to isometry. The families include all possible hyperbolic conics. Full article

In aero-engines, the rotor systems are frequently designed with multistage discs, in which the discs are fastened together through bolted joints. During operation, rotating machines are susceptible to rotor–stator rubbing faults. Those bolted joints are subjected to friction and impact forces during a rubbing event, leading to a dramatic change in mechanical properties at the contacting interfaces, influencing the rotor dynamics, which have attracted the attention of scholars. In the present work, a mathematical model, which considers the unbalance force, rotor dimensional properties, nonlinear oil-film force and rub-impact effect, is developed to study the bifurcation and stability characteristics of the bolted joint rotor system containing multi-discs subjected to the rub-impact effect. The time-domain waveforms of the system are obtained numerically by using the Runge–Kutta method, and a bifurcation diagram, time domain waveforms, spectrum plots, shaft orbits and Poincaré maps are adopted to reveal the rotor dynamics under the effect of the rub-impact. Additionally, the influences of rubbing position at the multi-discs on rotor dynamic properties are also examined through bifurcation diagrams. The numerical simulation results show that the segments of the rotating speeds for rubbing are wider and more numerous, and the middle disc is subjected to the rub-impact. When the rub-impact position is far away from disc 1, the rubbing force has little effect on the response of disc 1. The corresponding results can help to understand the bifurcation characteristics of a bolted joint rotor system containing multi-discs subjected to the rub-impact effect. Full article

This article is dedicated to investigating the nonlinear dynamical behaviors of the 8-pole rotor active magnetic bearing system. The rub and impact forces between the rotating disc and the pole-legs are included in the studied model for the first time. A new control scheme based on modifying the 8-pole positions has been introduced. The proposed control methodology is designed such that four poles only are located in the horizontal and vertical directions (i.e., in $+X,+Y,-X,-Y$ directions)$,$ while the other four poles are inserted in a way such that each pole makes 45° with two of the axes $+X,+Y,-X,-Y$. The control currents in the horizontal and vertical poles are suggested to be proportional to both the velocity and displacement of the rotor in the horizontal and vertical directions, respectively, while the control currents in the inclined poles are proposed to be dependent on the combination of both the displacement and velocity of the rotor in the horizontal and vertical directions. Accordingly, the whole-system mathematical model is derived. The derived discontinuous dynamical system is analyzed employing perturbation methods, Poincare maps, bifurcation diagrams, whirling orbits, and frequency spectrum. The obtained results demonstrated that the controller proportional control gain can play a significant role in changing the vibratory behaviors of the system, where the proposed control method can behave either as a cartesian control strategy or as a radial control one depending on the magnitude of the proportional gain. In addition, it is found that the rotor system can vibrate with periodic, periodic-n, quasiperiodic, or chaotic motion when the rub and/or impact forces occur. Moreover, it is reported for the first time that the rotor-AMB can oscillate symmetrically in $X$ and $Y$ directions either in full annular rub mode or quasiperiodic partial rub mode depending on the impact stiffness coefficient and the dynamic friction coefficient. Full article

This study numerically analyzes and investigates the effects of the bearing design parameters of a tilting pad journal bearing (TPJB) on the pad-pivot friction-induced nonlinear rotordynamic phenomena and bifurcations. The bearing parameters were set to the pad preload, pivot offset, spherical pivot radius, and bearing length to diameter (L/D) ratio. The Stribeck curve model (SCM) model was applied at the contact surface between the pad and the pivot, which varied to the boundary-mixed-fluid friction state depending on the friction condition. The rotor-bearing model was set up with a symmetrical five-pad TPJB system supporting a Jeffcott type rigid rotor. The fluid repelling force generated in the oil film between each pad and the shaft was calculated using a finite element method. The simulation recurrently conducted the transient numerical integration to obtain the Poincaré maps and phase states of the journal and pad with various bearing design variables, then the nonlinear properties of each condition were analyzed by expressing the bifurcation diagrams. As a result, the original findings of this study are: (1) The pad preload and pivot offset significantly influenced the emergence of Hopf bifurcations and the associated limit cycles. In contrast, (2) the pivot radius and L/D ratio contributed relatively less to the friction-induced instability. Resultantly, (3) all the effects diminished when the rotor operated under the larger mass eccentricity of the disc. Full article