Narrow Results

In this paper, we study the number of limit cycles $H(n)$ of the quadratic reversible system (r19) with the switching curve $y=0$ under the perturbations of piecewise polynomials of $x,y$ with degree $n$. By using the first-order Melnikov function, it is proved that $H(n)=2n+1$ for $n=0,1,2$, $H(n)=2n$ for $n=3,4,5$, and $2n\le H(n)\le 2n+1$ for $n\ge 6$. It is also proved that the exact cyclicity of system (r19) under the smooth perturbations of polynomials of $x,y$ with degree $n$ is $n$ by the first-order Melnikov function. The results in this paper improve the estimations about the limit cycles bifurcating from (r19) in the literature.

In this paper, we consider a class of near-Hamiltonian systems with a nilpotent center, and study the number of limit cycles including algebraic limit cycles. We prove that there are at most $n(m+1)+1$ large amplitude limit cycles if the first-order Melnikov function is not zero identically, including an algebraic limit cycle. Moreover, it can have $\frac{n(n+3)}{2}$ when $m\ge n$ and $\frac{m(2n-m+1)}{2}+n$ when $m<n$ small limit cycles. We also provide two examples as applications of our main results.

This paper deals with two classes of polynomial differential systems that are generalized rigid systems, i.e. differential systems whose orbits rotate with a constant angular velocity. We give bounds for the maximum number of limit cycles of such polynomial differential systems provided by the averaging theory of first-order.

Quantitative models may exhibit sophisticated behaviour that includes having multiple steady states, bistability, limit cycles, and period-doubling bifurcation. Such behaviour is typically driven by the numerical dynamics of the model, where the values of various numerical parameters play the crucial role. We introduce in this paper natural correspondents of these concepts to reaction systems modelling, a framework based on elementary set theoretical, forbidding/enforcing-based mechanisms. We construct several reaction systems models exhibiting these properties.

In this paper, we determine conditions for planar systems of the form

The solid particles with different sizes exist widely, like cell separation, food processing, water treatment, thus, investigating the motion of the solid particles with different sizes is important. This study investigates the motion of a pair of neutrally buoyant circular particles with different sizes in a lid-driven square cavity using the lattice Boltzmann method. The motion of the circular particles with different sizes and that of the circular particles with identical sizes are quite different. The steady trajectories of the circular particles with identical sizes are identical, which is not affected by the Reynolds number. Differently, the circular particles with different sizes orbit along different steady trajectories, namely, the steady trajectory of the small particle is closer to the walls of the square cavity, while that of the large particle shrinks toward the center of the square cavity, which may provide us a possible method to separate them. However, it is not always effective, if the Reynolds number is low, the velocity difference between the circular particles with different sizes is small, which may fail to separate them completely.

Understanding, predicting and controlling the motion of the solid particles became active research topics, like debris flows, sand storms, transport of volcanic ash, coal combustion, drug delivery and steel making. Thus, the motion of a neutrally buoyant elliptical particle in a double-lid-driven square cavity is studied with the lattice Boltzmann method, where the effects of the initial position, Reynolds number and aspect ratio are studied. The square cavity is divided into two parts by the two primary vortexes, and the motion of the elliptical particle is determined by the interaction of the two primary vortexes. The limit cycle is sensitive to the initial position of the elliptical particle, namely, placing the elliptical particle at different positions initially, the motion of the elliptical particle exhibits different modes. Especially, the elliptical particle is apt to cross the horizontal centerline of the square cavity at moderate Reynolds numbers. The effect of the Reynolds number on the motion of the elliptical particle is significant, with the increase of the Reynolds number, the effects of the two primary vortexes on the motion of the elliptical particle are different, which affects the trajectory of the elliptical particle fundamentally. With the increase of the aspect ratio, the elliptical particle becomes flatter, the hydrodynamic force and torque acting on the elliptical particle become unbalanced and the elliptical particle no longer moves along the 8-like limit cycle.

Understanding, predicting and controlling the motion of the solid particles in the square cavity with obstacles are important. This work performs a lattice Boltzmann study on the motion of a neutrally buoyant circular particle in a lid-driven square cavity with a semicircular obstacle, where the effects of Reynolds numbers and initial positions of the circular particle on the motion of the circular particle are investigated. The motion of the circular particle is affected significantly by the semicircular obstacle, which is quite different from the case without obstacles. With the increase of the Reynolds number, the motion of the circular particle is divided into three stages. At low Reynolds numbers ($500\le \mathrm{\text{Re}}\le 800$), similar to the case without obstacles, no matter where the initial position of the circular particle is, the limit cycle of the circular particle is the same. Beyond a critical Reynolds number (between 800 and 900), the limit cycle of the circular particle is dependent on the initial position of the circular particle obviously, which is attributed to the opposing centrifugal and wall-repulsion forces.

We study the bacterial respiration through the numerical solution of the Fairen–Velarde coupled nonlinear differential equations. The instantaneous concentrations of the oxygen and the nutrients are computed. The fixed point solution and the stable limit cycle are found in different parameter ranges as predicted by the linearized differential equations. In a specified range of parameters, it is observed that the system spends some time near the stable limit cycle and eventually reaches the stable fixed point. This metastability has been investigated systematically. Interestingly, it is observed that the system exhibits two distinctly different time scales in reaching the stable fixed points. The slow time scale of the metastable lifetime near the stable limit cycle and a fast time scale (after leaving the zone of limit cycle) in rushing towards the stable fixed point. The gross residence time, near the limit cycle (described by a slow time scale), can be reduced by varying the concentrations of nutrients. This idea can be used to control the harmful metastable lifespan of active bacteria.

Ken Wilson developed powerful renormalization group procedures for constructing effective theories and solving a broad class of difficult physical problems. His insights allowed him to later advance the Hamiltonian approach to quantum dynamics of particles and fields in the Minkowski space–time, motivated by QCD. The latter advances are described in this article, concluding with a remark on Ken's related interest in difficult systemic issues of society.

We demonstrate here that the potential can coexist with limit cycle in nonlinear dissipative dynamics, where the potential plays the driving role for dynamics and determines the final steady state distribution in a manner similar to other situations in physics. First, we show the existence of limit cycle from a typical physics setting by an explicit construction: the potential is of the Mexican-hat shape, the strength of the magnetic field scales with that of the potential gradient near the limit cycle, and the friction goes to zero faster than that of potential gradient when approaching to the limit cycle. The dynamics at the limit cycle is conserved in this limit. The diffusion matrix is nevertheless finite at the limit cycle. Second, based on the physics knowledge, we construct the potential in the dynamics with limit cycle in a typical dynamical systems setting. Third, we argue that such a construction can be, in principle, carried out in a general situation combined with a novel method. Our present result may be useful in many applications, such as in the discussion of metastability of limit cycle and in the construction of Hopfield potential in the neural network computation.

Our current understanding of routes to chaos is mainly based on torus bifurcation where new periods are generated, the period-doubling mechanism revealed in the logistic map, and intermittency where periodic and burst motion appear alternatively. We present a possible new route to chaos based on our geometric picture of the frequency-locking of limit-cycles in semiconductor superlattices. In the period-double route and/or its variations, the period increases exponentially with bifurcation order, whereas the period in the new route increases linearly with the order of bifurcations.

In this paper, two different kinds of methods are adopted to control Liu system — feedback method and nonfeedback method. On the one hand, direct feedback and adaptive time-delayed feedback are taken as examples for the study of feedback control. In the direct feedback method, Liu system can be stabilized at one equilibrium point or a limit cycle surrounding its equilibrium. In the adaptive time-delayed feedback method, feedback coefficient and delay time can be adjusted adaptively to stabilize Liu system at its original unstable periodic orbit. On the other hand, periodic parametric perturbation is used to control chaos in Liu system as a typical nonfeedback method. By changing the frequency of the perturbation signal, Liu system can be guided to not only periodic motion but also hyperchaos. Numerical simulations show the effectiveness of our methods.

The center problem for degenerate singular points of planar systems (the degenerate-center problem) is a poorly-understood problem in the qualitative theory of ordinary differential equations. It may be broken down into two problems: the monodromy problem, to decide if the singular point is of focus-center type, and the stability problem, to decide whether it is a focus or a center.

We present an outline on the status of the center problem for degenerate singular points, explaining the main techniques and obstructions arising in the study of the problem. We also present some new results. Our new results are the characterization of a family of vector fields having a degenerate monodromic singular point at the origin, and the computation of the generalized first focal value for this family V₁. This gives the solution of the stability problem in the monodromic case, except when V₁ = 1. Our approach relies on the use of the blow-up technique and the study of the blow-up geometry of singular points. The knowledge of the blow-up geometry is used to generate a bifurcation of a limit cycle.

The original Hilbert's 16th problem can be split into four parts consisting of Problems A–D. In this paper, the progress of study on Hilbert's 16th problem is presented, and the relationship between Hilbert's 16th problem and bifurcations of planar vector fields is discussed. The material is presented in eight sections.

Section 1: Introduction: what is Hilbert's 16th problem?

Section 2: The first part of Hilbert's 16th problem.

Section 3: The second part of Hilbert's 16th problem: introduction.

Section 4: Focal values, saddle values and finite cyclicity in a fine focus, closed orbit and homoclinic loop.

Section 5: Finiteness problem.

Section 6: The weakened Hilbert's 16th problem.

Section 7: Global and local bifurcations of Z_q–equivariant vector fields.

Section 8: The rate of growth of Hilbert number H(n) with n.

We study the center-focus problem as well as the number of limit cycles which bifurcate from a weak focus for several families of planar discontinuous ordinary differential equations. Our computations of the return map near the critical point are performed with a new method based on a suitable decomposition of certain one-forms associated with the expression of the system in polar coordinates. This decomposition simplifies all the expressions involved in the procedure. Finally, we apply our results to study a mathematical model of a mechanical problem, the movement of a ball between two elastic walls.

Database Tomography (DT) is a textual database analysis system consisting of two major components: (1) algorithms for extracting multiword phrase frequencies and phrase proximities (physical closeness of the multiword technical phrases) from any type of large textual database, to augment (2) interpretative capabilities of the expert human analyst. DT was used to derive technical intelligence from a Nonlinear Dynamics database derived from the Science Citation Index/Social Science Citation Index (SCI). Phrase frequency analysis by the technical domain experts provided the pervasive technical themes of the Nonlinear Dynamics database, and the phrase proximity analysis provided the relationships among the pervasive technical themes. Bibliometric analysis of the Nonlinear Dynamics literature supplemented the DT results with author/journal/institution publication and citation data.

The focus of the paper is mainly on the existence of limit cycles of a planar system with third-degree polynomial functions. A previously developed perturbation technique for computing normal forms of differential equations is employed to calculate the focus values of the system near equilibrium points. Detailed studies have been provided for a number of cases with certain restrictions on system parameters, giving rise to a complete classification for the local dynamical behavior of the system. In particular, a sufficient condition is established for the existence of k small amplitude limit cycles in the neighborhood of a high degenerate critical point. The condition is then used to show that the system can have eight and ten small amplitude (local) limit cycles for a set of particular parameter values.

A general explicit formula is derived for controlling bifurcations using nonlinear state feedback. This method does not increase the dimension of the system, and can be used to either delay (or eliminate) existing bifurcations or change the stability of bifurcation solutions. The method is then employed for Hopf bifurcation control. The Lorenz equation and Rössler system are used to illustrate the application of the approach. It is shown that a simple control can be obtained to simultaneously stabilize two symmetrical equilibria of the Lorenz system, and keep the symmetry of Hopf bifurcations from the equilibria. For the Rössler system, a control is also obtained to simultaneously stabilize two nonsymmetric equilibria and meanwhile stabilize possible Hopf bifurcations from the equilibria. Computer simulation results are presented to confirm the analytical predictions.

In this paper, we study the number of limit cycles in a family of polynomial systems. Using bifurcation methods, we obtain the maximal number of limit cycles in global bifurcation.