The Center Conditions and Hopf Cyclicity for a 3D Lotka-Volterra System

The main objective of this paper is not only to find necessary and The main objective of this paper is not only to find necessary and sufficient conditions for the existence of a center on a local center manifold for a three dimensional Lotka-Volterra system, but also to determine the maximum number of limit cycles that can bifurcate from the positive equilibrium as a fine focus. Firstly, the singular point quantities are computed and simplified to obtain necessary conditions for local integrability, and Darboux method is applied to show the sufficiency. Then, the Hopf bifurcation on the center manifold is investigated, from this, the conclusion of at most five small limit cycles generated in the vicinity of the equilibrium is obtained. To the best of our knowledge, this is the first case with five possible limit cycles for the cyclicity of 3D Lotka-Volterra systems.     

Multiple recurrent outbreak cycles in an autonomous epidemiological model due to multiple limit cycle bifurcation

Multiple recurrent outbreak cycles have been commonly observed in infectious diseases such as measles and chicken pox. This complex outbreak dynamics in epidemiologicals is rarely captured by deterministic models. In this paper, we investigate a simple 2-dimensional SI epidemiological model and propose that the coexistence of multiple attractors attributes to the complex outbreak patterns. We first determine the conditions on parameters for the existence of an isolated center, then properly perturb the model to generate Hopf bifurcation and obtain limit cycles around the center. We further analytically prove that the maximum number of the coexisting limit cycles is three, and provide a corresponding set of parameters for the existence of the three limit cycles. Simulation results demonstrate the case with the maximum coexisting attractors, which contains one stable disease free equilibrium and two stable endemic periodic solutions separated by one unstable periodic solution. Therefore, different disease outcomes can be predicted by a single nonlinear deterministic model based on different initial data.     

The 3-D Hopf bifurcation in bio-reactor when one competitor produces a toxin

In this paper, a model of competition in the bio-reactor of two competitors for a single nutrient where one of the competitors can produce toxin against its opponent is investigated. The conditions of the three dimensional Hopf bifurcation are obtained. The Hopf bifurcation implies the existence of limit cycles in the model that corresponds to the nonlinear oscillation in the reactor.     

Dynamics and bifurcation study on an extended Lorenz system

In this paper, we study dynamics and bifurcation of limit cycles in a recently developed new chaotic system, called extended Lorenz system. A complete analysis is provided for the existence of limit cycles bifurcating from Hopf critical points. The system has three equilibrium solutions: a zero one at the origin and two non-zero ones at two symmetric points. It is shown that the system can either have one limit cycle around the origin, or three limit cycles enclosing each of the two symmetric equilibria, giving a total six limit cycles. It is not possible for the system to have limit cycles simultaneously bifurcating from all the three equilibria. Simulations are given to verify the analytical predictions.     

A 3D competitive Lotka–Volterra system with three limit cycles: A falsification of a conjecture by Hofbauer and So

For three-dimensional competitive Lotka–Volterra systems, Zeeman [M.L. Zeeman, Hopf bifurcations in competitive three-dimensional Lotka–Volterra systems, Dyn. Stab. Syst. 8 (1993) 189–217] identified 33 stable nullcline equivalence classes. Among these, only classes 26–31 may have limit cycles. Hofbauer and So [J. Hofbauer, J.W.-H. So, Multiple limit cycles for three dimensional Lotka–Volterra equations, Appl. Math. Lett. 7 (1994) 65–70] conjectured that the number of limit cycles is at most two for these systems. In this paper, we construct three limit cycles for class 29 without a heteroclinic polycycle in Zeeman’s classification.     

Cycle Chains and the LPA Model

The LPA model is a three dimensional system of nonlinear difference equations that has found many applications in population dynamics and ecology. In this paper, we consider a special case of the model (a case that approximates that used in experimental bifurcation and chaos studies) and prove several theorems concerning the existence and stability of periodic cycles, invariant loops, and chaos. Key is the notion of synchronous orbits (i.e. orbits lying on the coordinate planes). The main result concerns the existence of an invariant loop of synchronous orbits that bifurcates, in a nongeneric way, from the trivial equilibrium. The geometry and dynamics of this invariant loop are characterized. Specifically, it is shown that the loop is a cycle chain consisting of synchronous heteroclinic orbits connecting the three temporal phases of a synchronous 3-cycle. We also show that a period doubling route to chaos occurs within the class of synchronous orbits.     

Rolling of a ball without spinning on a plane: the absence of an invariant measure in a system with a complete set of integrals

In the paper we consider a system of a ball that rolls without slipping on a plane. The ball is assumed to be inhomogeneous and its center of mass does not necessarily coincide with its geometric center. We have proved that the governing equations can be recast into a system of six ODEs that admits four integrals of motion. Thus, the phase space of the system is foliated by invariant 2-tori; moreover, this foliation is equivalent to the Liouville foliation encountered in the case of Euler of the rigid body dynamics. However, the system cannot be solved in terms of quadratures because there is no invariant measure which we proved by finding limit cycles.     

On the existence of a stable limit cycle to a piecewise linear system in R2

The present paper is devoted to the existence of limit cycles of planar piecewise linear (PWL) systems with two zones separated by a straight line and singularity of type “focus-focus” and “focus-center.” Our investigation is a supplement to the classification of Freire et al concerning the existence and number of the limit cycles depending on certain parameters. To prove existence of a stable limit cycle in the case “focus-center,” we use a pure geometric approach. In the case “focus-focus,” we prove existence of a special configuration of five parameters leading to the existence of a unique stable limit cycle, whose period can be found by solving a transcendent equation. An estimate of this period is obtained. We apply this theory on a two-dimensional system describing the qualitative behavior of a two-dimensional excitable membrane model.     

一个三维Chemostat竞争系统的Hopf分支和周期解

本文研究了一个三维Chemostat竞争系统的解的结构,分析了平衡点的稳定性和当系统的某一微生物物种处于竞争劣势趋于灭绝时另一微生物物种和养料的二维流形上极限环的存在性,以及系统的Hopf分支问题.文中用Friedrich方法得到了系统存在Hopf分支的条件,并判定了周期解的稳定性.     

Complex Dynamics in a Hysteretic Relay Feedback System with Delay

Relay feedback systems are used to control engineering devices. In practice the switching between different functional forms of the system is never instantaneous, but takes place after a small delay. In this paper we analyse the dynamics and bifurcations of a representative example of such systems. In the absence of delay, negative feedback results only in unimodal symmetric limit cycles, but positive feedback can lead to aperiodic trajectories and chaos. In the presence of delay, the system can behave as an equivalent system without delay, provided that the delay is small in a sense which we define precisely. For larger delays, we identify a new bifurcation phenomenon, an event collision, where the delayed switching manifold intersects the relay hysteretic lines. In this case the dynamics become much more complicated.     

KAM-Type Theorem on Resonant Surfaces for Nearly Integrable Hamiltonian Systems

Summary. In this paper, we consider analytic perturbations of an integrable Hamiltonian system in a given resonant surface. It is proved that, for most frequencies on the resonant surface, the resonant torus foliated by nonresonant lower dimensional tori is not destroyed completely and that there are some lower dimensional tori which survive the perturbation if the Hamiltonian satisfies a certain nondegenerate condition. The surviving tori might be elliptic, hyperbolic, or of mixed type. This shows that there are many orbits in the resonant zone which are regular as in the case of integrable systems. This behavior might serve as an obstacle to Arnold diffusion. The persistence of hyperbolic lower dimensional tori has been considered by many authors [5], [6], [15], [16], mainly for multiplicity one resonant case. To deal with the mechanisms of the destruction of the resonant tori of higher multiplicity into nonhyperbolic lower dimensional tori, we have to deal with some small coefficient matrices that are the generalization of small divisors. Received December 18, 1997; revised December 30, 1998; accepted June 21, 1999     

Ten limit cycles around a center-type singular point in a 3-d quadratic system with quadratic perturbation

In this paper, we show that perturbing a simple 3-d quadratic system with a center-type singular point can yield at least 10 small-amplitude limit cycles around a singular point. This result improves the 7 limit cycles obtained recently in a simple 3-d quadratic system around a Hopf singular point. Compared with Bautin’s result for quadratic planar vector fields, which can only have 3 small-amplitude limit cycles around an elementary center or focus, this result of 10 limit cycles is surprisingly high. The theory and methodology developed in this paper can be used to consider bifurcation of limit cycles in higher-dimensional systems.     

On the number of limit cycles by perturbing a piecewise smooth Hamilton system with two straight lines of separation

This paper deals with the problem of limit cycle bifurcations for a piecewise smooth Hamilton system with two straight lines of separation. By analyzing the obtained first order Melnikov function, we give upper and lower bounds of the number of limit cycles bifurcating from the period annulus between the origin and the generalized homoclinic loop. It is found that the first order Melnikov function is more complicated than in the case with one straight line of separation and more limit cycles can be bifurcated.     

一类异宿环分支极限环的唯一性

本文研究平面上一类两点或三点异宿环附近极限环的分支,在一简洁条件下证明了异宿环分支极限环的唯一性,并给出了极限环唯一存在的充要条件.作为对三维余维2分支的应用,解决了所出现的两点异宿环产生唯一极限环的问题.     

Bifurcation of limit cycles from a center in two-parameter system

In this paper we consider a two-parameter perturbated system which takes the systems discussed in [1], [2], [3] as its special case. The bifurcation region of the limit cycles is given on the parameter plane. The author also studies the stability of the limit cycles. At the end of this paper the author discusses the difference of the bifurcation of the limit cycles from a center between the case of two-parameter and the case of one-parameter.     

A concrete example with multiple limit cycles for three dimensional Lotka–Volterra systems

The number of limit cycles for three dimensional Lotka–Volterra systems is an open problem. Recently, Yu et al. (2016) constructed some examples with the possibility of the existence of four limit cycles. Unfortunately, multiple limit cycles are not visible by numerical simulations, because all of them are very close to the interior equilibrium and extremely small. We present a concrete example with multiple limit cycles for three dimensional Lotka–Volterra systems which we can confirm them by numerical simulations. First we prepare the modified formula to compute coefficients of the normal form for the generalized Hopf bifurcation. Applying this formula to three dimensional Lotka–Volterra competitive systems with the aid of the computer algebra system, we derive the critical parameter values explicitly such that the interior equilibrium is exactly an unstable weak focus. Also we show that the heteroclinic cycle on the boundary of ${\mathbb{R}}_{+}^3$ is repelling. This implies that there exists a stable limit cycle by the Poincare–Bendixson theorem. Then, adding some suitable perturbations to parameters, we generate additional two limit cycles near the interior equilibrium by the generalized Hopf bifurcation. Finally we confirm that there exist three limit cycles by numerical simulations.     

On the number of limit cycles of a Z 4-equivariant quintic near-Hamiltonian system

In this paper, we study the number of limit cycles of a near-Hamiltonian system having Z₄- equivariant quintic perturbations. Using the methods of Hopf and heteroclinic bifurcation theory, we find that the perturbed system can have 28 limit cycles, and its location is also given. The main result can be used to improve the lower bound of the maximal number of limit cycles for some polynomial systems in a previous work, which is the main motivation of the present paper.     

一类带毒的生化反应竞争模型的定性分析

有些微生物在连续培养中会产生毒素来抑制竞争者,同时竞争中也会产生一些振荡行为.本文研究两个微生物竞争同一营养,而其中一个竞争者会产生毒素抑制另一竞争者且产物系数为一般的形如δ1=A1+B1Sn,δ2=A2+B2Sm的函数时的生化反应模型.分析了系统平衡点的稳定性和当系统的某一微生物物种处于竞争劣势而趋于灭绝时另一微生物物种和营养的二维流形上极限环的存在性.     

Nonlinear oscillations in business cycle model with time lags

In this paper we analyse the dynamics of the Kaldor–Kalecki business cycle model. This model is based on the classical Kaldor model in which capital stock changes are caused by past investment decisions. This lag is connected with time delay needed for new capital to be installed. The dynamics of the model is reduced to the form of damped oscillator with negative feedback connected with lag parameter and next it is analysed in terms of bifurcation theory. We find conditions for existence and persistence of oscillatory behaviour which is represented by limit cycle on some central manifold in phase space, i.e., single Hopf bifurcation. We demonstrate that the Hopf cycles may be exhibited for nonzero measure set of the parameter space. The conditions for bifurcation of co-dimension two connected with interaction of bifurcations as well as bifurcation diagrams are also given. Finally, we obtain numerical values describing an amplitude and a period of oscillation for different parameter of the system. It is also proved that while the investment function is not nonlinear a quasi-periodic solution (a 1:2 resonant double Hopf point) can appear. The source of such a behaviour is rather a consequence of time lag than nonlinearity of the investment function. Our results confirm the existence of asymmetric (two periodic) cycles in the Kaldor–Kalecki model with time-to-build.     

Nongeneric Bifurcations Near a Nontransversal Heterodimensional Cycle

In this paper bifurcations of heterodimensional cycles with highly degenerate conditions are studied in three dimensional vector fields,where a nontransversal intersection between the two-dimensional manifolds of the saddle equilibria occurs.By setting up local moving frame systems in some tubular neighborhood of unperturbed heterodimensional cycles,the authors construct a Poincar′e return map under the nongeneric conditions and further obtain the bifurcation equations.By means of the bifurcation equations,the authors show that different bifurcation surfaces exhibit variety and complexity of the bifurcation of degenerate heterodimensional cycles.Moreover,an example is given to show the existence of a nontransversal heterodimensional cycle with one orbit flip in three dimensional system.