Cycles of two-dimensional systems: Computer calculations,proofs, and experiments

One of the central problems in studying small cycles in the neighborhood of equilibrium involves computation of Lyapunov’s quantities. While Lyapunov’s first and second quantities were computed in the general form in the 1940s–1950s, Lyapunov’s third quantity was calculated only for certain special cases. In the present work, we present general formulas for calculation of Lyapunov’s third quantity. Together with the classical Lyapunov method for calculation of Lyapunov’s quantities, which is based on passing to the polar coordinates, we suggest a method developed for the Euclidian coordinates and for the time domain. The calculation of Lyapunov’s quantities by two different analytic methods involving modern software tools for symbolic computing enables us to justify the formulas obtained for Lyapunov’s third quantity. For quadratic systems in which Lyapunov’s first and second quantities vanish, while the third one does not, large cycles were calculated. In the calculations, the quadratic system was reduced to the Liénard equation, which was used to evaluate the domain of parameters corresponding to the existence of four cycles (three “small” cycles and a “large” one). This domain extends the region of parameters obtained by S.L. Shi in 1980 for a quadratic system with four 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.     

Lyapunov quantities and limit cycles of two-dimensional dynamical systems. Analytical methods and symbolic computation

In the present work the methods of computation of Lyapunov quantities and localization of limit cycles are demonstrated. These methods are applied to investigation of quadratic systems with small and large limit cycles. The expressions for the first five Lyapunov quantities for general Lienard system are obtained. By the transformation of quadratic system to Lienard system and the method of asymptotical integration, quadratic systems with large limit cycles are investigated. The domain of parameters of quadratic systems, for which four limit cycles can be obtained, is determined.     

Algebraic limit cycles of degree 4 for quadratic systems

Yablonskii (Differential Equations 2 (1996) 335) and Filipstov (Differential Equations 9 (1973) 983) proved the existence of two different families of algebraic limit cycles of degree 4 in the class of quadratic systems. It was an open problem to know if these two algebraic limit cycles where all the algebraic limit cycles of degree 4 for quadratic systems. Chavarriga (A new example of a quartic algebraic limit cycle for quadratic sytems, Universitat de Lleida, Preprint 1999) found a third family of this kind of algebraic limit cycles. Here, we prove that quadratic systems have exactly four different families of algebraic limit cycles. The proof provides new tools based on the index theory for algebraic solutions of polynomial vector fields.     

Quadratic systems with maximum number of limit cycles

We suggest a method for obtaining quadratic systems with a given distribution of limit cycles. We use it to obtain a set of quadratic systems with the distributions (3, 1), (3, 0), and 3 of limit cycles and with different configurations of singular points. The distributions are justified with the use of a modified Dulac function in a natural domain of existence of limit cycles.     

Conditions for the Existence of Two Limit Cycles in a System with Hysteresis Nonlinearity

This work deals with a two-dimensional automatic control system containing a single nonlinear hysteretic element in the general form. The conditions sufficient for the existence of at least two limit cycles in the system are presented. To prove the existence of cycles, three closed contours embedded into each other are constructed on the phase manifold by “sewing” together pieces of the level lines of various Lyapunov functions. System trajectories cross the inner contour “from outside inwards” and the middle contour “from inside outwards.” The outer contour is crossed by system trajectories “from outside inwards.” The existence of these contours proves the presence of at least two limit cycles in the system. This paper is a continuation of our earlier published work “Conditions for the Global Stability of a Single System with Hysteresis Nonlinearity,” in which the conditions of global stability in this system are formulated.     

Relationships between limit cycles and algebraic invariant curves for quadratic systems

Algebraic limit cycles for quadratic systems started to be studied in 1958. Up to now we know 7 families of quadratic systems having algebraic limit cycles of degree 2, 4, 5 and 6. Here we present some new results on the limit cycles and algebraic limit cycles of quadratic systems. These results provide sometimes necessary conditions and other times sufficient conditions on the cofactor of the invariant algebraic curve for the existence or nonexistence of limit cycles or algebraic limit cycles. In particular, it follows from them that for all known examples of algebraic limit cycles for quadratic systems those cycles are unique limit cycles of the system.     

Limit cycle bifurcations by perturbing piecewise smooth Hamiltonian systems with multiple parameters

This paper is concerned with the problem of limit cycle bifurcation for piecewise smooth near-Hamiltonian systems with multiple parameters. By the first Melnikov function, some novel criteria have been established for the existence of multiple limit cycles. Furthermore, an example is included to validate the obtained results by considering the maximum number of limit cycles for a piecewise quadratic system studied in Llibre and Mereu (2014) [12]. Compared with the result in the above reference, one more limit cycle is found by our method.     

EXISTENCE OF LIMIT CYCLES FOR QUADRATIC SYSTEMS WITH ONE INFINITE SINGULAR POINT

This paper discusses the existence of limit cycles for quadratic systems with one infinite singular point. In particular, the author gives necessary anal sufficient conditions for the existence of a unique limit cycle of a class of quadratic systems with an integral line under certain conditions. As a special consequence, all the results of [-6] and [7] can be implied by means of affine transformations. The method used here is much simpler than those of [6] and [7].     

Centers and isochronous centers of a class of quasi-analytic switching systems

In this paper, we study the integrability and linearization of a class of quadratic quasi-analytic switching systems. We improve an existing method to compute the focus values and periodic constants of quasianalytic switching systems. In particular, with our method, we demonstrate that the dynamical behaviors of quasi-analytic switching systems are more complex than those of continuous quasi-analytic systems, by showing the existence of six and seven limit cycles in the neighborhood of the origin and infinity, respectively, in a quadratic quasi-analytic switching system. Moreover, explicit conditions are obtained for classifying the centers and isochronous centers of the system.     

Limit cycles for a perturbation of a quadratic center with symmetry

To estimate the number of limit cycles appearing under a perturbation of a quadratic system that has a center with symmetry, we use the method of generalized Dulac functions. To this end, we reduce the perturbed system to a Liénard system with a small parameter, for which we construct a Dulac function depending on the parameter. This permits one to estimate the number of limit cycles in the perturbed system for all sufficiently small parameter values. We find the Dulac function by solving a linear programming problem. The suggested method is used to analyze four specific perturbed systems that globally have exactly three limit cycles [i.e., the limit cycle distribution 3 or (3, 0)] and two systems that have the limit cycle distribution (3, 1) (i.e., one nest around each of the two foci).     

Limit cycles under perturbations of a quadratic Hamiltonian system

The perturbed quadratic Hamiltonian system is reduced to a Lienard system with a small parameter for which a Dulac function depending on it is constructed. This permits one to estimate the number of limit cycles of the perturbed system for all sufficiently small parameter values. To find the Dulac function, we use the solution of a linear programming problem. The suggested method is used for studying three specific perturbed systems that have exactly two limit cycles, i.e., the distribution 2 or (0, 2), and one system with distribution (1, 1).     

关于二次微分系统Ⅰ类方程的极限环的存在性

本文讨论二次微分系统Ⅰ类方程ｄｘｄｔ＝ － ｙ＋ δｘ＋ ｌｘ２＋ ｍ ｘｙ＋ ｎｙ２,ｄｙｄｔ＝ ｘ 的极限环的存在性问题,纠正了文［１］中讨论当ｌ＝ ０ 时大范围内存在极限环的缺陷     

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.     

Twelve limit cycles around a singular point in a planar cubic-degree polynomial system

In this paper, we prove the existence of 12 small-amplitude limit cycles around a singular point in a planar cubic-degree polynomial system. Based on two previously developed cubic systems in the literature, which have been proved to exhibit 11 small-amplitude limit cycles, we applied a different method to show 11 limit cycles. Moreover, we show that one of the systems can actually have 12 small-amplitude limit cycles around a singular point. This is the best result so far obtained in cubic planar vector fields around a singular point.     

The number and stability of limit cycles for planar piecewise linear systems of node–saddle type

The objective of this paper is to study the number and stability of limit cycles for planar piecewise linear (PWL) systems of node–saddle type with two linear regions. Firstly, we give a thorough analysis of limit cycles for Liénard PWL systems of this type, proving one is the maximum number of limit cycles and obtaining necessary and sufficient conditions for the existence and stability of a unique limit cycle. These conditions can be easily verified directly according to the parameters in the systems, and play an important role in giving birth to two limit cycles for general PWL systems. In this step, the tool of a Bendixon-like theorem is successfully employed to derive the existence of a limit cycle. Secondly, making use of the results gained in the first step, we obtain parameter regions where the general PWL systems have at least one, at least two and no limit cycles respectively. In addition for the general PWL systems, some sufficient conditions are presented for the existence and stability of a unique one and exactly two limit cycles respectively. Finally, some numerical examples are given to illustrate the results and especially to show the existence and stability of two nested limit cycles.     

THE POINCAR BIFURCATION OF QUADRATIC SYSTEMS HAVING A REGION CONSISTING OF PERIODIC CYCLES BOUNDED BY A HYPERBOLA AND AN ARC OF EQUATOR

The usual bifureation problems oeeurred in the study of the Planar differential systems having a region eonsisting of Periodie eycles are as follows:15 there any closed orbit whieh generates limit eyeles after a small perturbation?How many limit eyel     

Some criteria for the existence of limit cycles for quadratic vector fields

In this paper we consider real quadratic systems. We present new criteria for the existence and uniqueness of limit cycles for such systems by using Darbouxian particular solutions. Some results are based on the study of such systems in . We also generalize the well-know result of Bautin on the nonexistence of limit cycles for quadratic Lotka-Volterra systems.     

On solving the potential equation

We study radial solutions to the generalized Swift-Hohenberg equation on the plane with an additional quadratic term. We find stationary localized radial solutions that decay at infinity and solutions that tend to constants as the radius increases unboundedly (“droplets”). We formulate existence theorems for droplets and sketch the proofs employing the properties of the limit system as r → ∞. This system is a Hamiltonian system corresponding to a spatially one-dimensional stationary Swift-Hohenberg equation. We analyze the properties of this system and also discuss concentric-wave-type solutions. All the results are obtained by combining the methods of the theory of dynamical systems, in particular, the theory of homo-and heteroclinic orbits, and numerical simulation.     

THE POINCARE BIFURCATION OF QUADRATIC SYSTEMS HAVING A REGION CONSISTING OF PERIODIC CYCLES BOUNDED BY A HYPERBOLA AND AN ARC OF EQUATOR

In this paper, we discuss the Poincaré bifurcation for a class of quadratic systems having a region consisting of periodic cycles bounded by a hyperbola and an arc of equator. We prove that the system can at most generate two limit cycles after a small perturbation.