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.     

Existence of limit cycles in general planar piecewise linear systems of saddle–saddle dynamics

The existence and number of limit cycles in a class of general planar piecewise linear systems constituted by two linear subsystems with saddle–saddle dynamics are investigated. Using the Liénard-like canonical form with seven parameters, the parametric regions of the existence of limit cycles are given by constructing proper Poincaré maps. In particular, the existence of at least two limit cycles is proved and some parameter regions where two nested limit cycles exist are given.     

On the number of limit cycles of polynomial Liénard systems

Liénard systems are very important mathematical models describing oscillatory processes arising in applied sciences. In this paper, we study polynomial Liénard systems of arbitrary degree on the plane, and develop a new method to obtain a lower bound of the maximal number of limit cycles. Using the method and basing on some known results for lower degree we obtain new estimations of the number of limit cycles in the systems which greatly improve existing results.     

Limit cycles and global dynamics of planar piecewise linear refracting systems of focus–focus type

The limit cycle problem of planar piecewise linear refracting systems has been solved except the focus–focus (FF) type. Here we concern this remaining FF type, and prove that such systems have at most one limit cycle and have 14 topologically different global phase portraits.     

On the application of the homotopy analysis method to limit cycles’ approximation in planar self-excited systems

In this paper, the homotopy analysis method is applied to deduce analytical approximations of limit cycles and their frequencies in general planar self-excited systems with strong nonlinearity. After changing general planar self-excited systems to the canonical forms by several linear transformations, the auxiliary linear operators and the initial guess of solutions are introduced. Hence, the homotopy analysis solving is set up. Importantly, in solving the higher-order deformation equations, the idea of a perturbation procedure of limit cycles’ approximation proposed in the setting of second-order self-excited equations is embedded. As an application, a Rosenzweig–MacArthur predator–prey model is studied in detail. By choosing the suitable convergence-control parameters, the accurately analytical approximations of the large amplitude limit cycles and their frequency of the model are obtained. The high accuracy of the analytical results are illustrated by comparing with those of numerical integrations.     

On the limit cycles of a quintic planar vector field

This paper concerns the number and distributions of limit cycles in a Z_2-equivariant quintic planar vector field.25 limit cycles are found in this special planar polynomial system and four different configurations of these limit cycles are also given by using the methods of the bifurcation theory and the qualitative analysis of the differential equation.It can be concluded that H(5)≥25=5~2, where H(5)is the Hilbert number for quintic polynomial systems.The results obtained are useful to study the weakened 16th Hilbert problem.     

Global phase portraits of planar piecewise linear refracting systems of saddle–saddle type

This paper deal with the global dynamics of planar piecewise linear refracting systems of saddle–saddle type with a straight line of separation. We investigate the singularities, limit cycles, homoclinic orbits, heteroclinic orbits and make the classification of global phase portraits in the Poincaré disk for the refracting systems. We prove that these systems have 18 topologically different global phase portraits.     

Bounding the number of limit cycles of discontinuous differential systems by using Picard–Fuchs equations

In this paper, by using Picard–Fuchs equations and Chebyshev criterion, we study the upper bounds of the number of limit cycles given by the first order Melnikov function for discontinuous differential systems, which can bifurcate from the periodic orbits of quadratic reversible centers of genus one (r19): $\dot{x}=y?12x^2+16y^2$, $\dot{y}=?x?16xy$, and (r20): $\dot{x}=y+4x^2$, $\dot{y}=?x+16xy$, and the periodic orbits of the quadratic isochronous centers $(S_1):\dot{x}=?y+x^2?y^2$, $\dot{y}=x+2xy$, and $(S_2):\dot{x}=?y+x^2$, $\dot{y}=x+xy$. The systems (r19) and (r20) are perturbed inside the class of polynomial differential systems of degree n and the system $(S_1)$ and $(S_2)$ are perturbed inside the class of quadratic polynomial differential systems. The discontinuity is the line $y=0$. It is proved that the upper bounds of the number of limit cycles for systems (r19) and (r20) are respectively $4n?3(n\ge 4)$ and $4n+3(n\ge 3)$ counting the multiplicity, and the maximum numbers of limit cycles bifurcating from the period annuluses of the isochronous centers $(S_1)$ and $(S_2)$ are exactly 5 and 6 (counting the multiplicity) on each period annulus respectively.     

Dulac–Cherkas criterion for exact estimation of the number of limit cycles of autonomous systems on a plane

The problem of exact nonlocal estimation of the number of limit cycles surrounding one point of rest in a simply connected domain of the real phase space is considered for autonomous systems of differential equations with continuously differentiable right-hand sides. Three approaches to solving this problem are proposed that are based on sequential two-step usage of the Dulac–Cherkas criterion, which makes it possible to find closed transversal curves dividing the connected domain in doubly connected subdomains that surround the point of rest, with the system having precisely one limit cycle in each of them. The effectiveness of these approaches is exemplified with polynomial Liènard systems, a generalized van der Pol system, and a perturbed Hamiltonian system. For some systems, the derived estimate holds true in the entire phase space.     

A linear bound on the Euler number of threefolds of Calabi–Yau and of general type

Let X⊂ℙ ^N be either a threefold of Calabi–Yau or of general type (embedded with r K _X ). In this article we give lower and upper bounds, linear on the degree of X and N, for the Euler number of X. As a corollary we obtain the boundedness of the region described by the Chern ratios of threefolds with ample canonical bundle and a new upper bound for the number of nodes of a complete intersection threefold. Received: 26 April 2000 / Revised version: 20 November 2000     

Further study on Horozov-Iliev’s method of estimating the number of limit cycles

In the study of the number of limit cycles of near-Hamiltonian systems, the first order Melnikov function plays an important role. This paper aims to generalize Horozov-Iliev’s method to estimate the upper bound of the number of zeros of the function.

     

Small-amplitude limit cycles of polynomial Liénard systems

In this paper,we study the number of limit cycles appeared in Hopf bifurcations of a Linard system with multiple parameters.As an application to some polynomial Li’enard systems of the form x=y,y=gm(x)-fn(x)y,we obtain a new lower bound of maximal number of limit cycles which appear in Hopf bifurcation for arbitrary degrees m and n.     

Large Ericksen number limit for the 2D general Ericksen–Leslie system

In this paper, we address the issue of the so-called large Ericksen number limit for the general Ericksen–Leslie system of liquid crystals. By the standard nondimensionalization, we first prove that there exists a unique local smooth solution to the limiting system. Then we provide a qualitative estimate for the difference between the solutions of the general Ericksen–Leslie system and the limiting system, which corresponds to the partial decoupling effect in the limit of large Ericksen number.     

On the action of the complex Monge-Ampère operator on piecewise linear functions

The action of the mixed complex Monge-Ampère operator (h ₁, …, h _k ) ? dd ^c h ₁ ∧ … ∧ dd ^c h _k on piecewise linear functions h _i is considered. The language of Monge-Ampère operators is used to transfer results on mixed volumes and tropical varieties to a broader context, which arises under the passage from polynomials to exponential sums. In particular, it is proved that the value of the Monge-Ampère operator depends only on the product of the functions h _i .     

Small-amplitude limit cycles of some Liénard-type systems

As we know, the Liénard system and its generalized forms are classical and important models of nonlinear oscillators, and have been widely studied by mathematicians and scientists. The main problem considered by most people is the number of limit cycles. In this paper, we investigate two kinds of Liénard systems and obtain the maximal number (i.e. the least upper bound) of limit cycles appearing in Hopf bifurcations by applying some known bifurcation theorems with technical analysis.     

Visualization of the $$\varepsilon $$-subdifferential of piecewise linear–quadratic functions

Computing explicitly the \(\varepsilon \)-subdifferential of a proper function amounts to computing the level set of a convex function namely the conjugate minus a linear function. The resulting theoretical algorithm is applied to the the class of (convex univariate) piecewise linear–quadratic functions for which existing numerical libraries allow practical computations. We visualize the results in a primal, dual, and subdifferential views through several numerical examples. We also provide a visualization of the Brøndsted–Rockafellar theorem.