Planar Polynomial Research Articles

Normal Forms of Planar Polynomial Differential Systems

Using the invariant theory, we develop an algorithmic method, which is based on the construction of a matrix of linear transformation, to compute normal forms of planar polynomial differential systems. We illustrate our method in the case where the planar polynomial differential system is cubic.

Bifurcation of small limit cycles in cubic integrable systems using higher-order analysis

In this paper, we present a method of higher-order analysis on bifurcation of small limit cycles around an elementary center of integrable systems under perturbations. This method is equivalent to higher-order Melinikov function approach used for studying bifurcation of limit cycles around a center but simpler. Attention is focused on planar cubic polynomial systems and particularly it is shown that the system studied by Żoła̧dek (1995) [24] can indeed have eleven limit cycles under perturbations at least up to 7th order. Moreover, the pattern of numbers of limit cycles produced near the center is discussed up to 39th-order perturbations, and no more than eleven limit cycles are found.

Global bifurcation analysis of the Kukles cubic system

In this paper, we carry out the global bifurcation analysis of the Kukles system representing a planar polynomial dynamical system with arbitrary linear and cubic right-hand sides and having an anti-saddle at the origin. Using our geometric approach and the Wintner-Perko termination principle, we solve the problem on the maximum number and distribution of limit cycles in this system.

Global bifurcation analysis of the Kukles cubic system

In this paper, we carry out the global bifurcation analysis of the Kukles system representing a planar polynomial dynamical system with arbitrary linear and cubic right-hand sides and having an anti-saddle at the origin. Using our geometric approach and the Wintner-Perko termination principle, we solve the problem on the maximum number and distribution of limit cycles in this system.

A topological classification of plane polynomial systems having a globally attracting singular point

In this paper, plane polynomial systems having a singular point attracting all orbits in positive time are classified up to topological equivalence. This is done by assigning a combinatorial invariant to the system (a so-called consisting of finitely many vectors with components in the set $\{n/3: n=0,1,2,\ldots\}$), so that two such systems are equivalent if and only if (after appropriately fixing an orientation in $\mathbb{R}^2$ and a heteroclinic separatrix) they have the same feasible set. In fact, this classification is achieved in the more general setting of continuous flows having finitely many separatrices. Polynomial representatives for each equivalence class are found, although in a non-constructive way. Since, to the best of our knowledge, the literature does not provide any concrete polynomial system having a non-trivial globally attracting singular point, an explicit example is given as well.

Zero Counting and Invariant Sets of Differential Equations

Abstract Consider a polynomial vector field $\xi$ in ${\mathbb C}^n$ with algebraic coefficients, and $K$ a compact piece of a trajectory. Let $N(K,d)$ denote the maximal number of isolated intersections between $K$ and an algebraic hypersurface of degree $d$. We introduce a condition on $\xi$ called constructible orbits and show that under this condition $N(K,d)$ grows polynomially with $d$. We establish the constructible orbits condition for linear differential equations over ${\mathbb C}(t)$, for planar polynomial differential equations and for some differential equations related to the automorphic $j$-function. As an application of the main result, we prove a polylogarithmic upper bound for the number of rational points of a given height in planar projections of $K$ following works of Bombieri–Pila and Masser.

The dynamical Manin–Mumford problem for plane polynomial automorphisms

Let f be a polynomial automorphism of the affine plane. In this paper we consider the possibility for it to possess infinitely many periodic points on an algebraic curve C . We conjecture that this happens if and only if f admits a time-reversal symmetry; in particular the Jacobian Jac (f) must be a root of unity. As a step towards this conjecture, we prove that the Jacobian and all its Galois conjugates lie on the unit circle in the complex plane. Under mild additional assumptions we are able to conclude that indeed Jac (f) is a root of unity. We use these results to show in various cases that any two automorphisms sharing an infinite set of periodic points must have a common iterate, in the spirit of recent results by Baker–DeMarco and Yuan–Zhang.

Sum of Squares Certificates for Stability of Planar, Homogeneous, and Switched Systems

We show that existence of a global polynomial Lyapunov function for a homogeneous polynomial vector field or a planar polynomial vector field (under a mild condition) implies existence of a polynomial Lyapunov function that is a sum of squares (sos) and that the negative of its derivative is also a sum of squares. This result is extended to show that such sos-based certificates of stability are guaranteed to exist for all stable switched linear systems. For this class of systems, we further show that if the derivative inequality of the Lyapunov function has an sos certificate, then the Lyapunov function itself is automatically a sum of squares. These converse results establish cases where semidefinite programming is guaranteed to succeed in finding proofs of Lyapunov inequalities. Finally, we demonstrate some merits of replacing the sos requirement on a polynomial Lyapunov function with an sos requirement on its top homogeneous component. In particular, we show that this is a weaker algebraic requirement in addition to being cheaper to impose computationally.

A uniqueness criterion of limit cycles for planar polynomial systems with homogeneous nonlinearities

This paper is devoted to study the planar polynomial system:x˙=ax−y+Pn(x,y),y˙=x+ay+Qn(x,y), where a∈R and Pn,Qn are homogeneous polynomials of degree n≥2. Denote ψ(θ)=cos⁡(θ)⋅Qn(cos⁡(θ),sin⁡(θ))−sin⁡(θ)⋅Pn(cos⁡(θ),sin⁡(θ)). We prove that the system has at most 1 limit cycle surrounding the origin provided (n−1)aψ(θ)+ψ˙(θ)≠0. Furthermore, this upper bound is sharp. This is maybe the first uniqueness criterion, which only depends on a (linear) condition of ψ, for the limit cycles of this kind of systems. We show by examples that in many cases, the criterion is applicable while the classical ones are invalid. The tool that we mainly use is a new estimate for the number of limit cycles of Abel equation with coefficients of indefinite signs. Employing this tool, we also obtain another geometric criterion which allows the system to possess at most 2 limit cycles surrounding the origin.

Classical Planar Algebraic Curves Realizable by Quadratic Polynomial Differential Systems

In this paper, we show planar quadratic polynomial differential systems exhibiting as solutions some famous planar invariant algebraic curves. Also we place particular attention to the Darboux integrability of these differential systems.

Estimate for the number of limit cycles of Abel equation via a geometric criterion on three curves

This paper is devoted to the investigation of Abel equation $$\dot{x}=S(t,x)=\sum ^3_{i=0}a_i(t)x^i$$ , where $$a_i\in \mathrm C^{\infty }([0,1])$$ . A solution x(t) with $$x(0)=x(1)$$ is called a periodic solution. And an orbit $$x=x(t)$$ is called a limit cycle if x(t) is a isolated periodic solution. By means of Lagrange interpolation formula, we give a criterion to estimate the number of limit cycles of the equation. This criterion is only concerned with S(t, x) on three non-intersecting curves. Applying our main result, we prove that the maximum number of limit cycles of the equation is 4 if $$a_2(t)a_0(t)<0$$ . To the best of our knowledge, this is a nontrivial supplement for a classical result which says that the equation has at most 3 limit cycles when $$a_2(t)\ne 0$$ and $$a_0(t)\equiv 0$$ . We also study a planar polynomial system with homogeneous nonlinearities: $$\begin{aligned} \dot{x}=ax-y+P_n(x,y),\ \ \dot{y}=x+ay+Q_n(x,y), \end{aligned}$$ where $$a\in \mathbb R$$ and $$P_n, Q_n$$ are homogeneous polynomials of degree $$n\ge 2$$ . Denote by $$\psi (\theta )=\cos (\theta )\cdot Q_n\big (\cos (\theta ),\sin (\theta )\big )-\sin (\theta )\cdot P_n\big (\cos (\theta ),\sin (\theta )\big )$$ . We prove that if $$(n-1)a\psi (\theta )+\dot{\psi }(\theta )\ne 0$$ , then the polynomial system has at most 1 limit cycle surrounding the origin, and the multiplicity is no more than 2.

Polynomial Hamiltonian systems of degree 3 with symmetric nilpotent centers

We provide normal forms and the global phase portraits in the Poincaré disk for all Hamiltonian planar polynomial vector fields of degree 3 symmetric with respect to the x-axis having a nilpotent center at the origin.

Center problem for trigonometric Liénard systems

We give a complete algebraic characterization of the non-degenerated centers for planar trigonometric Liénard systems. The main tools used in our proof are the classical results of Cherkas on planar analytic Liénard systems and the characterization of some subfields of the quotient field of the ring of trigonometric polynomials. Our results are also applied to some particular subfamilies of planar trigonometric Liénard systems. The results obtained are reminiscent of the ones for planar polynomial Liénard systems but the proofs are different.

On the Bifurcation of Limit Cycles Due to Polynomial Perturbations of Hamiltonian Centers

We study the number of limit cycles bifurcating from the period annulus of a real planar polynomial Hamiltonian ordinary differential system with a center at the origin when it is perturbed in the class of polynomial vector fields of a given degree.

Bifurcation diagrams for Hamiltonian nilpotent centers of linear plus cubic homogeneous polynomial vector fields

Following the work done in [8] we provide the bifurcation diagrams for the global phase portraits in the Poincaré disk of all Hamiltonian nilpotent centers of linear plus cubic homogeneous planar polynomial vector fields.

Global behaviour of the period function of the sum of two quasi-homogeneous vector fields

We study the global behaviour of the period function on the period annulus of degenerate centres for two families of planar polynomial vector fields. These families are the quasi-homogeneous vector fields and the vector fields given by the sum of two quasi-homogeneous Hamiltonian ones. In the first case we prove that the period function is globally decreasing, extending previous results that deal either with the Hamiltonian quasi-homogeneous case or with the general homogeneous situation. In the second family, and after adding some more additional hypotheses, we show that the period function of the origin is either decreasing or has at most one critical period and that both possibilities may happen. This result also extends some previous results that deal with the situation where both vector fields are homogeneous and the origin is a non-degenerate centre.

Averaging approach to cyclicity of hopf bifurcation in planar linear-quadratic polynomial discontinuous differential systems

It is well known that the cyclicity of a Hopf bifurcation in continuous quadratic polynomial differential systems in $\mathbb{R}^2$ is $. In contrast here we consider discontinuous differential systems in $\mathbb{R}^2$ defined in two half-planes separated by a straight line. In one half plane we have a general linear center at the origin of $\mathbb{R}^2$, and in the other a general quadratic polynomial differential system having a focus or a center at the origin of $\mathbb{R}^2$. Using averaging theory, we prove that the cyclicity of a Hopf bifurcation for such discontinuous differential systems is at least 5. Our computations show that only one of the averaged functions of fifth order can produce 5 limit cycles and there are no more limit cycles up to sixth order averaged function.

First integrals and phase portraits of planar polynomial differential cubic systems with invariant straight lines of total multiplicity eight

First integrals and phase portraits of planar polynomial differential cubic systems with invariant straight lines of total multiplicity eight

EFFECTIVE CONSTRUCTION OF POINCAR&amp;#201;-BENDIXSON REGIONS

This paper deals with the problem of location and existence of limit cycles for real planar polynomial differential systems. We provide a method to construct Poincare-Bendixson regions by using transversal curves, that enables us to prove the existence of a limit cycle that has been numerically detected. We apply our results to several known systems, like the Brusselator one or some Lienard systems, to prove the existence of the limit cycles and to locate them very precisely in the phase space. Our method, combined with some other classical tools can be applied to obtain sharp bounds for the bifurcation values of a saddle-node bifurcation of limit cycles, as we do for the Rychkov system.

Normal forms and global phase portraits of quadratic and cubic integrable vector fields having two nonconcentric circles as invariant algebraic curves

ABSTRACTIn this paper, we give the normal form of all planar polynomial vector fields of degree d ≤ 3 having two nonconcentric circles C1 and C2 as invariant algebraic curves and the function , with α and β real values, as first integral. Moreover, we classify all global phase portraits on the Poincaré disc of a subclass of these vector fields.