Period Annulus Research Articles

Limit Cycles Bifurcating from a Periodic Annulus in Discontinuous Planar Piecewise Linear Hamiltonian Differential System with Three Zones

In this paper, we study the number of limit cycles that can bifurcate from a periodic annulus in a discontinuous planar piecewise linear Hamiltonian differential system with three zones separated by two parallel straight lines. We prove that if the central subsystem, i.e. the system defined between the two parallel lines, has a real center and the other subsystems have centers or saddles, then we have at least three limit cycles that appear after perturbations of the periodic annulus. For this, we study the number of zeros of a Melnikov function for piecewise Hamiltonian system and present a normal form for this system in order to simplify the computations.

On the Number of Limit Cycles Bifurcating from the Linear Center with an Algebraic Switching Curve

This paper studies the family of piecewise linear differential systems in the plane with two pieces separated by a switching curve $y=x^{m}$, where $m>1$ is an arbitrary positive. By analysing the first order Melnikov function, we give an upper bound and an lower bound of the maximum number of limit cycles which bifurcate from the period annulus around the origin under polynomial perturbations of degree $n$. The results shows that the degree of switching curves affect the number of limit cycles.

The criticality of reversible quadratic centers at the outer boundary of its period annulus

This paper deals with the period function of the reversible quadratic centersXν=−y(1−x)∂x+(x+Dx2+Fy2)∂y, where ν=(D,F)∈R2. Compactifying the vector field to S2, the boundary of the period annulus has two connected components, the center itself and a polycycle. We call them the inner and outer boundary of the period annulus, respectively. We are interested in the bifurcation of critical periodic orbits from the polycycle Πν at the outer boundary. A critical period is an isolated critical point of the period function. The criticality of the period function at the outer boundary is the maximal number of critical periodic orbits of Xν that tend to Πν0 in the Hausdorff sense as ν→ν0. This notion is akin to the cyclicity in Hilbert's 16th Problem. Our main result (Theorem A) shows that the criticality at the outer boundary is at most 2 for all ν=(D,F)∈R2 outside the segments {−1}×[0,1] and {0}×[0,2]. With regard to the bifurcation from the inner boundary, Chicone and Jacobs proved in their seminal paper on the issue that the upper bound is 2 for all ν∈R2. In this paper the techniques are different because, while the period function extends analytically to the center, it has no smooth extension to the polycycle. We show that the period function has an asymptotic expansion near the polycycle with the remainder being uniformly flat with respect to ν and where the principal part is given in a monomial scale containing a deformation of the logarithm, the so-called Écalle-Roussarie compensator. More precisely, Theorem A follows by obtaining the asymptotic expansion to fourth order and computing its coefficients, which are not polynomial in ν but transcendental. Theorem A covers two of the four quadratic isochrones, which are the most delicate parameters to study because its period function is constant. The criticality at the inner boundary in the isochronous case is bounded by the number of generators of the ideal of all the period constants but there is no such approach for the criticality at the outer boundary. A crucial point to study it in the isochronous case is that the flatness of the remainder in the asymptotic expansion is preserved after the derivation with respect to parameters. We think that this constitutes a novelty that is of particular interest also in the study of similar problems for limit cycles in the context of Hilbert's 16th Problem. Theorem A also reinforces the validity of a long standing conjecture by Chicone claiming that the quadratic centers have at most two critical periodic orbits. A less ambitious goal is to prove the existence of a uniform upper bound for the number of critical periodic orbits in the family of quadratic centers. By a compactness argument this would follow if one can prove that the criticality of the period function at the outer boundary of any quadratic center is finite. Theorem A leaves us very close to this existential result.

On the number of limit cycles of a class of Liénard–Rayleigh oscillators

This paper is devoted to the study of limit cycle bifurcations for a class of cubic near-Hamiltonian systems, which are in the form of Liénard–Rayleigh oscillators. By using geometric techniques to the first order Melnikov functions, we provide a complete study on the maximum number of limit cycles in Poincaré bifurcation from a period annulus.

Partially-covered fractal induced turbulence on fins thermal dissipation

The impacts of partially-covered fractal grids induced turbulence on the forced convective heat transfer across plate-fin heat sink at Reynolds number ReDh = 22.0 × 103 were numerically and experimentally investigated. Results showed that partially covered grids rendered a higher thermal dissipation performance, with partially-covered square fractal grid (PCSFG) registering an outstanding increase of 43% in Nusselt number relative to the no grid configuration. The analyzation via an in-house developed single particle tracking velocimetry (SPTV) system displayed the findings of unique “Turbulence Annulus” formation, which provided a small degree of predictivity in the periodic annulus oscillations. Further assessments on PCSFG revealed the preferred inter-fin flow dynamics of (i) high flow velocity, (ii) strong turbulence intensity, (iii) vigorous flow fluctuations, (iv) small turbulence length scale, and (v) heightened decelerated flow events. These features stemmed from the coupling effects of multilength-scale fractal bar thicknesses in generating a veracity of eddy sizes, and a vertical segmentation producing heightened mass flow rate while inducing favourable wake-flow structures to penetrate inter-fin regions. Teeming effects of such energetic eddies within plate-fin array unveiled a powerful vortex shedding effect, with PCSFG achieving fluctuation frequency f = 18.5 Hz close to an optimal magnitude. The coaction of such traits limits the growth of fin boundary layers, providing superior thermal transfer capabilities which benefits the community in developing for higher efficiency heat transfer systems.

Upper Bound on the Number of Zeros of Abelian Integrals for Hyperbolic Hamiltonian Systems of Degree Seven

In this paper, we study the bound on the number of isolated zeros of Abelian integrals associated with the seventh-degree hyperbolic Hamiltonian system. When the unperturbed system has a unique period annulus bounded by a heteroclinic loop, a series of results on the upper bound of the number of zeros of the related Abelian integrals are obtained. However, there are still some open problems left about the exact bound. Furthermore, we give some results for the system when the parameters in the Hamiltonian system are fixed to certain values. The main tools involve algebraic symbolic computations such as counting the zeros of semi-algebraic systems.

The Number of Limit Cycles for a Class of Cubic Systems with Multiple Parameters

In this paper, we study the number of limit cycles for a class of cubic systems with general quadratic polynomial perturbations. By using the Melnikov function theory we obtain that five limit cycles can be bifurcated from a period annulus. We also study the Hopf bifurcation at the center surrounded by the annulus.

An estimate of the number of limit cycles bifurcating from a planar integrable system

This paper is concerned with the number of limit cycles bifurcating from a planar integrable system, which has a center at the origin and two invariant lines orthogonal with the switching line. By using the first order averaging method, we present an estimate of the maximum number of limit cycles bifurcating from the period annulus around the center via perturbations of piecewise polynomials.

Bifurcation of Limit Cycles by Perturbing a Piecewise Linear Hamiltonian System

In this paper, we study limit cycle bifurcations for planar piecewise smooth near-Hamiltonian systems with nth-order polynomial perturbation. The piecewise smooth linear differential systems with two centers formed in two ways, one is that a center-fold point at the origin, the other is a center-fold at the origin and another unique center point exists. We first explore the expression of the first order Melnikov function. Then by using the Melnikov function method, we give estimations of the number of limit cycles bifurcating from the period annulus. For the latter case, the simultaneous occurrence of limit cycles near both sides of the homoclinic loop is partially addressed.

The Exact Bound of the Number of Limit Cycles Bifurcating from the Global Nilpotent Center with a Nonlinear Switching Curve

In this paper, we study the bifurcations of a class of planar Hamiltonian systems having a global nilpotent center under the perturbations of polynomials of degree [Formula: see text] with a nonlinear switching curve. We obtain the exact bound of the number of limit cycles bifurcating from the period annulus if the first order Melnikov function is not identically zero. We also give some examples to illustrate our results.

Bifurcations of limit cycles in piecewise smooth Hamiltonian system with boundary perturbation

In this paper, the general planar piecewise smooth Hamiltonian system with period annulus around the center at the origin is considered. We obtain the expressions for the first order and the second order Melnikov functions of it's general second order perturbation, which can be used to find the number of limit cycles bifurcated from periodic orbits. Further, we have shown that the number of limit cycles of the system $\dot{X}=\begin{cases} (H_y^+,-H_x^+) & \mbox{if}~y>\varepsilon (H_y^-,-H_x^-) & \mbox{if}~y<\varepsilon f(x) \end{cases}$ equals to the number of positive zeros of $f$ when at $\varepsilon=0$ the system has a period annulus around the origin.

Bifurcation for a class of piecewise cubic systems with two centers

In this paper, a class of symmetric cubic planar piecewise polynomial systems are presented, which have two symmetric centers corresponding to two period annuli. By perturbation and considering piecewise first order Melnikov function, we show the existence of 18 limit cycles (not small-amplitude limit cycles) with the configuration ( 9 , 9 ) bifurcating from the two period annuli and 22 small-amplitude limit cycles with the configuration ( 11 , 11 ) , respectively.

Limit Cycles Appearing From Piecewise Smooth Perturbations to a Reversible Nonlinear Center

This paper deals with the limit cycle bifurcation from a reversible differential center of degree [Formula: see text] due to small piecewise smooth homogeneous polynomial perturbations. By using the averaging theory for discontinuous systems and the complex method based on the Argument Principle, we obtain lower and upper bounds for the maximum number of limit cycles bifurcating from the period annulus around the center of the unperturbed system.

Fifteen Limit Cycles Bifurcating from a Perturbed Cubic Center

In this work, we study the bifurcation of limit cycles from the period annulus surrounding the origin of a class of cubic polynomial differential systems; when they are perturbed inside the class of all polynomial differential systems of degree six, we obtain at most fifteenth limit cycles by using the averaging theory of first order.

ON THE MAXIMUM NUMBER OF PERIOD ANNULI FOR SECOND ORDER CONSERVATIVE EQUATIONS

We consider a second order scalar conservative differential equation whose potential function is a Morse function with a finite number of critical points and is unbounded at infinity. We give an upper bound for the number of nonglobal nontrivial period annuli of the equation and prove that the upper bound obtained is sharp. We use tree theory in our considerations.

Cyclicity of a Class of Hamiltonian Systems under Perturbations of Piecewise Smooth Polynomials

In the study of the weakened Hilbert’s 16th problem, there usually appears period annulus surrounding more than one singular point, implying that there are often more than three generate functions in the Abelian integrals (or the first order Melnikov functions). It is also known that there are usually more than three generate functions in the study of the bifurcation of limit cycles in piecewise smooth Hamiltonian systems. This paper aims to explore a method so that we can estimate the upper bound of limit cycles bifurcating from the period annulus if the unperturbed system has some good properties such as Picard–Fuchs equations. This method is based on [Gavrilov &amp; Iliev, 2009] and is useful for smooth and piecewise smooth polynomial vector fields of a general degree. Finally, we present an example to illustrate an application of the theoretical results.

Bifurcation of limit cycles in a piecewise smooth near-integrable system

In this paper, we study the bifurcation of limit cycles in a class of piecewise smooth quadratic integrable systems under small polynomial perturbations of degree n. By using the first order Melnikov function, we derive a lower bound for the number of limit cycles which bifurcate from the period annulus.

Limit Cycles from Perturbing a Piecewise Smooth System with a Center and a Homoclinic Loop

We study bifurcations of limit cycles arising after perturbations of a special piecewise smooth system, which has a center and a homoclinic loop. By using the Picard–Fuchs equation, we give an upper bound of the maximum number of limit cycles bifurcated from the period annulus between the center and the homoclinic loop. Furthermore, by applying the method of first-order Melnikov function we obtain a lower bound of the maximum number of limit cycles bifurcated from the center.

Co-existence of a Period Annulus and a Limit Cycle in a Class of Predator–Prey Models with Group Defense

For a family of two-dimensional predator–prey models of Gause type, we investigate the simultaneous occurrence of a center singularity and a limit cycle. The family is characterized by the fact that the functional response is nonanalytical and exhibits group defense. We prove the existence and uniqueness of the limit cycle using a new theorem for Liénard systems. The new theorem gives conditions for the uniqueness of a limit cycle which surrounds a period annulus. The results of this paper provide a mechanism for studying the global behavior of solutions to Gause systems through bifurcation of an integrable system which contains a center and a limit cycle.

Number of Limit Cycles from a Class of Perturbed Piecewise Polynomial Systems

In this paper, we study Poincaré bifurcation of a class of piecewise polynomial systems, whose unperturbed system has a period annulus together with two invariant lines. The main concerns are the number of zeros of the first order Melnikov function and the estimation of the number of limit cycles which bifurcate from the period annulus under piecewise polynomial perturbations of degree [Formula: see text].