Shape Of Limit Cycle Research Articles

Numerical investigation on buffeting and lock-in phenomenon for biplane airfoils at [formula omitted

Biplane airfoils at high angles of attack have interesting nonlinear characteristics, the purpose of this work is to evaluate the impacts of spacing ratio (λ=ds/c=0.25, 0.5, 1, 2 and 4, where ds is the vertical distance between biplane airfoils and c is chord length) and Reynolds number (0.87×105≤Re≤4.35×105) on buffeting and lock-in phenomenon at angle of attack α=40∘ throughout numerical investigation. Present simulation results show that biplane configuration usually can reduce buffeting frequency slightly comparing to the single airfoil, yet reasonable spacing λ=2 is helpful to suppress buffeting and continuous acceleration also promotes stability. Particular attention is paid to stagnation point region that moves away from leading edge of the upper airfoil with increasing λ. However, the lower airfoil corresponds to the fixed stagnation point region. Additionally, limit cycle oscillation (LCO) of biplane airfoils is determined by pitching angles (θ), and in consideration of spacing effect, limit cycles of the lower airfoil move downward but those of the upper airfoil move in the opposite direction. Moreover, the shape of limit cycle for the later first collapses then inflates. Unique 8-shaped hysteresis of the upper airfoil in λ=1 is caused by sporadically wake interference. On the whole, V-shaped lock-in region will expand and shift left to some extent based on λ in contrast to the single airfoil. In terms of area, lock-in region shrinks then expands with rising λ, and the proportion of the left part increases as λ increases.

On the Shape of Limit Cycles That Bifurcate from Isochronous Center

New idea and algorithm are proposed to compute asymptotic expression of limit cycles bifurcated from the isochronous center. Compared with known inverse integrating factor method, new algorithm to analytically computing shape of limit cycle proposed in this paper is simple and easy to apply. The applications of new algorithm to some examples are also given.

Shape and period of limit cycles bifurcating from a class of Hamiltonian period annulus

In this work we are concerned with the problem of shape and period of isolated periodic solutions of perturbed analytic radial Hamiltonian vector fields in the plane. Françoise developed a method to obtain the first non vanishing Poincaré–Pontryagin–Melnikov function. We generalize this technique and we apply it to know, up to any order, the shape of the limit cycles bifurcating from the period annulus of the class of radial Hamiltonians. We write any solution, in polar coordinates, as a power series expansion in terms of the small parameter. This expansion is also used to give the period of the bifurcated periodic solutions. We present the concrete expression of the solutions up to third order of perturbation of Hamiltonians of the form H=H(r). Necessary and sufficient conditions that show if a solution is simple or double are also presented.

Limit cycles of polynomial differential systems with homogeneous nonlinearities of degree 4 via the averaging method

By using the averaging method, we study the limit cycles for a class of quartic polynomial differential systems as well as their global shape in the plane. More specifically, we analyze the global shape of limit cycles bifurcating from a Hopf bifurcation and also from periodic orbits with linear center x ˙ = - y , y ˙ = x . The perturbation of these systems is made inside the class of quartic polynomial differential systems without quadratic and cubic terms.

The shape of limit cycles that bifurcate from non-Hamiltonian centers

The shape of limit cycles that bifurcate from non-Hamiltonian centers

On the shape of limit cycles that bifurcate from Hamiltonian centers

On the shape of limit cycles that bifurcate from Hamiltonian centers