Nowadays, environmental energy resources, especially mechanical vibrations, have attracted the attention of researchers to provide energy for low-power electronic circuits. A common method for environmental mechanical energy harvesting involves using piezoelectric materials. In this study, a spiral multimode piezoelectric energy harvester was designed and fabricated. To achieve wide bandwidth in low frequencies (below 15 Hz), the first three resonance frequencies of the beam were designed to be close to each other. To do this, the five lengths of the substrate layer were optimized by the Taguchi method, using an L27 orthogonal array. Each experiment of the Taguchi method was then simulated in ANSYS software. Next, the optimum level of each design variable was obtained. A test rig was then constructed based on the optimum design values and some experimental investigations were conducted. A good correlation was observed between measured and the finite element results.

The problem of distinguish between a focus and a center is one of the classical problems in the qualitative theory of planar differential systems. In this paper we provide a new family of centers of polynomial differential systems of arbitrary even degree. Moreover, we classify the global phase portraits in the Poincaré disc of the centers of this family having degree 2, 4 and 6.

In this paper we study the dynamics of the Higgins-Selkov systeṁ x = 1 − xy γ ,ẏ = αy(xy γ−1 − 1), where α is a real parameter and γ > 1 is an integer. We classify the phase portraits of this system for γ = 3, 4, 5, 6, in the Poincaré disc for all the values of the parameter α. Moreover, we determine in function of the parameter α the regions of the phase space with biological meaning.

Consider the class of reversible quadratic systems [Formula: see text] with [Formula: see text]. These quadratic polynomial differential systems have a center at the point [Formula: see text] and the circle [Formula: see text] is one of the periodic orbits surrounding this center. These systems can be written into the form [Formula: see text] with [Formula: see text]. For all [Formula: see text] we prove that the averaging theory up to seventh order applied to this last system perturbed inside the whole class of quadratic polynomial differential systems can produce at most two limit cycles bifurcating from the periodic orbits surrounding the center (0,0) of that system. Up to now this result was only known for [Formula: see text] (see Li, 2002; Liu, 2012).

In this paper, we study the asymptotic expansion of the first order Melnikov function near a 3-polycycle connecting a cusp (of order one or two) to two hyperbolic saddles for a near-Hamiltonian system in the plane. The formulas for the first coefficients of the expansion are given as well as the method of bifurcation of limit cycles. Then we use the results to study two Hamiltonian systems with this 3-polycycle and determine the number and distribution of limit cycles that can bifurcate from the perturbed systems. Moreover, a sharp upper bound for the number of limit cycles bifurcated from the whole periodic annulus is found when there is a cusp of order one.