Limit Cycles of Planar Piecewise Smooth Quadratic Systems with Focus-Parabolic Type Critical Points

It is very important to determine the maximum number of limit cycles of planar piecewise smooth quadratic systems and it has become a focal subject in recent years. Almost all of the previous studies on this problem focused on systems with focus–focus type critical points. In this paper, we consider planar piecewise smooth quadratic systems with focus-parabolic type critical points. By using the generalized polar coordinates to compute the corresponding Lyapunov constants, we construct a class of planar piecewise smooth quadratic systems with focus-parabolic type critical points having six limit cycles. Our results improve the results obtained by Coll, Gasull and Prohens in 2001, who constructed a class of such systems with four limit cycles.

Computing Poincaré-Lyapunov Constants via Carleman Linearization

Using Carleman linearization an approximation is given for the solution of a system at Hopf bifurcation. The values of the Poincaré-Lyapunov constants (whether they are zero or not) affect the linear algebraic properties of the Carleman matrix and they appear in solvability conditions (through the Fredholm alternative). We provide a linear algebra based algorithm to compute the Poincaré-Lyapunov constants.

Limit Cycles and Analytic Centers for a Family of4n-1Degree Systems with Generalized Nilpotent Singularities

With the aid of computer algebra systemMathematica8.0 and by the integral factor method, for a family of generalized nilpotent systems, we first compute the first several quasi-Lyapunov constants, by vanishing them and rigorous proof, and then we get sufficient and necessary conditions under which the systems admit analytic centers at the origin. In addition, we present that seven amplitude limit cycles can be created from the origin. As an example, we give a concrete system with seven limit cycles via parameter perturbations to illustrate our conclusion. An interesting phenomenon is that the exponent parameterncontrols the singular point type of the studied system. The main results generalize and improve the previously known results in Pan.

Stochastic and Chaotic Assessment of Human Involuntary Movements

According to theory of chaos and self-organization calculation of chaotic dynamics of postural tremor parameters is presented. We have shown that stochastic approach, calculation of distribution function for reiterations of measurements, tremorograms in one subject exhibits chaotic dynamics of these functions f(x). Otherwise 15 measurements by 5 seconds tremerograms show impossibility of coincidence f(x) at pairwise comparison (105 pairs) of tremerograms. Functions f(x) can coincide (for pairs of tremorograms) less than 2-5% from the general number without effect on a person. However, physical load increases the coincidence to 10-15%. Simultaneously, all the amplitude-frequency characteristics do not coincide, Lyapunov constants cannot be calculated, but autocorrelation functions do not reach zero. All the stochastic parameters exhibit constant changes. Calculation of quasi-attractors can provide real distinction between biomechanical system before static load and after. Sizes of quasi-attractor (square and volume) can show distinctions in physiological body states for continuous motions x(t), i.e. for dx/dt≠ 0.

The Center Conditions and Bifurcation of Limit Cycles at the Degenerate Singularity of a Three-Dimensional System

We investigate multiple limit cycles bifurcation and center-focus problem of the degenerate equilibrium for a three-dimensional system. By applying the method of symbolic computation, we obtain the first four quasi-Lyapunov constants. It is proved that the system can generate 3 small limit cycles from nilpotent critical point on center manifold. Furthermore, the center conditions are found and as weak foci the highest order is proved to be the fourth; thus we obtain at most 3 small limit cycles from the origin via local bifurcation. To our knowledge, it is the first example of multiple limit cycles bifurcating from a nilpotent singularity for the flow of a high-dimensional system restricted to the center manifold.

Fourteen Limit Cycles in a Seven-Degree Nilpotent System

Center conditions and the bifurcation of limit cycles for a seven-degree polynomial differential system in which the origin is a nilpotent critical point are studied. Using the computer algebra system Mathematica, the first 14 quasi-Lyapunov constants of the origin are obtained, and then the conditions for the origin to be a center and the 14th-order fine focus are derived, respectively. Finally, we prove that the system has 14 limit cycles bifurcated from the origin under a small perturbation. As far as we know, this is the first example of a seven-degree system with 14 limit cycles bifurcated from a nilpotent critical point.