Piecewise Smooth Differential Systems Research Articles

Non-isochronicity on piecewise Hamiltonian differential systems with homogeneous nonlinearities

In this paper, we prove the non-isochronicity of Σ-centres for a class of planar piecewise smooth differential systems with a straight switching line, whose two sub-systems are Hamiltonian differential systems with a non-degenerated centre and only homogeneous nonlinearities. This result generalizes a classical result of non-isochronicity on the smooth Hamiltonian systems with homogeneous nonlinearities.

Global Phase Portraits of Piecewise Quadratic Differential Systems with a Pseudo-Center

This paper deals with the global dynamics of planar piecewise smooth differential systems constituted by two different vector fields separated by one straight line that passes through the origin. From a quasi-canonical family of piecewise quadratic differential systems with a pseudo-focus point at the origin, which has six parameters, we investigate the subfamilies where the origin is indeed a pseudo-center. For such subfamilies, we classify its global phase portraits in the Poincaré disk and the associated bifurcation sets.

The center problem on piecewise smooth differential systems with two zones

The center problem on piecewise smooth differential systems with two zones

Bifurcating Limit Cycles with a Perturbation of Systems Composed of Piecewise Smooth Differential Equations Consisting of Four Regions

Systems composed of piecewise smooth differential (PSD) mappings have quantitatively been searched for answers to a substantial issue of limit cycle (LC) bifurcations. In this paper, LC numbers (LCNs) of a PSD system (PSDS) consisting of four regions are dealt with. A Melnikov mapping whose order is one is implicitly obtained by finding its originators when the system is perturbed under any nth degree of real polynomials. Then, the approach employing the Picard–Fuchs mapping is utilized to attain a higher boundary of bifurcation LCNs of systems composed of PSD functions with a global center. The method we used could be implemented to examine the problems related to the LC of other PSDS.

Limit Cycles of Continuous Piecewise Smooth Differential Systems

Limit Cycles of Continuous Piecewise Smooth Differential Systems

Bifurcation from Periodic Orbits of n-Dimensional Piecewise Smooth Differential Systems with Two Switching Planes

In this paper, the general perturbation problem of piecewise smooth integrable differential systems with two switching planes is considered. Firstly, when the unperturbed system has a family of periodic orbits, we obtain the first order Melnikov vector function which can be used to study the number of periodic orbits bifurcated from the period annuli. Then, by using the obtained Melnikov vector function, we get an upper bound of the number of periodic orbits of a concrete [Formula: see text]-dimensional piecewise smooth differential system.

The Monotonicity of the Ratio of Two Line Integrals in Piecewise Smooth Differential Systems

The Monotonicity of the Ratio of Two Line Integrals in Piecewise Smooth Differential Systems

Melnikov functions of arbitrary order for piecewise smooth differential systems in [formula omitted] and applications

In this paper we develop an arbitrary order Melnikov function to study limit cycles bifurcating from a periodic submanifold for autonomous piecewise smooth differential systems in Rn with two zones separated by a hyperplane. This result not only extends some of the known results on the Melnikov theory in dimension and order but also compensates for some defects of the averaging theory in studying the limit cycle bifurcation of autonomous systems from a periodic submanifold. To demonstrate the application of our theoretical result and its superiority for some systems to the existing averaging theory, we study the maximum number of limit cycles bifurcating from an n-dimensional periodic submanifold caused by non-smooth centers of the fold-fold type, providing an upper bound for any order piecewise polynomial perturbations of degree m. Concerning the planar case of the unperturbed system, a piecewise Hamiltonian system, we obtain a better upper bound for piecewise polynomial Hamiltonian perturbations up to order two. The realizability of these upper bounds is also discussed.

Limit cycles appearing from a generalized heteroclinic loop with a cusp and a nilpotent saddle

This paper studies the limit cycle bifurcation problem of a class of piecewise smooth differential polynomial systems of degree n by perturbing a piecewise cubic polynomial system having a generalized heteroclinic loop with a cusp and a nilpotent saddle. First, we provide all possible phase portraits of the unperturbed system on the plane with crossing periodic orbits and obtain a condition for the appearance of a generalized heteroclinic loop with a cusp and a nilpotent saddle by qualitative theoretical knowledge. Then, we investigate the algebraic structure of the first order Melnikov function and give its asymptotic expansion near the generalized heteroclinic loop with the help of analytical skills. Finally, we employ the expansion together with its coefficients to obtain the existence of at least 3n−1 limit cycles.

The Number of Limit Cycles Bifurcating from a Degenerate Center of Piecewise Smooth Differential Systems

For two families of planar piecewise smooth polynomial differential systems, whose unperturbed system has a degenerate center at the origin, we study the biggest lower bound for the maximum number of limit cycles bifurcating from the periodic orbits of the center. These results are extensions of the known ones on unperturbed nondegenerate [Formula: see text]-center, derived from a nonsmooth harmonic oscillator model, to degenerate [Formula: see text]-center. Our study involves some new computational treatments. The main tools are the generalized polar coordinate change and the generalized Lyapunov polar coordinate change together with an averaging theory for one-dimensional piecewise smooth differential equations. Finally, we present two Maple programs for computing the averaging functions and consequently the biggest lower bound on the maximum number of limit cycles of degenerate [Formula: see text]-center under general polynomial perturbations of degree [Formula: see text].

On the maximum number of limit cycles for a piecewise smooth differential system

In this paper, we consider an upper bound for the maximum number of limit cycles bifurcating from the periodic solutions of x¨+x=0 under piecewise smooth perturbation. By using the first order average method, we prove that the maximum number is at most n and n can be reached.

Sliding Shilnikov connection in Filippov-type predator–prey model

Recently, a piecewise smooth differential system was derived as a model of a 1 predator-2 prey interaction where the predator feeds adaptively on its preferred prey and an alternative prey. In such a model, strong evidence of chaotic behavior was numerically found. Here, we revisit this model and prove the existence of a Shilnikov sliding connection when the parameters are taken in a codimension one submanifold of the parameter space. As a consequence of this connection, we conclude, analytically, that the model behaves chaotically for an open region of the parameter space.

Piecewise-Smooth Slow–Fast Systems

We deal with piecewise-smooth differential systems $\dot {z}=X(z), z=(x,y)\in \mathbb {R}\times \mathbb {R}^{n-1},$ with switching occurring in a codimension one smooth surface Σ. A regularization of X is a 1-parameter family of smooth vector fields Xδ,δ > 0, satisfying that Xδ converges pointwise to X in $\mathbb {R}^{n}\setminus {\Sigma }$ , when $\delta \rightarrow 0$ . The regularized system $\dot {z}=X^{\delta }(z)$ is a slow–fast system. We work with two known regularizations: the classical one proposed by Sotomayor and Teixeira and its generalization, using transition functions without imposing the monotonicity condition. Minimal sets of regularized systems are studied with tools of the geometric singular perturbation theory. Moreover, we analyzed the persistence of the sliding region of piecewise-smooth slow–fast systems by singular perturbations.

Study of Periodic Orbits in Periodic Perturbations of Planar Reversible Filippov Systems Having a Twofold Cycle

We study the existence of periodic solutions in a class of planar Filippov systems obtained from non-autonomous periodic perturbations of reversible piecewise smooth differential systems. It is assumed that the unperturbed system presents a simple two-fold cycle, which is characterized by a closed trajectory connecting a visible two-fold singularity to itself. It is shown that under certain generic conditions the perturbed system has sliding and crossing periodic solutions. In order to get our results, Melnikov's ideas were applied together with tools from the geometric singular perturbation theory. Finally, a study of a perturbed piecewise Hamiltonian model is performed.

Limit cycles of piecewise polynomial perturbations of higher dimensional linear differential systems

The averaging theory has been extensively employed for studying periodic solutions of smooth and nonsmooth differential systems. Here, we extend the averaging theory for studying periodic solutions a class of regularly perturbed non-autonomous $n$-dimensional discontinuous piecewise smooth differential system. As a fundamental hypothesis, it is assumed that the unperturbed system has a manifold $\mathcal{Z}\subset\mathbb{R}^n$ of periodic solutions satisfying $\dim(\mathcal{Z}) < n.$ Then, we apply this result to study limit cycles bifurcating from periodic solutions of linear differential systems, $x'=Mx$, when they are perturbed inside a class of discontinuous piecewise polynomial differential systems with two zones. More precisely, we study the periodic solutions of the following differential system: $$ x'=Mx+ \varepsilon F\_1^n(x)+\varepsilon^2 F\_2^n(x), $$ in $\mathbb{R}^{d+2}$, where $\varepsilon$ is a small parameter, $M$ is a $(d+2)\times(d+2)$ matrix having one pair of pure imaginary conjugate eigenvalues, $m$ zeros eigenvalues, and $d-m$ non-zero real eigenvalues.

Global dynamics of a mechanical system with dry friction

The global dynamics of a mechanical system with dry friction is completely analyzed. This mechanical system is a class of discontinuous and transcendental piecewise smooth differential systems. Moreover, it can exhibit rich and complex dynamical phenomena, such as Hopf bifurcation, grazing bifurcation, grazing-sliding bifurcation and bifurcations of limit cycles. Finally, all global phase portraits of the system are presented on the Poincaré disc.

Analysis of the Dynamics of Piecewise Linear Memristors

In this paper, we consider a class of flux controlled memristive circuits with a piecewise linear memristor (i.e. the characteristic curve of the memristor is given by a piecewise linear function). The mathematical model is described by a discontinuous planar piecewise smooth differential system, which is defined on three zones separated by two parallel straight lines [Formula: see text] (called as discontinuity lines in discontinuous differential systems). We first investigate the stability of equilibrium points and the existence and uniqueness of a crossing limit cycle for the memristor-based circuit under self-excited oscillation. We then analyze the existence of periodic orbits of forced nonlinear oscillation for the memristive circuit with an external exciting source. Finally, we give numerical simulations to show good matches between our theoretical and simulation results.

Limit cycles for perturbing Hamiltonian system inside piecewise smooth polynomial differential system

In this paper, we obtain the first-order Melnikov function of piecewise smooth polynomial perturbation of a Hamiltonian system. As application, we consider the number of limit cycles for perturbing the global center and truncated pendulum inside a piecewise smooth cubic polynomial differential system. Our results show that a piecewise smooth differential system can bifurcate more limit cycles than the smooth one.

On the Limit Cycles Bifurcating from Piecewise Quasi-Homogeneous Differential Center

In this paper, a class of piecewise smooth quasi-homogeneous differential systems are considered. Using the first order Melnikov function derived in [Liu &amp; Han, 2010], we obtain a lower bound of the maximum number of limit cycles which bifurcate from the periodic annulus of the center under polynomial perturbation. The results reveal that piecewise smooth quasi-homogeneous differential systems can bifurcate more limit cycles than the smooth systems.

Bifurcation Analysis of Planar Piecewise Smooth Systems with a Line of Discontinuity

In this paper, we consider bifurcations of a class of planar piecewise smooth differential systems constituted by a general linear system and a quadratic Hamiltonian system. The linear system has four parameters. When the parameters vary in different regions, the left linear system can have a saddle, a node or a focus. For each case, we provide a completely qualitative analysis of the dynamical behavior for this piecewise smooth system. Our results generalize and improve the results in this direction.