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Register a new userGlobal dynamics of planar quasi-homogeneous differential systems
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In this paper we provide a new method to study global dynamics of planar quasi--homogeneous differential systems. We first prove that all planar quasi--homogeneous polynomial differential systems can be translated into homogeneous differential systems and show that all quintic quasi--homogeneous but non--homogeneous systems can be reduced to four homogeneous ones. Then we present some properties of homogeneous systems, which can be used to discuss the dynamics of quasi--homogeneous systems. Finally we characterize the global topological phase portraits of quintic quasi--homogeneous but non--homogeneous differential systems.
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In this paper we research global dynamics and bifurcations of planar piecewise smooth quadratic quasi--homogeneous but non-homogeneous polynomial differential systems. We present sufficient and necessary conditions for the existence of a center in piecewise smooth quadratic quasi--homogeneous systems. Moreover, the center is global and non-isochronous if it exists, which cannot appear in smooth quadratic quasi-homogeneous systems. Then the global structures of piecewise smooth quadratic quasi--homogeneous but non-homogeneous systems are studied. Finally we investigate limit cycle bifurcations of the piecewise smooth quadratic quasi-homogeneous center and give the maximal number of limit cycles bifurcating from the periodic orbits of the center by applying the Melnikov method for piecewise smooth near-Hamiltonian systems.
By applying a singular perturbation approach, canard limit cycles exhibited by a general family of singularly perturbed planar piecewise linear (PWL) differential systems are analyzed. The performed study involves both hyperbolic and non-hyperbolic canard limit cycles appearing after both a supercritical and a subcritical Hopf bifurcation. The obtained results are completely comparable with those obtained for smooth vector fields. In some sense, the manuscript can be understood as an extension towards the PWL framework of the results obtained for smooth systems by Krupa and Szmolyan [18]. In addition, some novel slow-fast behaviors are obtained. In particular, in the supercritical case, and under suitable conditions, it is proved that the limit cycles are organized along a curve exhibiting two folds. Each of these folds corresponds to a saddle-node bifurcation of canard limit cycles, one involving headless canard cycles, whereas the other involving canard cycles with head. This configuration allows the coexistence of three canard limit cycles.
In this paper, the symmetry group of a differential system of n quadratic homogeneous first order ODEs of n variables is studied. For this purpose, we consider the action of both point and contact transformations to signify the corresponding Lie algebras. We also find the independent differential invariants of these actions.
The main purpose of this paper is to investigate commuting flows and integrable systems on the configuration spaces of planar linkages. Our study leads to the definition of a natural volume form on each configuration space of planar linkages, the notion of cross products of integrable systems, and also the notion of multi-Nambu integrable systems. The first integrals of our systems are functions of Bott-Morse type, which may be used to study the topology of configuration spaces. Dedicated to Anatoly T. Fomenko on the occasion of his 75th birthday
Attractors of cooperative dynamical systems are particularly simple; for example, a nontrivial periodic orbit cannot be an attractor. This paper provides characterizations of attractors for the wider class of coherent systems, defined by the property that no directed feedback loops are negative. Several new results for cooperative systems are obtained in the process.
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