Let G be either a finite cyclic group of prime order or S. We show that if G acts on a manifold or, more generally, on a Poincaré duality space M , then each term of the Leray spectral sequence of the map M×GEG → BG satisfies a properly defined “Poincaré duality.” As a consequence of this fact we obtain new results relating the cohomology groups of M and M. We apply our results to study group a...

We investigate the existence and multiplicity of nontrivial periodic bouncing solutions for linear and asymptotically linear impact oscillators by applying a generalized version of the Poincaré–Birkhoff theorem to an adequate Poincaré section called the successor map. The main theorem includes a generalization of a related result by Bonheure and Fabry and provides a sufficient condition for the...

We show how to construct a symplectic approximation to the Poincaré map, using data from a symplectic integrator. We illustrate by producing a full-turn map for a realistic model of the Large Hadron Collider. Mapping of one turn is typically faster by a factor of 60 than direct integration. This allows one to follow orbits over times comparable to the required storage time of the beam, on a wor...

Hybrids systems are combinations of continuous and discrete systems. The bouncing ball is an extensively studied hybrid system, for which many solid Lyapunov-based tools are now available. Toward applying these tools to walking robots, where a hybrid dynamical system framework is also a natural fit, the rimless wheel provides a salient dynamic model because it shares commonalities with both bou...

This paper presents an approach to extracting the separation surfaces from periodic 2D time-dependent vector fields based on a recently introduced path line oriented topology. This topology is based on critical path lines which repeat the same spatial cycle per time period. Around those path lines there are areas of similar asymptotic flow behavior (basins) which are captured by a 2D Poincaré m...

This paper deals with the control of chaos exhibited by an impacting mechanical oscillator of one-degree-of-freedom. Its mathematical model is represented by an impulsive hybrid non-autonomous linear dynamics. This dynamics can display attractive nonlinear phenomena such as bifurcations and chaos. In this paper, we propose an approach to control chaos. The proposed strategy is based primarily o...

Self-similar trajectories play an important role in deterministic feedback control systems that possess a symmetry group of Fuller type. We consider self-similar trajectories in multi-input systems and the linear part of the associated Poincaré maps in orbit space with respect to the symmetry group. We show that Fuller groups contract the symplectic structure of the system’s phase space and der...

The transition from periodicity to chaos in a DC-DC Buck power converter is studied in this paper. The converter is controlled through a direct Pulse Width Modulation (PWM) in order to regulate the error dynamics at zero. Results show robustness with low output error and a fixed switching frequency. Furthermore, some rich dynamics appear as the constant associated with the first order error dyn...

This paper introduces a new simple system with a butterfly chaotic attractor. This system has rich and complex dynamics. With some typical parameters, its Lyapunov dimension is greater than other known three dimensional chaotic systems. It exhibits chaotic behavior over a large range of parameters, and the divergence of flow of this system is not a constant. The dynamics of this new system are ...