Locating the sets of exceptional points in dissipative systems and the self-stability of bicycles

Sets in the parameter space corresponding to complex exceptional points have high codimension and by this reason they are difficult objects for numerical location. However, complex EPs play an important role in the problems of stability of dissipative systems where they are frequently considered as precursors to instability. We propose to locate the set of complex EPs using the fact that the global minimum of the spectral abscissa of a polynomial is attained at the EP of the highest possible order. Applying this approach to the problem of self-stabilization of a bicycle we find explicitly the EP sets that suggest scaling laws for the design of robust bikes that agree with the design of the known experimental machines

Robust synchronization of heterogeneous robot swarms on the sphere

Synchronization on the sphere is important to certain control applications in swarm robotics. Of recent interest is the Lohe model, which generalizes the Kuramoto model from the circle to the sphere. The Lohe model is mainly studied in mathematical physics as a toy model of quantum synchronization. The model makes few assumptions, wherefore it is well-suited to represent a swarm. Previous work on this model has focused on the cases of complete and acyclic networks or the homogeneous case where all oscillator frequencies are equal. This paper concerns the case of heterogeneous oscillators connected by a non-trivial network. We show that any undesired equilibrium is exponentially unstable if the frequencies satisfy a given bound. This property can also be interpreted as a robustness result for small model perturbations of the homogeneous case with zero frequencies. As such, the Lohe model is a good choice for control applications in swarm robotics

Optimal synchronization of Kuramoto oscillators: a dimensional reduction approach

A recently proposed dimensional reduction approach for studying synchronization in the Kuramoto model is employed to build optimal network topologies to favor or to suppress synchronization. The approach is based in the introduction of a collective coordinate for the time evolution of the phase locked oscillators, in the spirit of the Ott-Antonsen ansatz. We show that the optimal synchronization of a Kuramoto network demands the maximization of the quadratic function $\omega^T L \omega$, where $\omega$ stands for the vector of the natural frequencies of the oscillators, and $L$ for the network Laplacian matrix. Many recently obtained numerical results can be re-obtained analytically and in a simpler way from our maximization condition. A computationally efficient {hill climb} rewiring algorithm is proposed to generate networks with optimal synchronization properties. Our approach can be easily adapted to the case of the Kuramoto models with both attractive and repulsive interactions, and again many recent numerical results can be rederived in a simpler and clearer analytical manner.Comment: 6 pages, 6 figures, final version to appear in PR

Hamiltonian dynamics and spectral theory for spin-oscillators

We study the Hamiltonian dynamics and spectral theory of spin-oscillators. Because of their rich structure, spin-oscillators display fairly general properties of integrable systems with two degrees of freedom. Spin-oscillators have infinitely many transversally elliptic singularities, exactly one elliptic-elliptic singularity and one focus-focus singularity. The most interesting dynamical features of integrable systems, and in particular of spin-oscillators, are encoded in their singularities. In the first part of the paper we study the symplectic dynamics around the focus-focus singularity. In the second part of the paper we quantize the coupled spin-oscillators systems and study their spectral theory. The paper combines techniques from semiclassical analysis with differential geometric methods.Comment: 32 page

Improving the Spectral Efficiency of Nonlinear Satellite Systems through Time-Frequency Packing and Advanced Processing

We consider realistic satellite communications systems for broadband and broadcasting applications, based on frequency-division-multiplexed linear modulations, where spectral efficiency is one of the main figures of merit. For these systems, we investigate their ultimate performance limits by using a framework to compute the spectral efficiency when suboptimal receivers are adopted and evaluating the performance improvements that can be obtained through the adoption of the time-frequency packing technique. Our analysis reveals that introducing controlled interference can significantly increase the efficiency of these systems. Moreover, if a receiver which is able to account for the interference and the nonlinear impairments is adopted, rather than a classical predistorter at the transmitter coupled with a simpler receiver, the benefits in terms of spectral efficiency can be even larger. Finally, we consider practical coded schemes and show the potential advantages of the optimized signaling formats when combined with iterative detection/decoding.Comment: 8 pages, 8 figure

Chaotic Dynamics Enhance the Sensitivity of Inner Ear Hair Cells

Hair cells of the auditory and vestibular systems are capable of detecting sounds that induce sub-nanometer vibrations of the hair bundle, below the stochastic noise levels of the surrounding fluid. Hair bundles of certain species are also known to oscillate without external stimulation, indicating the presence of an underlying active mechanism. We propose that chaotic dynamics enhance the sensitivity and temporal resolution of the hair bundle response, and provide experimental and theoretical evidence for this effect. By varying the viscosity and ionic composition of the surrounding fluid, we are able to modulate the degree of chaos observed in the hair bundle dynamics in vitro. We consistently find that the hair bundle is most sensitive to a stimulus of small amplitude when it is poised in the weakly chaotic regime. Further, we show that the response time to a force step decreases with increasing levels of chaos. These results agree well with our numerical simulations of a chaotic Hopf oscillator and suggest that chaos may be responsible for the sensitivity and temporal resolution of hair cells