Analysis of interconnected oscillators by dissipativity theory

This paper employs dissipativity theory for the global analysis of limit cycles in particular dynamical systems of possibly high dimension. Oscillators are regarded as open systems that satisfy a particular dissipation inequality. It is shown that this characterization has implications for the global stability analysis of limit cycle oscillations: i) in isolated oscillators, ii) in interconnections of oscillators, and iii) for the global synchrony analysis in interconnections of identical oscillators. © 2007 IEEE.

Interaction of additive and quantization noises in digital PLLs

Phase-locked loops (PLLs) are a crucial component in modern communications systems. Comprising of a phase-detector, linear filter, and controllable oscillator, they are widely used in radio receivers to retrieve the information content from remote signals. As such, they are capable of signal demodulation, phase and carrier recovery, frequency synthesis, and clock synchronization. Continuous-time PLLs are a mature area of study, and have been covered in the literature since the early classical work by Viterbi [1] in the 1950s. With the rise of computing in recent decades, discrete-time digital PLLs (DPLLs) are a more recent discipline; most of the literature published dates from the 1990s onwards. Gardner [2] is a pioneer in this area. It is our aim in this work to address the difficulties encountered by Gardner [3] in his investigation of the DPLL output phase-jitter where additive noise to the input signal is combined with frequency quantization in the local oscillator. The model we use in our novel analysis of the system is also applicable to another of the cases looked at by Gardner, that is the DPLL with a delay element integrated in the loop. This gives us the opportunity to look at this system in more detail, our analysis providing some unique insights into the variance `dip' seen by Gardner in [3]. We initially provide background on the probability theory and stochastic processes. These branches of mathematics are the basis for the study of noisy analogue and digital PLLs. We give an overview of the classical analogue PLL theory as well as the background on both the digital PLL and circle map, referencing the model proposed by Teplinsky et al. [4, 5]. For our novel work, the case of the combined frequency quantization and noisy input from [3] is investigated first numerically, and then analytically as a Markov chain via its Chapman-Kolmogorov equation. The resulting delay equation for the steady-state jitter distribution is treated using two separate asymptotic analyses to obtain approximate solutions. It is shown how the variance obtained in each case matches well to the numerical results. Other properties of the output jitter, such as the mean, are also investigated. In this way, we arrive at a more complete understanding of the interaction between quantization and input noise in the first order DPLL than is possible using simulation alone. We also do an asymptotic analysis of a particular case of the noisy first-order DPLL with delay, previously investigated by Gardner [3]. We show a unique feature of the simulation results, namely the variance `dip' seen for certain levels of input noise, is explained by this analysis. Finally, we look at the second-order DPLL with additive noise, using numerical simulations to see the effects of low levels of noise on the limit cycles. We show how these effects are similar to those seen in the noise-free loop with non-zero initial conditions.

Traveling waves in the Lethargic Crab Disease

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Modelling the lethargic crab disease.

The lethargic crab disease (LCD) is an emergent infirmity that has decimated native populations of the mangrove land crab (Ucides cordatus, Decapoda: Ocypodidae) along the Brazilian coast. Several potential etiological agents have been linked with LCD, but only in 2005 was it proved that it is caused by an ascomycete fungus. This is the first attempt to develop a mathematical model to describe the epidemiological dynamics of LCD. The model presents four possible scenarios, namely, the trivial equilibrium, the disease-free equilibrium, endemic equilibrium, and limit cycles arising from a Hopf bifurcation. The threshold values depend on the basic reproductive number of crabs and fungi, and on the infection rate. These scenarios depend on both the biological assumptions and the temporal evolution of the disease. Numerical simulations corroborate the analytical results and illustrate the different temporal dynamics of the crab and fungus populations.

Universal aspects of light halo nuclei

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Some remarks on bifurcation analysis of a nonlinear vibrating system excited by a shape memory alloy material (SMA)

In this paper, the dynamical response of a coupled oscillator is investigated, taking in consideration the nonlinear behavior of a SMA spring coupling the two oscillators. Due to the nonlinear coupling terms, the system exhibits both regular and chaotic motions. The Poincaré sections for different sets of coupling parameters are verified. © 2011 World Scientific Publishing Company.

Closed poly-trajectories and Poincaré index of non-smooth vector fields on the plane

This paper is concerned with closed orbits of non-smooth vector fields on the plane. For a class of non-smooth vector fields we provide necessary and sufficient conditions for the existence of closed poly-trajectorie. By means of a regularization process we prove that hyperbolic closed poly-trajectories are limit sets of a sequence of limit cycles of smooth vector fields. In our approach the Poincaré Index for non-smooth vector fields is introduced. © 2013 Springer Science+Business Media New York.

Piecewise linear perturbations of a linear center

This paper is mainly devoted to the study of the limit cycles that can bifurcate from a linear center using a piecewise linear perturbation in two zones. We consider the case when the two zones are separated by a straight line Σ and the singular point of the unperturbed system is in Σ. It is proved that the maximum number of limit cycles that can appear up to a seventh order perturbation is three. Moreover this upper bound is reached. This result confirms that these systems have more limit cycles than it was expected. Finally, center and isochronicity problems are also studied in systems which include a first order perturbation. For the latter systems it is also proved that, when the period function, defined in the period annulus of the center, is not monotone, then it has at most one critical period. Moreover this upper bound is also reached.

Estudo de bifurcações e aplicações em análise de sistemas de energia elétrica

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Estudo de ciclos limites em sistemas diferenciais lineares por partes

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Bifurcações em sistemas dinâmicos suaves por partes

Pós-graduação em Matemática - IBILCE

Análise global de sistemas quadráticos e cúbicos com duas circunferências não-concêntricas invariantes

Pós-graduação em Matematica Aplicada e Computacional - FCT

Ciclos limite em sistemas lineares por partes apresentando dois focos virtuais

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Conjuntos minimais de sistemas lineares por partes

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Classificação de centros e estudo de ciclos limite para sistemas lineares por partes em duas zonas no plano

Pós-graduação em Matemática - IBILCE