This thesis examines how novel invariants called the "stability index" as proposed by Podvigina and Ashwin can be used to characterize the local geometry of riddled basins of attraction for both skew and non-skew product systems. In particular, it would be interesting to understand how the stability index behaves on the basin boundary between multiple basins of attraction. Then we can ask this question: How can we identify when a basin is riddled? To answer this, we present three models with the presence of riddled basins. In the first model, we present a skew product system of a simple example of a piecewise linear map. We prove that the riddled basin occurs within a certain range of parameter and calculate the stability index analytically for this map. Our results for the stability index at a point show that for Lebesgue almost all points in the map, the index is positive and for some points the index may be negative. We verify these results with our numerical computation for this index. We also make a corollary claiming that the formula for the stability index at a point can be expressed in terms of the stability index for an attractor and Lyapunov exponents for this map. This suggests that this index could be useful as a diagnostic tool to study bifurcation of the riddled basins of attraction. In the second model, we refer to a skew product map studied by Keller. Previously, Keller computed the stability index for an attractor in his map whereas in this thesis, we use an alternative way to compute the index; that is on the basins of attraction for Keller's map, found by inverting his map. Using the same map, we also verify maximum and minimum measures as obtained in his paper by studying Birkhoff averages on periodic points of Markov map in his system. We also conjecture result by Keller and Otani on the dimension of zero sets of invariant graph (i.e. basin boundary) that appears in Keller's map to a complete range of a parameter in the map. The last model is a non-skew product map which is also has a riddled basin. For this map, we compute the stability index for an attractor on the baseline of the map. The result indicates that the index is positive for Lebesgue almost all points whenever the riddled basin occurs.

Um estudo de bacias crivadas e um estudo de séries de batimentos cardíacos através de ferramentas não lineares são apresentados. Bacias crivadas ocorrem em sistemas não lineares onde a simetria do espaço de fase permite a existência de um subespaço invariante capaz de atrair e repelir órbitas. Como conseqüência para todo ponto pertencente a bacia de atração do atirador existirá um ponto não pertencente numa distância arbitrariamente próxima. Pode-se verificar a presença de bacias crivadas pela análise do espaço de fase e dos expoentes máximos transversais de Lyapunov de tempo finito. A caracterização do fenômeno pode ainda ser complementada pelas leis de escala provenientes de um modelo para as flutuações dos expoentes máximos transversais de Lyapunov de tempo finito. O crivamento é analisado para um sistema mecânico e para um modelo ecológico. Comparamos para os dois sistemas as previsões teóricas, dadas por um modelo stocástico, com os resultados numéricos. No estudo de séries de batimentos cardíacos diversos grupos de dados são submetidos a diferentes análises a fim de determinar ´ndices que permitam, dado um paciente, decidir a qual grupo ele pertence. Expoentes de Lyapunov, análise depurada de flutuações e segmentação das séries foram empregados na análise das séries de intervalos RR e pressão arterial. Desses métodos empregados, nenhum foi conclusivo no sentido de caracterizar os grupos. Porém, uma nova formulação do método de segmentação das séries mostrou ser possível a caracterização através de um parâmetro, que todavia, exige séries longas de observação. / A study of riddled basins of attraction and a study of heart rate variability through nonlinear dynamics tools are presented. Riddled basins occur in nonlinear systems whose phase space symmetry allows an invariant subspace with an chaotic attractor. This invariant subspace can either attract or repel orbits. As a consequence, for every point belonging to the basin of attraction there is another point, arbitrarily close, that does not belong to the basin of attraction. The presence of riddled basins is verified by analyzing the maximal transversal Lyapunov exponent and the maximal transversal finite time Lyapunov exponent. The characterization of riddling is complemented by the calculation of scaling laws provided by a stochastic model of the transversal finite time Lyapunov exponents. Riddling is analyzed for a mechanical system and for an ecological model. The results are compared with the theoretical prediction given by the stochastic model. In the study of heart rate variability, time series of different groups were analyzed in order to determine quantifiers of healthiness and sickness, in the sense that given a patient one can say if the patient belongs to a healthy group or not. Lyapunov exponents, detrended fluctuation analysis and time series segmentation were applied to RR-intervals and blood pressure time series. These methods were not able to characterize the groups. However, a new formulation of the segmentation method indicates that it is possible to find a quantifier, although this quantifier requires long time series of observation.

Um estudo de bacias crivadas e um estudo de séries de batimentos cardíacos através de ferramentas não lineares são apresentados. Bacias crivadas ocorrem em sistemas não lineares onde a simetria do espaço de fase permite a existência de um subespaço invariante capaz de atrair e repelir órbitas. Como conseqüência para todo ponto pertencente a bacia de atração do atirador existirá um ponto não pertencente numa distância arbitrariamente próxima. Pode-se verificar a presença de bacias crivadas pela análise do espaço de fase e dos expoentes máximos transversais de Lyapunov de tempo finito. A caracterização do fenômeno pode ainda ser complementada pelas leis de escala provenientes de um modelo para as flutuações dos expoentes máximos transversais de Lyapunov de tempo finito. O crivamento é analisado para um sistema mecânico e para um modelo ecológico. Comparamos para os dois sistemas as previsões teóricas, dadas por um modelo stocástico, com os resultados numéricos. No estudo de séries de batimentos cardíacos diversos grupos de dados são submetidos a diferentes análises a fim de determinar ´ndices que permitam, dado um paciente, decidir a qual grupo ele pertence. Expoentes de Lyapunov, análise depurada de flutuações e segmentação das séries foram empregados na análise das séries de intervalos RR e pressão arterial. Desses métodos empregados, nenhum foi conclusivo no sentido de caracterizar os grupos. Porém, uma nova formulação do método de segmentação das séries mostrou ser possível a caracterização através de um parâmetro, que todavia, exige séries longas de observação. / A study of riddled basins of attraction and a study of heart rate variability through nonlinear dynamics tools are presented. Riddled basins occur in nonlinear systems whose phase space symmetry allows an invariant subspace with an chaotic attractor. This invariant subspace can either attract or repel orbits. As a consequence, for every point belonging to the basin of attraction there is another point, arbitrarily close, that does not belong to the basin of attraction. The presence of riddled basins is verified by analyzing the maximal transversal Lyapunov exponent and the maximal transversal finite time Lyapunov exponent. The characterization of riddling is complemented by the calculation of scaling laws provided by a stochastic model of the transversal finite time Lyapunov exponents. Riddling is analyzed for a mechanical system and for an ecological model. The results are compared with the theoretical prediction given by the stochastic model. In the study of heart rate variability, time series of different groups were analyzed in order to determine quantifiers of healthiness and sickness, in the sense that given a patient one can say if the patient belongs to a healthy group or not. Lyapunov exponents, detrended fluctuation analysis and time series segmentation were applied to RR-intervals and blood pressure time series. These methods were not able to characterize the groups. However, a new formulation of the segmentation method indicates that it is possible to find a quantifier, although this quantifier requires long time series of observation.