2001 |
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| 3. | Negi, Surendra Singh; Ramaswamy, Ramakrishna Plethora of strange nonchaotic attractors Journal Article Pramana – Journal of Physics, 56 (1), pp. 47–56, 2001, ISSN: 03044289. Abstract | Links | BibTeX | Tags: 10, 42, 45, b, Fractals, Lyapunov exponent, nonchaotic, pacs nos 05, strange attractors @article{Negi2001, title = {Plethora of strange nonchaotic attractors}, author = {Surendra Singh Negi and Ramakrishna Ramaswamy}, url = {https://ramramaswamy.org/papers/079.pdf}, doi = {10.1007/s12043-001-0140-7}, issn = {03044289}, year = {2001}, date = {2001-01-01}, journal = {Pramana – Journal of Physics}, volume = {56}, number = {1}, pages = {47–56}, abstract = {We show that it is possible to devise a large class of skew-product dynamical systems which have strange nonchaotic attractors (SNAs): the dynamics is asymptotically on fractal attractors and the largest Lyapunov exponent is non-positive. Furthermore, we show that quasiperiodic forcing, which has been a hallmark of essentially all hitherto known examples of such dynamics is not necessary for the creation of SNAs.}, keywords = {10, 42, 45, b, Fractals, Lyapunov exponent, nonchaotic, pacs nos 05, strange attractors}, pubstate = {published}, tppubtype = {article} } We show that it is possible to devise a large class of skew-product dynamical systems which have strange nonchaotic attractors (SNAs): the dynamics is asymptotically on fractal attractors and the largest Lyapunov exponent is non-positive. Furthermore, we show that quasiperiodic forcing, which has been a hallmark of essentially all hitherto known examples of such dynamics is not necessary for the creation of SNAs. |
1992 |
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| 2. | B Tadi´c U Nowak, Usadel Ramaswamy K R; Padlewski, S Scaling behaviour in disordered sandpile automata Journal Article Physical Review A, 45 (12), pp. 8536–8545, 1992. Abstract | Links | BibTeX | Tags: Criticality, Fractals, Sandpile, scaling @article{Tadi´c1992, title = {Scaling behaviour in disordered sandpile automata}, author = {B Tadi´c, U Nowak, K Usadel, R Ramaswamy and S Padlewski }, url = {https://link.aps.org/doi/10.1103/PhysRevA.45.8536}, doi = {10.1103/PhysRevA.45.8536}, year = {1992}, date = {1992-06-01}, journal = {Physical Review A}, volume = {45}, number = {12}, pages = {8536–8545}, abstract = {We study numerically the scaling behavior of disordered sandpile automata with preferred direction on a two-dimensional square lattice. We consider two types of bulk defects that modify locally the dynamic rule: (i) a random distribution of holes, through which sand grains may leave the system, and (ii) several models with a random distribution of critical heights. We find that at large time and length scales the self-organized critical behavior, proved exactly in the pure model, is lost for any finite concentration of defects both in the model of random holes and in those models of random critical heights in which the dynamic rule violates the height conservation law. In the case of the random critical height model with the height-conserving dynamics, we find that self-organized criticality holds for the entire range of concentrations of defects, and it belongs to the same universality class as the pure model. In the case of random holes we analyze the scaling properties of the probability distributions P(T,p,L) and D(s,p,L) of avalanches of duration larger than T and size larger than s, respectively, at lattices with linear size L and concentration of defect sites p. We find that in general the following scaling forms apply: P(T)= T − α scrP(T/x,T/L) and D(s)= s − τ scrD(s/m,s/ L ν ), where x≡x(p) and m≡m(p) are the characteristic duration (length) and the characteristic size (mass) of avalanches for a given concentration of defects. The power-law behavior of the distributions still persists for length scales T≪x(p) and mass scales s≪m(p). The characteristic length x(p) and mass m(p) are finite for small concentrations of defects and diverge at p→0 according to the power law x(p)∼ p − μ x and m(p)∼ p − μ m , with the numerically determined values of the exponents close to μ x =1 and μ m =1.5. The finite size of the lattice may affect the measured probability distributions if for a given concentration of defects the characteristic length x(p) exceeds the lattice size L. A finite-size scaling analysis for the mass distribution yields the exponent ν=1.5, while the duration of the avalanches scales linearly with the size. We also determine the exponent D=1.5 that connects the mass and the duration of avalanches.}, keywords = {Criticality, Fractals, Sandpile, scaling}, pubstate = {published}, tppubtype = {article} } We study numerically the scaling behavior of disordered sandpile automata with preferred direction on a two-dimensional square lattice. We consider two types of bulk defects that modify locally the dynamic rule: (i) a random distribution of holes, through which sand grains may leave the system, and (ii) several models with a random distribution of critical heights. We find that at large time and length scales the self-organized critical behavior, proved exactly in the pure model, is lost for any finite concentration of defects both in the model of random holes and in those models of random critical heights in which the dynamic rule violates the height conservation law. In the case of the random critical height model with the height-conserving dynamics, we find that self-organized criticality holds for the entire range of concentrations of defects, and it belongs to the same universality class as the pure model. In the case of random holes we analyze the scaling properties of the probability distributions P(T,p,L) and D(s,p,L) of avalanches of duration larger than T and size larger than s, respectively, at lattices with linear size L and concentration of defect sites p. We find that in general the following scaling forms apply: P(T)= T − α scrP(T/x,T/L) and D(s)= s − τ scrD(s/m,s/ L ν ), where x≡x(p) and m≡m(p) are the characteristic duration (length) and the characteristic size (mass) of avalanches for a given concentration of defects. The power-law behavior of the distributions still persists for length scales T≪x(p) and mass scales s≪m(p). The characteristic length x(p) and mass m(p) are finite for small concentrations of defects and diverge at p→0 according to the power law x(p)∼ p − μ x and m(p)∼ p − μ m , with the numerically determined values of the exponents close to μ x =1 and μ m =1.5. The finite size of the lattice may affect the measured probability distributions if for a given concentration of defects the characteristic length x(p) exceeds the lattice size L. A finite-size scaling analysis for the mass distribution yields the exponent ν=1.5, while the duration of the avalanches scales linearly with the size. We also determine the exponent D=1.5 that connects the mass and the duration of avalanches. |
1987 |
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| 1. | Ramaswamy, R; Swaminathan, S Fractal eigenfunctions in (classically) nonintegrable hamiltonian systems Journal Article EPL, 4 (2), pp. 127–131, 1987, ISSN: 12864854. Abstract | Links | BibTeX | Tags: Fractals, Nonintegrable Hamiltonian Systems @article{Ramaswamy1987a, title = {Fractal eigenfunctions in (classically) nonintegrable hamiltonian systems}, author = {R Ramaswamy and S Swaminathan}, url = {https://ramramaswamy.org/papers/033.pdf}, doi = {10.1209/0295-5075/4/2/001}, issn = {12864854}, year = {1987}, date = {1987-01-01}, journal = {EPL}, volume = {4}, number = {2}, pages = {127–131}, abstract = {Bound-state eigenfunctions for a (classically) nonintegrable two degrees of freedom Hamiltonian system are studied. Between the de Broglie wavelength and a localization length, the probability density has a statistically fractal structure in some eigenstates. This novel characterization of eigenstates is intrinsically basis-set and coordinate independent and might therefore provide an objective approach to the question of quantum-chaotic behaviour. textcopyright IOP Publishing Ltd.}, keywords = {Fractals, Nonintegrable Hamiltonian Systems}, pubstate = {published}, tppubtype = {article} } Bound-state eigenfunctions for a (classically) nonintegrable two degrees of freedom Hamiltonian system are studied. Between the de Broglie wavelength and a localization length, the probability density has a statistically fractal structure in some eigenstates. This novel characterization of eigenstates is intrinsically basis-set and coordinate independent and might therefore provide an objective approach to the question of quantum-chaotic behaviour. textcopyright IOP Publishing Ltd. |