The brain has to process and react to an enormous amount of information from the senses in real t... more The brain has to process and react to an enormous amount of information from the senses in real time. How is all this information represented and processed within the nervous system? A proposal of nonlinear and complex systems research is that dynamical attractors may form the basis of neural information processing. Here we show that this idea can be successfully applied to the human auditory system, and can explain our perception of pitch.
We investigate the hierarchical structure of three-frequency resonances in nonlinear dynamical sy... more We investigate the hierarchical structure of three-frequency resonances in nonlinear dynamical systems with three interacting frequencies. We hypothesize an ordering of these resonances based on a generalization of the Farey tree organization from two frequencies to three. In experiments and numerical simulations we demonstrate that our hypothesis describes the hierarchies of three-frequency resonances in representative dynamical systems. We conjecture that this organization may be universal across a large class of three-frequency systems.
Proceedings of the National Academy of Sciences, 2001
Two and a half millennia ago Pythagoras initiated the scientific study of the pitch of sounds; ye... more Two and a half millennia ago Pythagoras initiated the scientific study of the pitch of sounds; yet our understanding of the mechanisms of pitch perception remains incomplete. Physical models of pitch perception try to explain from elementary principles why certain physical characteristics of the stimulus lead to particular pitch sensations. There are two broad categories of pitch-perception models: place or spectral models consider that pitch is mainly related to the Fourier spectrum of the stimulus, whereas for periodicity or temporal models its characteristics in the time domain are more important. Current models from either class are usually computationally intensive, implementing a series of steps more or less supported by auditory physiology. However, the brain has to analyze and react in real time to an enormous amount of information from the ear and other senses. How is all this information efficiently represented and processed in the nervous system? A proposal of nonlinear a...
The dependence on u of the period doubling scaling indices for unimodal maps with a critical poin... more The dependence on u of the period doubling scaling indices for unimodal maps with a critical point of the form 1x1 " is numerically investigated. A new symbolic dynamics based computation technique working in configuration space is introduced. The existence of an upper bound for 6 (u + m) is numerically verified. An accurate estimate of 29.8 is given for this limit. Moreover, the global functional form of J (u) is shown to have an interesting symmetry.
The stroboscopic map of symmetric self-oscillators driven by pulses of alternating sign is constr... more The stroboscopic map of symmetric self-oscillators driven by pulses of alternating sign is constructed, for the regime of strong relaxation, by means of the so-called phase-transition curve. An ordering for the symmetries of the phase-locked orbits is found which is conjectured to be universal. Both results are illustrated by an exactly solvable example of such a system. When the approximation fails, unusual period doubling from symmetric orbits and strange attractors of twodimensional structure are encountered.
We apply results from nonlinear dynamics to an old problem in acoustical physics: the mechanism o... more We apply results from nonlinear dynamics to an old problem in acoustical physics: the mechanism of the perception of the pitch of sounds, especially the sounds known as complex tones that are important for music and speech intelligibility.
We take a dynamical-systems approach to study the qualitative dynamical aspects of the tidal lock... more We take a dynamical-systems approach to study the qualitative dynamical aspects of the tidal locking of the rotation of secondary celestial bodies with their orbital motion around the primary. We introduce a minimal model including the essential features of gravitationally induced elastic deformation and tidal dissipation that demonstrates the details of the energy transfer between the orbital and rotovibrational degrees of freedom. Despite its simplicity, our model can account for both synchronization into the 1:1 spin-orbit resonance and the circularization of the orbit as the only true asymptotic attractors, together with the existence of relatively long-lived metastable orbits with the secondary in p : q synchronous rotation.
Brinicles are hollow tubes of ice from centimetres to metres in length that form under floating s... more Brinicles are hollow tubes of ice from centimetres to metres in length that form under floating sea ice in the polar oceans when dense, cold brine drains downwards from sea ice into sea water close to its freezing point. When this extremely cold brine leaves the ice it freezes the water it comes into contact with; a hollow tube of ice-a brinicle-growing downwards around the plume of descending brine. We show that brinicles can be understood as a form of the self-assembled tubular precipitation structures termed chemical gardens, plant-like structures formed on placing together a soluble metal salt, often in the form of a seed crystal, and an aqueous solution of one of many anions, often silicate. On one hand, in the case of classical chemical gardens, an osmotic pressure difference across a semipermeable precipitation membrane that filters solutions by rejecting the solute leads to an inflow of water and to its rupture. The internal solution, generally being lighter than the external solution, flows up through the break, and as it does so a tube grows upwards by precipitation around the jet of internal solution. Such chemical-garden
At the 2007 Helmholtz Workshop in Berlin, two seemingly disparate papers were presented. One of t... more At the 2007 Helmholtz Workshop in Berlin, two seemingly disparate papers were presented. One of these, by Julyan Cartwright, Diego González, and Oreste Piro, dealt with a nonlinear dynamical model for pitch perception based on frequency ratios and forced oscillators, while the other, by Jack Douthett and Richard Krantz, focused on musical scales, maximally even (ME) sets, and their relationship to the one-dimensional antiferromagnetic Ising model. Both these approaches lead to a fractal structure involving Farey series known as a devil's staircase. Why is this? What is the connection between them? The ME sets approach is related to the Ising model of statistical physics; on the other hand, the forced oscillator model relates to the circle map of dynamical systems. Thus we find ourselves facing a deeper question: what are the links between these two paradigms, the Ising model and the circle map, that are fundamental to statistical physics on the one hand and to dynamical systems on the other? Here we present the two halves of the work side by side, so that the construction of the two arguments that arrive at a devil's staircase in music can be seen together.
International Journal of Bifurcation and Chaos, 1999
We investigate numerically and experimentally dynamical systems having three interacting frequenc... more We investigate numerically and experimentally dynamical systems having three interacting frequencies: a discrete mapping (a circle map), an exactly solvable model (a system of coupled ordinary differential equations), and an experimental device (an electronic oscillator). We compare the hierarchies of three-frequency resonances we find in each of these systems. All three show similar qualitative behaviour, suggesting the existence of generic features in the parameter-space organization of three-frequency resonances.
Chaos: An Interdisciplinary Journal of Nonlinear Science, 1993
A model of a hard oscillator with analytic solution is presented. Its behavior under periodic kic... more A model of a hard oscillator with analytic solution is presented. Its behavior under periodic kicking, for which a closed form stroboscopic map can be obtained, is studied. It is shown that the general structure of such an oscillator includes four distinct regions; the outer two regions correspond to very small or very large amplitude of the external force and match the corresponding regions in soft oscillators (invertible degree one and degree zero circle maps, respectively). There are two new regions for intermediate amplitude of the forcing. Region 3 corresponds to moderate high forcing, and is intrinsic to hard oscillators; it is characterized by discontinuous circle maps with a flat segment. Region 2 (low moderate forcing) has a certain resemblance to a similar region in soft oscillators (noninvertible degree one circle maps); however, the limit set of the dynamics in this region is not a circle, but a branched manifold, obtained as the tangent union of a circle and an interval...
IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 2000
We construct an experimental model of a nonlinear dynamical system with three frequencies. With t... more We construct an experimental model of a nonlinear dynamical system with three frequencies. With this analog electronic circuit made up of quasiperiodically forced coupled phase-locked loops, we investigate the structure of the scaling of three-frequency resonances or lockings in the dynamics. We hypothesize and confirm experimentally that for weak coupling the three-frequency resonance with the largest frequency-locked plateau in the space of parameters in the interval between two adjacent resonances
The ability of the auditory system to perceive the fundamental frequency of a sound even when thi... more The ability of the auditory system to perceive the fundamental frequency of a sound even when this frequency is removed from the stimulus is an interesting phenomenon related to the pitch of complex sounds. This capability is known as residue or virtual pitch perception and was first reported last century in the pioneering work of Seebeck. It is residue perception that allows one to listen to music with small transistor radios, which in general have a very poor and sometimes negligible response to low frequencies. The first attempt, due to von ...
The brain has to process and react to an enormous amount of information from the senses in real t... more The brain has to process and react to an enormous amount of information from the senses in real time. How is all this information represented and processed within the nervous system? A proposal of nonlinear and complex systems research is that dynamical attractors may form the basis of neural information processing. Here we show that this idea can be successfully applied to the human auditory system, and can explain our perception of pitch.
We investigate the hierarchical structure of three-frequency resonances in nonlinear dynamical sy... more We investigate the hierarchical structure of three-frequency resonances in nonlinear dynamical systems with three interacting frequencies. We hypothesize an ordering of these resonances based on a generalization of the Farey tree organization from two frequencies to three. In experiments and numerical simulations we demonstrate that our hypothesis describes the hierarchies of three-frequency resonances in representative dynamical systems. We conjecture that this organization may be universal across a large class of three-frequency systems.
Proceedings of the National Academy of Sciences, 2001
Two and a half millennia ago Pythagoras initiated the scientific study of the pitch of sounds; ye... more Two and a half millennia ago Pythagoras initiated the scientific study of the pitch of sounds; yet our understanding of the mechanisms of pitch perception remains incomplete. Physical models of pitch perception try to explain from elementary principles why certain physical characteristics of the stimulus lead to particular pitch sensations. There are two broad categories of pitch-perception models: place or spectral models consider that pitch is mainly related to the Fourier spectrum of the stimulus, whereas for periodicity or temporal models its characteristics in the time domain are more important. Current models from either class are usually computationally intensive, implementing a series of steps more or less supported by auditory physiology. However, the brain has to analyze and react in real time to an enormous amount of information from the ear and other senses. How is all this information efficiently represented and processed in the nervous system? A proposal of nonlinear a...
The dependence on u of the period doubling scaling indices for unimodal maps with a critical poin... more The dependence on u of the period doubling scaling indices for unimodal maps with a critical point of the form 1x1 " is numerically investigated. A new symbolic dynamics based computation technique working in configuration space is introduced. The existence of an upper bound for 6 (u + m) is numerically verified. An accurate estimate of 29.8 is given for this limit. Moreover, the global functional form of J (u) is shown to have an interesting symmetry.
The stroboscopic map of symmetric self-oscillators driven by pulses of alternating sign is constr... more The stroboscopic map of symmetric self-oscillators driven by pulses of alternating sign is constructed, for the regime of strong relaxation, by means of the so-called phase-transition curve. An ordering for the symmetries of the phase-locked orbits is found which is conjectured to be universal. Both results are illustrated by an exactly solvable example of such a system. When the approximation fails, unusual period doubling from symmetric orbits and strange attractors of twodimensional structure are encountered.
We apply results from nonlinear dynamics to an old problem in acoustical physics: the mechanism o... more We apply results from nonlinear dynamics to an old problem in acoustical physics: the mechanism of the perception of the pitch of sounds, especially the sounds known as complex tones that are important for music and speech intelligibility.
We take a dynamical-systems approach to study the qualitative dynamical aspects of the tidal lock... more We take a dynamical-systems approach to study the qualitative dynamical aspects of the tidal locking of the rotation of secondary celestial bodies with their orbital motion around the primary. We introduce a minimal model including the essential features of gravitationally induced elastic deformation and tidal dissipation that demonstrates the details of the energy transfer between the orbital and rotovibrational degrees of freedom. Despite its simplicity, our model can account for both synchronization into the 1:1 spin-orbit resonance and the circularization of the orbit as the only true asymptotic attractors, together with the existence of relatively long-lived metastable orbits with the secondary in p : q synchronous rotation.
Brinicles are hollow tubes of ice from centimetres to metres in length that form under floating s... more Brinicles are hollow tubes of ice from centimetres to metres in length that form under floating sea ice in the polar oceans when dense, cold brine drains downwards from sea ice into sea water close to its freezing point. When this extremely cold brine leaves the ice it freezes the water it comes into contact with; a hollow tube of ice-a brinicle-growing downwards around the plume of descending brine. We show that brinicles can be understood as a form of the self-assembled tubular precipitation structures termed chemical gardens, plant-like structures formed on placing together a soluble metal salt, often in the form of a seed crystal, and an aqueous solution of one of many anions, often silicate. On one hand, in the case of classical chemical gardens, an osmotic pressure difference across a semipermeable precipitation membrane that filters solutions by rejecting the solute leads to an inflow of water and to its rupture. The internal solution, generally being lighter than the external solution, flows up through the break, and as it does so a tube grows upwards by precipitation around the jet of internal solution. Such chemical-garden
At the 2007 Helmholtz Workshop in Berlin, two seemingly disparate papers were presented. One of t... more At the 2007 Helmholtz Workshop in Berlin, two seemingly disparate papers were presented. One of these, by Julyan Cartwright, Diego González, and Oreste Piro, dealt with a nonlinear dynamical model for pitch perception based on frequency ratios and forced oscillators, while the other, by Jack Douthett and Richard Krantz, focused on musical scales, maximally even (ME) sets, and their relationship to the one-dimensional antiferromagnetic Ising model. Both these approaches lead to a fractal structure involving Farey series known as a devil's staircase. Why is this? What is the connection between them? The ME sets approach is related to the Ising model of statistical physics; on the other hand, the forced oscillator model relates to the circle map of dynamical systems. Thus we find ourselves facing a deeper question: what are the links between these two paradigms, the Ising model and the circle map, that are fundamental to statistical physics on the one hand and to dynamical systems on the other? Here we present the two halves of the work side by side, so that the construction of the two arguments that arrive at a devil's staircase in music can be seen together.
International Journal of Bifurcation and Chaos, 1999
We investigate numerically and experimentally dynamical systems having three interacting frequenc... more We investigate numerically and experimentally dynamical systems having three interacting frequencies: a discrete mapping (a circle map), an exactly solvable model (a system of coupled ordinary differential equations), and an experimental device (an electronic oscillator). We compare the hierarchies of three-frequency resonances we find in each of these systems. All three show similar qualitative behaviour, suggesting the existence of generic features in the parameter-space organization of three-frequency resonances.
Chaos: An Interdisciplinary Journal of Nonlinear Science, 1993
A model of a hard oscillator with analytic solution is presented. Its behavior under periodic kic... more A model of a hard oscillator with analytic solution is presented. Its behavior under periodic kicking, for which a closed form stroboscopic map can be obtained, is studied. It is shown that the general structure of such an oscillator includes four distinct regions; the outer two regions correspond to very small or very large amplitude of the external force and match the corresponding regions in soft oscillators (invertible degree one and degree zero circle maps, respectively). There are two new regions for intermediate amplitude of the forcing. Region 3 corresponds to moderate high forcing, and is intrinsic to hard oscillators; it is characterized by discontinuous circle maps with a flat segment. Region 2 (low moderate forcing) has a certain resemblance to a similar region in soft oscillators (noninvertible degree one circle maps); however, the limit set of the dynamics in this region is not a circle, but a branched manifold, obtained as the tangent union of a circle and an interval...
IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 2000
We construct an experimental model of a nonlinear dynamical system with three frequencies. With t... more We construct an experimental model of a nonlinear dynamical system with three frequencies. With this analog electronic circuit made up of quasiperiodically forced coupled phase-locked loops, we investigate the structure of the scaling of three-frequency resonances or lockings in the dynamics. We hypothesize and confirm experimentally that for weak coupling the three-frequency resonance with the largest frequency-locked plateau in the space of parameters in the interval between two adjacent resonances
The ability of the auditory system to perceive the fundamental frequency of a sound even when thi... more The ability of the auditory system to perceive the fundamental frequency of a sound even when this frequency is removed from the stimulus is an interesting phenomenon related to the pitch of complex sounds. This capability is known as residue or virtual pitch perception and was first reported last century in the pioneering work of Seebeck. It is residue perception that allows one to listen to music with small transistor radios, which in general have a very poor and sometimes negligible response to low frequencies. The first attempt, due to von ...
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Papers by Diego Gonzalez