Equation (7) is incorrect: it should be replaced by I [ f ∗ φeff=π/2 ] (η) = 0. This does not, ho... more Equation (7) is incorrect: it should be replaced by I [ f ∗ φeff=π/2 ] (η) = 0. This does not, however, affect any result of the paper. Indeed, there is no sense in considering the impulse for the case φeff = π/2 (nor for φeff = 3π/2) because, unlike f ∗ φeff=0 (t) (equation (8)), the normalized function f ∗ φeff=π/2 (t) = fφeff=π/2 (t) /[2M(η)] (equation (6)) does not present an ηdependent ‘load’ (constant force) term. Clearly, this is because, unlike the maxima, M (η), and minima, m (η), of fφeff=0 (t) ≡ η cos t + (1 − η) cos (2t), those of fφeff=π/2 (t) ≡ η cos t − (1 − η) sin (2t) are symmetric, i.e., m (η) = −M (η) (compare figures 1(a) and 2(a)). This ultimately comes from the fact that the waveform of fφeff=π/2 (t) fits (for η = 2/3) that of one of the four equivalent expressions of the biharmonic universal excitation
General results concerning maintenance or enhancement of chaos are presented for dissipative syst... more General results concerning maintenance or enhancement of chaos are presented for dissipative systems subjected to two harmonic perturbations (one chaos inducing and the other chaos enhancing). The connection with previous results on chaos suppression is also discussed in a general setting. It is demonstrated that, in general, a second harmonic perturbation can reliably play an enhancer or inhibitor role by solely adjusting its initial phase. Numerical results indicate that general theoretical findings concerning periodic chaos-inducing perturbations also work for aperiodic chaos-inducing perturbations, and in arrays of identical chaotic coupled oscillators.
We demonstrate the effectiveness of Jacobian elliptic functions (JEFs) for inquiring into the res... more We demonstrate the effectiveness of Jacobian elliptic functions (JEFs) for inquiring into the reshaping effect of quasiperiodic forces in nonlinear nonautonomous systems exhibiting strange nonchaotic attractors (SNAs). Specifically, we characterize analytically and numerically some reshaping-induced transitions starting from SNAs in the context of quasiperiodically forced systems. We found similar scenarios of SNAs from the analysis of two representative examples: a quasiperiodically forced damped pendulum and a two-dimensional map. This clearly well-suited and advantageous use of the JEFs, which in their own right lie at the heart of nonlinear physics, may encourage students at intermediate university levels to study them in depth.
World scientific series on nonlinear science, series A, 2005
Page 1. WORLD SCIENTIFIC SERIES ON ONLINEAR SCIENC Series Editor: Leon O. Chua HIKK Ricardo Chaco... more Page 1. WORLD SCIENTIFIC SERIES ON ONLINEAR SCIENC Series Editor: Leon O. Chua HIKK Ricardo Chacon World Scientific Page 2. Page 3. CONTROL OF HOMOCLINIC CHAOS BV WERK PERIODIC PERTURBATIONS Page 4. ...
Communications in Nonlinear Science and Numerical Simulation, Apr 1, 2020
Abstract We experimentally, numerically, and theoretically characterize the effectiveness of inco... more Abstract We experimentally, numerically, and theoretically characterize the effectiveness of incommensurate excitations at suppressing chaos in damped driven systems. Specifically, we consider an inertial Brownian particle moving in a prototypical two-well potential and subjected to a primary (chaos-inducing) harmonic excitation and a suppressory incommensurategeneric (non-harmonic) excitation. We show that the effective amplitude of the suppressory excitation is minimal when the impulse transmitted by it is near its maximum, while its value is rather insensitive to higher-order convergents of the irrational ratio between the involved driving periods. Remarkably, the number and values of the effective initial phase difference between the two excitations are independent of the impulse while they critically depend on each particular convergent in a complex way involving both the approximate frustration of chaos-inducing homoclinic bifurcations and the maximum survival of relevant spatio-temporal symmetries of the dynamical equation.
International Journal of Bifurcation and Chaos, Dec 1, 1996
This paper studies the effect of continuous and discontinuous time dependent forcings onto dynami... more This paper studies the effect of continuous and discontinuous time dependent forcings onto dynamical systems. We compare these different forcings in the context of laminar chaotic mixing. It is shown that the response of a Hamiltonian two-dimensional system to a time periodic sinusoidal forcing differs qualitatively and quantitatively from the response to a square wave function of the same frequency. Consequently, the mixing efficiency of both types of forcings are different. Also a periodic function of the same shape as that of the velocity of the unperturbed system is tested as a forcing, its mixing efficiency being intermediate.
Music is presented as a setting for teaching nonlinear dynamics, showing how different sequences ... more Music is presented as a setting for teaching nonlinear dynamics, showing how different sequences of notes may illustrate ideas such as the sensitivity to initial conditions, and the dynamics and chaotic behaviour connected with fixed-point and limit-cycle attractors. The aim is not music composition, but a first approach to an interdisciplinary tool suitable for a single session class at preuniversity
We study a parametrically damped two-well Duffing oscillator, subjected to a periodic string of s... more We study a parametrically damped two-well Duffing oscillator, subjected to a periodic string of symmetric pulses. The order-chaos threshold when altering solely the width of the pulses is investigated theoretically through Melnikov analysis. We show analytically and numerically that most of the results appear independent of the particular wave form of the pulses provided that the transmitted impulse is the same. By using this property, the stability boundaries of the stationary solutions are determined to first approximation by means of an elliptic harmonic balance method. Finally, the bifurcation behavior at the stability boundaries is determined numerically.
Equation (7) is incorrect: it should be replaced by I [ f ∗ φeff=π/2 ] (η) = 0. This does not, ho... more Equation (7) is incorrect: it should be replaced by I [ f ∗ φeff=π/2 ] (η) = 0. This does not, however, affect any result of the paper. Indeed, there is no sense in considering the impulse for the case φeff = π/2 (nor for φeff = 3π/2) because, unlike f ∗ φeff=0 (t) (equation (8)), the normalized function f ∗ φeff=π/2 (t) = fφeff=π/2 (t) /[2M(η)] (equation (6)) does not present an ηdependent ‘load’ (constant force) term. Clearly, this is because, unlike the maxima, M (η), and minima, m (η), of fφeff=0 (t) ≡ η cos t + (1 − η) cos (2t), those of fφeff=π/2 (t) ≡ η cos t − (1 − η) sin (2t) are symmetric, i.e., m (η) = −M (η) (compare figures 1(a) and 2(a)). This ultimately comes from the fact that the waveform of fφeff=π/2 (t) fits (for η = 2/3) that of one of the four equivalent expressions of the biharmonic universal excitation
General results concerning maintenance or enhancement of chaos are presented for dissipative syst... more General results concerning maintenance or enhancement of chaos are presented for dissipative systems subjected to two harmonic perturbations (one chaos inducing and the other chaos enhancing). The connection with previous results on chaos suppression is also discussed in a general setting. It is demonstrated that, in general, a second harmonic perturbation can reliably play an enhancer or inhibitor role by solely adjusting its initial phase. Numerical results indicate that general theoretical findings concerning periodic chaos-inducing perturbations also work for aperiodic chaos-inducing perturbations, and in arrays of identical chaotic coupled oscillators.
We demonstrate the effectiveness of Jacobian elliptic functions (JEFs) for inquiring into the res... more We demonstrate the effectiveness of Jacobian elliptic functions (JEFs) for inquiring into the reshaping effect of quasiperiodic forces in nonlinear nonautonomous systems exhibiting strange nonchaotic attractors (SNAs). Specifically, we characterize analytically and numerically some reshaping-induced transitions starting from SNAs in the context of quasiperiodically forced systems. We found similar scenarios of SNAs from the analysis of two representative examples: a quasiperiodically forced damped pendulum and a two-dimensional map. This clearly well-suited and advantageous use of the JEFs, which in their own right lie at the heart of nonlinear physics, may encourage students at intermediate university levels to study them in depth.
World scientific series on nonlinear science, series A, 2005
Page 1. WORLD SCIENTIFIC SERIES ON ONLINEAR SCIENC Series Editor: Leon O. Chua HIKK Ricardo Chaco... more Page 1. WORLD SCIENTIFIC SERIES ON ONLINEAR SCIENC Series Editor: Leon O. Chua HIKK Ricardo Chacon World Scientific Page 2. Page 3. CONTROL OF HOMOCLINIC CHAOS BV WERK PERIODIC PERTURBATIONS Page 4. ...
Communications in Nonlinear Science and Numerical Simulation, Apr 1, 2020
Abstract We experimentally, numerically, and theoretically characterize the effectiveness of inco... more Abstract We experimentally, numerically, and theoretically characterize the effectiveness of incommensurate excitations at suppressing chaos in damped driven systems. Specifically, we consider an inertial Brownian particle moving in a prototypical two-well potential and subjected to a primary (chaos-inducing) harmonic excitation and a suppressory incommensurategeneric (non-harmonic) excitation. We show that the effective amplitude of the suppressory excitation is minimal when the impulse transmitted by it is near its maximum, while its value is rather insensitive to higher-order convergents of the irrational ratio between the involved driving periods. Remarkably, the number and values of the effective initial phase difference between the two excitations are independent of the impulse while they critically depend on each particular convergent in a complex way involving both the approximate frustration of chaos-inducing homoclinic bifurcations and the maximum survival of relevant spatio-temporal symmetries of the dynamical equation.
International Journal of Bifurcation and Chaos, Dec 1, 1996
This paper studies the effect of continuous and discontinuous time dependent forcings onto dynami... more This paper studies the effect of continuous and discontinuous time dependent forcings onto dynamical systems. We compare these different forcings in the context of laminar chaotic mixing. It is shown that the response of a Hamiltonian two-dimensional system to a time periodic sinusoidal forcing differs qualitatively and quantitatively from the response to a square wave function of the same frequency. Consequently, the mixing efficiency of both types of forcings are different. Also a periodic function of the same shape as that of the velocity of the unperturbed system is tested as a forcing, its mixing efficiency being intermediate.
Music is presented as a setting for teaching nonlinear dynamics, showing how different sequences ... more Music is presented as a setting for teaching nonlinear dynamics, showing how different sequences of notes may illustrate ideas such as the sensitivity to initial conditions, and the dynamics and chaotic behaviour connected with fixed-point and limit-cycle attractors. The aim is not music composition, but a first approach to an interdisciplinary tool suitable for a single session class at preuniversity
We study a parametrically damped two-well Duffing oscillator, subjected to a periodic string of s... more We study a parametrically damped two-well Duffing oscillator, subjected to a periodic string of symmetric pulses. The order-chaos threshold when altering solely the width of the pulses is investigated theoretically through Melnikov analysis. We show analytically and numerically that most of the results appear independent of the particular wave form of the pulses provided that the transmitted impulse is the same. By using this property, the stability boundaries of the stationary solutions are determined to first approximation by means of an elliptic harmonic balance method. Finally, the bifurcation behavior at the stability boundaries is determined numerically.
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Papers by Ricardo Chacón