Homoclinic phenomena in opto-thermal bistability with localized absorption

Rosell, Joan; Farjas, Jordi; Herrero, R.; Pi, F.; Orriols, G.

doi:10.1016/0167-2789(95)00077-h

Cited by 21 publications

(29 citation statements)

References 26 publications

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“…13 The linear stability analysis of the stationary solution points out clearly an effective dynamical dimension equal to the number of spacing layers and it has been shown that the partial differential equation may be reduced to the following dimensionless low-order model:…”

Section: Nonlinear Systemmentioning

confidence: 99%

“…A() describes the light interference within the absorbing film and it is a positive-defined almostsinusoidal function depending only on the mirror parameters. 13 It may be written in the following closed form:…”

Section: Nonlinear Systemmentioning

confidence: 99%

“…The nonlinear systems are based on the so-called optothermal bistability with localized absorption ͑BOITAL͒ 11 and they have been described in detail elsewhere from both the experimental 12 and mathematical 13,14 points of view. A BOITAL device consists of a Fabry-Pérot cavity in which the input mirror is partially absorbing and the spacer is a multilayer of transparent materials with alternatively opposite thermo-optic effects.…”

Section: Nonlinear Systemmentioning

confidence: 99%

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N-dimensional dynamical systems exploiting instabilities in full

Rius

Figueras

Herrero

et al. 2000

Chaos: An Interdisciplinary Journal of Nonlinear Science

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We report experimental and numerical results showing how certain N-dimensional dynamical systems are able to exhibit complex time evolutions based on the nonlinear combination of N-1 oscillation modes. The experiments have been done with a family of thermo-optical systems of effective dynamical dimension varying from 1 to 6. The corresponding mathematical model is an N-dimensional vector field based on a scalar-valued nonlinear function of a single variable that is a linear combination of all the dynamic variables. We show how the complex evolutions appear associated with the occurrence of successive Hopf bifurcations in a saddle-node pair of fixed points up to exhaust their instability capabilities in N dimensions. For this reason the observed phenomenon is denoted as the full instability behavior of the dynamical system. The process through which the attractor responsible for the observed time evolution is formed may be rather complex and difficult to characterize. Nevertheless, the well-organized structure of the time signals suggests some generic mechanism of nonlinear mode mixing that we associate with the cluster of invariant sets emerging from the pair of fixed points and with the influence of the neighboring saddle sets on the flow nearby the attractor. The generation of invariant tori is likely during the full instability development and the global process may be considered as a generalized Landau scenario for the emergence of irregular and complex behavior through the nonlinear superposition of oscillatory motions. © 2000 American Institute of Physics. ͓S1054-1500͑00͒01004-1͔Oscillatory phenomena are ubiquitous in natural and social systems and their investigation is currently done within the context of nonlinear dynamics. The oscillations in a given system may appear associated either with intrinsically sustained mechanisms or with externally modulated inputs. Processes with different time scales often coexist and in certain cases the interrelation of oscillations produces complex evolutions in which, however, cyclic repetitions are usually apparent. For instance, in biology, a direct example of this behavior is found in the bursting response of small neural networks in the stomatogastric nervous system of crustacea, 1 while a more involved example could be the wake-sleep cycle of a brain. Leaving apart the case of external modulations, we find it interesting to understand how a nonlinear system can produce different characteristic frequencies and how it can mix the corresponding oscillations to yield complex time evolutions. Although the problem looks basic and simple its answer is pending. The paradigm of chaos is not useful here and among the known mechanisms of nonlinear dynamics only the Landau scenario can be invoked. The present work is a contribution to the enlightenment of this problem. We report experimental and numerical results showing the emergence of complexity through the nonlinear superposition of oscillatory motions in a dynamical system. The phenomenon develops in a generalized Landa...

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Section: Nonlinear Systemmentioning

confidence: 99%

Section: Nonlinear Systemmentioning

confidence: 99%

Section: Nonlinear Systemmentioning

confidence: 99%

See 1 more Smart Citation

N-dimensional dynamical systems exploiting instabilities in full

Rius

Figueras

Herrero

et al. 2000

Chaos: An Interdisciplinary Journal of Nonlinear Science

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“…The BOITAL cavity is well described by the homogeneous heat equation subject to a nonlocal and nonlinear boundary condition [11], and it can be reduced to a dimensionless model of order equal to the number of layers [12]. In this model the variables, c j , are the light phase shifts due to temperature rises in each layer.…”

mentioning

confidence: 99%

“…The nonlinear system is based on the so-called optothermal bistability with localized absorption (BOITAL) and consists of a Fabry-Pérot cavity where the input mirror is a partially absorbing film, the rear mirror is a high reflection dielectric coating, and the spacer is made of two transparent layers with opposite thermo-optic effects [11]. Such a kind of system presents a multiple steady-state solution associated with the periodic interferometric function and can experience a supercritical Hopf bifurcation due to the competition and time delay between the contributions of the two layers to the light phase shift [11]. With this configuration and increasing the input power, the system destabilizes at the first stable branch yielding a limit cycle, which grows by approaching the neighboring saddle point and a homoclinic connection may occur.…”

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confidence: 99%

Experimental Observation of the Amplitude Death Effect in Two Coupled Nonlinear Oscillators

et al. 2000

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The amplitude death phenomenon has been experimentally observed with a pair of thermo-optical oscillators linearly coupled by heat transfer. A parametric analysis has been done and compared with numerical simulations of a time delayed model. The role of the coupling strength is also discussed from experimental and numerical results. [1] are some of the observed behaviors that can be described by phase models considering a weak coupling. On the other hand, effects associated with the amplitude of the different units are expected for strong couplings. One of the most intriguing effects is the so-called amplitude death, dealing with the absence of oscillations for the coupled system while each subsystem oscillates when isolated. Two kinds of amplitude death have been considered. In one of the types the coupling creates a saddle-node pair of fixed points on the limit cycle of the entrained oscillations [5], while the second one is associated with a more common bifurcation-the Hopf bifurcation-in which the oscillation amplitudes are damped out up to reach a steady state [6]. The relevance of the second kind of amplitude death and the important role assumed by the time delay in the coupling of oscillators have been recently noted due to its possible application in physics, chemistry, medicine, and biology [7]. The effect has been predicted to occur for two or more identical or different coupled oscillators [6,[8][9][10], but it has not been experimentally observed until now [7].Here, we present the experimental observation of the amplitude death phenomenon with two optothermal oscillators, located on the same interferometric device and thermally coupled. The relation with the coupling effects on the Hopf bifurcation of the whole system is clearly stated by means of a parametric analysis as a function of the coupling strength and the relative light irradiation of the coupled units. The experimental results are compared with simulations of a time delayed model. The nonlinear system is based on the so-called optothermal bistability with localized absorption (BOITAL) and consists of a Fabry-Pérot cavity where the input mirror is a partially absorbing film, the rear mirror is a high reflection dielectric coating, and the spacer is made of two transparent layers with opposite thermo-optic effects [11]. Such a kind of system presents a multiple steady-state solution associated with the periodic interferometric function and can experience a supercritical Hopf bifurcation due to the competition and time delay between the contributions of the two layers to the light phase shift [11]. With this configuration and increasing the input power, the system destabilizes at the first stable branch yielding a limit cycle, which grows by approaching the neighboring saddle point and a homoclinic connection may occur. After the connection the limit cycle is destroyed and the flow escapes towards an upper node branch where a similar behavior takes place.For a transversally extended optical device, different oscillators separated by a distan...

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Reproducing kernel method to solve parabolic partial differential equations with nonlocal conditions

Allahviranloo

Sahihi

2020

Numerical Methods Partial

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In this study, the parabolic partial differential equations with nonlocal conditions are solved. To this end, we use the reproducing kernel method (RKM) that is obtained from the combining fundamental concepts of the Galerkin method, and the complete system of reproducing kernel Hilbert space that was first introduced by Wang et al. who implemented RKM without Gram-Schmidt orthogonalization process. In this method, first the reproducing kernel spaces and their kernels such that satisfy the nonlocal conditions are constructed, and then the RKM without Gram-Schmidt orthogonalization process on the considered problem is implemented. Moreover, convergence theorem, error analysis theorems, and stability theorem are provided in detail. To show the high accuracy of the present method several numerical examples are solved.

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Homoclinic phenomena in opto-thermal bistability with localized absorption

Cited by 21 publications

References 26 publications

N-dimensional dynamical systems exploiting instabilities in full

N-dimensional dynamical systems exploiting instabilities in full

Experimental Observation of the Amplitude Death Effect in Two Coupled Nonlinear Oscillators

Reproducing kernel method to solve parabolic partial differential equations with nonlocal conditions

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