<i>Nonlinear Differential Equations</i>

Struble, Raimond A.; Balise, Peter L.

doi:10.1063/1.3058323

Cited by 83 publications

(12 citation statements)

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“…The fixed points of this dynamical system are determined by the value of the scaled detuning ∆ [20,23]: the exact linear resonance, like in many other instances of nonlinear resonance.…”

Section: A Inviscid Fluidmentioning

confidence: 99%

Parametric autoresonance in Faraday waves

Assaf

Meerson

2005

Phys. Rev. E

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We develop a theory of parametric excitation of weakly nonlinear standing gravity waves in a tank, which is under vertical vibrations with a slowly time-dependent ("chirped") vibration frequency. We show that, by using a negative chirp, one can excite a steadily growing wave via parametric autoresonance. The method of averaging is employed to derive the governing equations for the primary mode. These equations are solved analytically and numerically, for typical initial conditions, for both inviscid and weakly viscous fluids. It is shown that, when passing through resonance, capture into resonance always occurs when the chirp rate is sufficiently small. The critical chirp rate, above which breakdown of autoresonance occurs, is found for different initial conditions. The autoresonance excitation is expected to terminate at large amplitudes, when the underlying constantfrequency system ceases to exhibit a non-trivial stable fixed point.

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“…The fixed points of this dynamical system are determined by the value of the scaled detuning ∆ [20,23]: the exact linear resonance, like in many other instances of nonlinear resonance.…”

Section: A Inviscid Fluidmentioning

confidence: 99%

Parametric autoresonance in Faraday waves

Assaf

Meerson

2005

Phys. Rev. E

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“…Two definitions seem particularly useful in the examination of the stability of vibrations-in-unison: If stability in the sense of Liapunov," or L-stability and "stability in the sense of Poincare" or P-stability [39]. To define the former, let…”

Section: Definitionsmentioning

confidence: 99%

“…In these cases, Hsu has developed criteria for determining the stability [43,44]; his results are based on the application of the method of Strubel [39] in a slightly modified form. A comprehensible summary of Hsu's methods and results is too lengthy to be reproduced here.…”

Section: Forced Vibrationsmentioning

confidence: 99%

On Nonlinear Vibrations of Systems with Many Degrees of Freedom

Rosenberg

1966

Advances in Applied Mechanics

381

279

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“…The fundamental theorem of stability (2, 9,10) states that if the linear approximation ( 2 ) is asymptotically stable at x,, then the nonlinear system (1) must be asymptotically stable in some region about x.. Indeed, for every t > 0, there exists a 8 > 0, such that I l y ( t ) 11 4 8 implies…”

Section: The F U N D a M E N T A L T H E O R E Mmentioning

confidence: 96%

The stability of nonlinear systems in the region of linear dominance

Gura

Perlmutter

1965

AIChE Journal

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A technique has been developed for finding quantitative regions of asymptotic stability for nonlinear systems by using a known proof of the theorem which substantiates the linearization.In a sense, the method defines a region in which the first-order approximation is dominant from the point of view of stability. While the computations necessary can be somewhat involved, the procedure is essentially identical in all situations, and once established, can easily be used for a vast class of autonomous systems. The practical problems of both reversible and irreversible chemical reactions occurring in a continuous flow stirred vessel have been analyzed. Stability regions were obtained which are sufficiently extensive to be practical from the process viewpoint. P A R T IThe analysis and design of control systems have given practical engineering importance to the mathematical theorems having to do with the stability of ordinary differential equations. For linear systems a variety of techniques are available for graphical or analytic solutions. Developments for nonlinear systems in particular have followed either of two paths. By the direct method, a Liapunov function is sought which will be sufficient to establish a region of asymptotic stability in phase space. This approach utilizes the precise nature of the nonlinearity and may yield global results. However, an important drawback is the lack of a systematic, widely applicable procedure; usable engineering results for each new problem still depend on the ingenuity of the investigator, although a number of generalizations have been made (6, Alternatively, the stability of a nonlinear system may be related to the stability of a more tractable linear system. This linearization, sometimes called Liapunov's first method, is based on the expectation that first-order approximations can adequately describe the behavior of the nonlinear functions. Strictly speaking, this is valid only for small deviations from the point of linearization; as commonly formulated ( 2 , 9, l o ) , the stability established in this way is local stability in some neighborhood of unknown size. Results based on such considerations are of limited use because of their qualitative nature.In this paper it will be shown that quantitative results of practical significance can be obtained from these ideas. In effect, a region will be determined so that for all initial conditions within its boundary, the linearized version of a nonlinear differential equation will be dominant from the point of view of stability. Since the necessary and sufficient conditions for the stability of linear systems are well known, the method is systematic and applicable to a very wide class of nonlinear problems. ) . THE F U N D A M E N T A L T H E O R E MConsider a system that can be written as the vector equation order partial derivatives in some neighborhood of x,, the steady state of interest, and J 8 is the Jacobian matrix [ a F , / a x , ] evaluated at the steady state. By defining y = x -x,, the linearized version of (1) is E...

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Nonlinear Differential Equations

Cited by 83 publications

References 0 publications

Parametric autoresonance in Faraday waves

Parametric autoresonance in Faraday waves

On Nonlinear Vibrations of Systems with Many Degrees of Freedom

The stability of nonlinear systems in the region of linear dominance

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