On the equations of Poizat and Liénard

Freitag, James; Jaoui, Rémi; Marker, David R.; Nagloo, Joel

doi:10.48550/arxiv.2201.03838

Cited by 3 publications

(3 citation statements)

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“…While (6) allows for the exact solution discussed earlier, only a limited number of exact solutions to the general Liénard equation ( 15) are known (see, e.g., [15]- [17]). However, these known solutions are not directly related to circuit engineering problems and, therefore, are not discussed in this work.…”

Section: B the Fast-slow System Representation And The Van Der Pol Eq...mentioning

confidence: 99%

Understanding Relaxation Oscillator Circuits Using Fast-Slow System Representations

Voglhuber-Brunnmaier,

Jakoby

2023

IEEE Access

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We demonstrate how to utilize a fundamental principle of nonlinear dynamics, namely Liénard-type representations of ordinary differential equations, also referred to as fast-slow systems, to describe and understand relaxation oscillations in electronic circuits. Relaxation oscillations are characterized by periods of slow signal changes followed by fast, sudden transitions. They are generated either intentionally by means of usually simple circuits or occur often unintentionally where they would not have been expected, such as in circuits where there is only one dominant energy storage device. The second energy storage required to promote oscillatory solutions of the governing equations can also be provided by spurious elements or mechanisms. Conditions distinguishing harmonic from (anharmonic) relaxation oscillations are discussed by considering the underlying eigenvalues of the system. Subsequently, we shown how to intuitively understand the relaxation oscillations through analyses in the phase diagram based on the fast-slow system representation of the nonlinear differential equation. Practical examples of oscillators, including RC and LR op-amp circuits and the so-called "Joule thief" circuit, are discussed to illustrate the principle. The method's applicability is not limited to electrical circuits but extends to a variety of disciplines, such as chemistry, biology, geology, meteorology, and the social sciences.

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Section: B the Fast-slow System Representation And The Van Der Pol Eq...mentioning

confidence: 99%

Understanding Relaxation Oscillator Circuits Using Fast-Slow System Representations

Voglhuber-Brunnmaier,

Jakoby

2023

IEEE Access

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“…(7) Blázquez-Sanz et al [BCFN20] prove the strong minimality of certain general Schwarzian differential equations. (8) Freitag et al [FJMN22] show that various equations of Liénard-type are strongly minimal, using techniques from valuation theory.…”

Section: Introductionmentioning

confidence: 99%

Generic differential equations are strongly minimal

DeVilbiss¹,

Freitag²

2023

Compositio Math.

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In this paper we develop a new technique for showing that a nonlinear algebraic differential equation is strongly minimal based on the recently developed notion of the degree of non-minimality of Freitag and Moosa. Our techniques are sufficient to show that generic order $h$ differential equations with non-constant coefficients are strongly minimal, answering a question of Poizat (1980).

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“…From there, we use Nishioka's theorem 1.2 with a model theoretic study of fibrations inspired by the work of [16] to prove the strong minimality of Schwarz triangle functions. The motivation for establishing the strong minimality of certain differential equations has been discussed many places and ranges from functional transcendence theorems [3] to diophantine results [9,10,11] to understanding solutions of equations from special classes of functions [5,6].…”

Section: Introductionmentioning

confidence: 99%