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Related Concept Videos

Root Loci for Positive-Feedback Systems01:23

Root Loci for Positive-Feedback Systems

The Hartley oscillator is a positive feedback system that sustains oscillations by feeding the output back to the input in phase, thereby reinforcing the signal. Positive feedback systems can be viewed as negative feedback systems with inverted feedback signals. In these systems, the root locus encompasses all points on the s-plane where the angle of the system transfer function equals 360 degrees.
The construction rules for the root locus in positive feedback systems are similar to those in...
Classification of Systems-II01:31

Classification of Systems-II

Continuous-time systems have continuous input and output signals, with time measured continuously. These systems are generally defined by differential or algebraic equations. For instance, in an RC circuit, the relationship between input and output voltage is expressed through a differential equation derived from Ohm's law and the capacitor relation,
Second-Order Circuits01:17

Second-Order Circuits

Integrating two fundamental energy storage elements in electrical circuits results in second-order circuits, encompassing RLC circuits and circuits with dual capacitors or inductors (RC and RL circuits). Second-order circuits are identified by second-order differential equations that link input and output signals.
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Pole and System Stability01:24

Pole and System Stability

The transfer function is a fundamental concept representing the ratio of two polynomials. The numerator and denominator encapsulate the system's dynamics. The zeros and poles of this transfer function are critical in determining the system's behavior and stability.
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Second Order systems II01:18

Second Order systems II

In an underdamped second-order system, where the damping ratio ζ is between 0 and 1, a unit-step input results in a transfer function that, when transformed using the inverse Laplace method, reveals the output response. The output exhibits a damped sinusoidal oscillation, and the difference between the input and output is termed the error signal. This error signal also demonstrates damped oscillatory behavior. Eventually, as the system reaches a steady state, the error diminishes to zero.
If  ζ...
Second-order Op Amp Circuits01:19

Second-order Op Amp Circuits

Implementing second-order low-pass filters in audio systems is crucial in refining audio signals by eliminating undesirable high-frequency noise. These filters typically involve second-order op-amp circuits configured as voltage followers, encompassing two nodes with distinct storage elements.
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Stable periodicity and negative circuits in differential systems.

Adrien Richard1, Jean-Paul Comet

  • 1I3S, UMR 6070 CNRS, Université de Nice-Sophia Antipolis, 2000 route des Lucioles, 06903 Sophia Antipolis, France. richard@unice.fr

Journal of Mathematical Biology
|December 2, 2010
PubMed
Summary

We disprove René Thomas's conjecture linking negative feedback circuits to stable periodicity in ordinary differential equations. A weaker version of the conjecture is proven using Snoussi's theorem.

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Area of Science:

  • Systems biology
  • Mathematical biology
  • Theoretical biology

Background:

  • René Thomas's conjecture proposed a link between negative feedback circuits and stable periodicity in ordinary differential equation (ODE) systems.
  • Understanding this relationship is crucial for modeling biological oscillations and dynamics.
  • Previous studies have explored the conditions under which ODE systems exhibit stable periodic behavior.

Purpose of the Study:

  • To provide a counter-example to René Thomas's conjecture.
  • To investigate the relationship between negative feedback and stable periodicity in ODE systems.
  • To prove a weakened version of the conjecture.

Main Methods:

  • We constructed a specific counter-example to challenge the conjecture.
  • We employed a theorem by Snoussi to establish a weaker result.
  • Analysis involved the qualitative theory of ordinary differential equations.

Main Results:

  • A concrete counter-example demonstrating the conjecture's invalidity was presented.
  • A modified or weaker version of the conjecture was proven to hold under certain conditions.
  • The findings refine our understanding of feedback mechanisms in biological ODE models.

Conclusions:

  • René Thomas's conjecture, in its original form, is not universally true for ODE systems.
  • The study highlights the complexity of predicting stable periodicity solely based on negative feedback.
  • The proven weak version offers valuable insights into the behavior of biological regulatory networks.