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Linear systems are characterized by two main properties: superposition and homogeneity. Superposition allows the response to multiple inputs to be the sum of the responses to each individual input. Homogeneity ensures that scaling an input by a scalar results in the response being scaled by the same scalar.
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The coupling interactions of nuclei across four or more bonds are usually weak, with J values less than 1 Hz. While these are usually not observed in spectra, the presence of multiple bonds along the coupling pathway can result in observable long-range coupling.
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In the standard form, the transfer function is shown in constant gain, poles/zeros at origin, simple poles/zeros, and quadratic poles/zeros; each contributing uniquely to the system's overall response. The term represents the magnitude of the simple zero:
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Vicinal or three-bond coupling is commonly observed between protons attached to adjacent carbons. Here, nuclear spin information is primarily transferred via electron spin interactions between adjacent C‑H bond orbitals. This generally favors the antiparallel arrangement of spins, so 3J values are usually positive.
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Inductive circuits present intriguing challenges in electrical engineering, particularly during the transition from the time domain to the frequency domain. This transformation involves converting inductors into impedances and utilizing phasor representation.
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Nonlinear phase coupling functions: a numerical study.

Michael Rosenblum1,2, Arkady Pikovsky1,2

  • 1Institute of Physics and Astronomy, University of Potsdam, Karl-Liebknecht-Strasse 24/25, 14476 Potsdam-Golm, Germany.

Philosophical Transactions. Series A, Mathematical, Physical, and Engineering Sciences
|October 29, 2019
PubMed
Summary
This summary is machine-generated.

Researchers explored advanced phase coupling functions for forced oscillators. These nonlinear functions accurately predict synchronization regions in complex systems, enhancing oscillator analysis.

Keywords:
coupling functionphase approximationphase response curve

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Area of Science:

  • Nonlinear dynamics
  • Complex systems analysis
  • Theoretical physics

Background:

  • Phase reduction is a key method for analyzing coupled oscillators.
  • Existing models often use simplified, first-order approximations for phase coupling.
  • Understanding nonlinear interactions is crucial for accurate system prediction.

Purpose of the Study:

  • To investigate phase coupling functions beyond the first-order approximation.
  • To analyze coupling functions up to the fourth order in force strength.
  • To demonstrate the predictive power of nonlinear coupling functions for synchronization.

Main Methods:

  • Utilized the periodically forced Stuart-Landau oscillator as a model system.
  • Determined phase coupling functions analytically up to fourth order.
  • Performed numerical analysis of the derived coupling functions.

Main Results:

  • Derived novel nonlinear phase coupling functions.
  • Validated the accuracy of these functions in predicting synchronization.
  • Showcased the importance of higher-order terms in coupling analysis.

Conclusions:

  • Higher-order nonlinear phase coupling functions offer improved accuracy.
  • These functions are valuable tools for predicting synchronization in forced oscillators.
  • The findings advance the understanding of dynamical interaction mechanisms.