Liénard systems and potential-Hamiltonian decomposition I – methodology

Demongeot, Jacques; Glade, Nicolas; Forest, Loïc

doi:10.1016/j.crma.2006.10.016

Cited by 33 publications

(36 citation statements)

References 18 publications

(26 reference statements)

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“…x(1-x 2 -y 2 ), dy/dt = -x ? y(1-x 2 -y 2 ) and their homogeneous solutions (described in Demongeot et al 2007a) are: xðtÞ ¼ q 0 e t cosðtÀh 0 Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 1þq 2 0 ðe 2t À1Þ p ; yðtÞ ¼ Àq 0 e t sinðtÀh 0 Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 1þq 2 0 ðe 2t À1Þ p with (q 0 , h 0 ) the polar coordinates corresponding to (x(0), y(0)). Let w the limit cycle of the system.…”

Section: Analytical Resolution Of the Isochrons Of The Anharmonic Penmentioning

confidence: 99%

The Isochronal Fibration: Characterization and Implication in Biology

Amor

Glade²,

Lobos³

et al. 2010

Acta Biotheor

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Limit cycles, because they are constituted of a periodic succession of states (discrete or continuous) constitute a good manner to store information. From any points of the state space reached after a perturbation or stimulation of the cognitive system storing this information, one can aim to join through a more or less long return trajectory a precise neighbourhood of the asymptotic trajectory at a specific moment (or a specific place) on the limit cycle, i.e. where the information of interest stands. We propose that the isochronal fibration, initially imagined and described by A. T. Winfree may be an excellent way to connect directly those two locations. Each isochron is indeed the set of points in temporal phase with one single point of the attractor. The characterisation of the isochronal fibration of various dynamical systems is not easy and until now has principally only been done numerically but not analytically. By integrating the homogeneous solutions of the dynamical system we can solve this fibration in the case of the well known anharmonic pendulum. Other isochronal fibration on classical examples such as the van der Pol system and the non-symmetrical PFK limit cycle are obtained numerically and we also provide the first numerical study on 3-dimentional systems like the anharmonic pendulum with a linear relaxation on its third variable and the Lorenz attractor. The empirical approach seems us useful for dealing with the isochronal fibration which could constitute a powerful tool for understanding and controlling the dynamics of biological or biological-inspired systems.

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Section: Analytical Resolution Of the Isochrons Of The Anharmonic Penmentioning

confidence: 99%

The Isochronal Fibration: Characterization and Implication in Biology

Amor

Glade²,

Lobos³

et al. 2010

Acta Biotheor

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“…Liénard systems [9] are 2-dimensional ordinary differential equations (2D ODEs) defined by dx/dt = y, dy/dt = g(x) + yf (x), where g and f are polynomials. Liénard systems have in general as asymptotics a unique limit- cycle [15].…”

Section: Introductionmentioning

confidence: 99%

“…a Liénard system called van der Pol system, if p(x) = μx(1 − x 2 /3), c = −1, a = b = 0. Then to approach algebraically its limit-cycle, we use the PH-decomposition proposed in Theorem 1 of [9], by remarking that:…”

Section: Introductionmentioning

confidence: 99%

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Liénard systems and potential-Hamiltonian decomposition II – algorithm

Demongeot

Glade

Forest

2007

Comptes Rendus Mathematique

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We show here how to approach with an increasing precision the limit-cycles of Liénard systems, bifurcating from a stable stationary state, by contour lines of Hamiltonian systems derived from a potential-Hamiltonian decomposition of the Liénard flow. We evoke the case (non polynomial) of pure potential systems (n-switches) and pure Hamiltonian systems (2D Lotka-Volterra), and we show that, with the proposed approximation, we can deal with the case of mixed systems (van der Pol or FitzHugh-Nagumo) frequently used for modelling oscillatory systems in biology. We suggest finally that the proposed algorithm, generic for PHdecomposition, can be used for estimating the isochronal fibration in some specific cases near the pure potential or Hamiltonian systems. In a following Note, we will give applications in biology of the potential-Hamiltonian decomposition. To cite this article: J. Demongeot et al., C. R. Acad. Sci. Paris, Ser. I 344 (2007). RésuméSystèmes de Liénard et décomposition potentielle-hamiltonienne II -algorithme. Nous montrons ici comment approcher, avec une précision croissante, les cycles limites des systèmes de Liénard par les courbes de niveau du système hamiltonien obtenu à partir d'une décomposition potentielle-hamiltonienne. Nous présentons des cas non polynomiaux de systèmes purement potentiels (type n-switch) et purement hamiltoniens (type Lotka-Volterra 2D), puis nous donnons des exemples de décomposition potentiellehamiltonienne pour des systèmes de type van der Pol (mixtes) qui sont d'usage fréquent en modélisation des systèmes biologiques oscillants. Nous suggérons enfin que l'algorithme proposé, générique pour la décomposition potentielle-hamiltonienne, soit utilisé pour estimer la fibration isochrone, dans le cas de systèmes voisins des systèmes purs (potentiels ou hamiltoniens). Dans une Note future, nous décrirons des applications biologiques précises de la décomposition potentielle-hamiltonienne. Pour citer cet article : J. Demongeot et al., C. R. Acad. Sci. Paris, Ser. I 344 (2007).

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“…The use of Liénard systems is universal in biological modelling, especially for physiological processes (Demongeot et al 2007). …”

Section: (I) the Central Respiratory Pattern Generatormentioning

confidence: 99%

A model of mechanical interactions between heart and lungs

Jallon

Abdulhay

Calabrese

et al. 2009

Phil. Trans. R. Soc. A.

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To study the mechanical interactions between heart, lungs and thorax, we propose a mathematical model combining a ventilatory neuromuscular model and a model of the cardiovascular system, as described by Smith et al. 1022616701863)); using a Liénard oscillator, it allows the activity of the respiratory centres, the respiratory muscles and rib cage internal mechanics to be simulated. The minimal haemodynamic system model of Smith includes the heart, as well as the pulmonary and systemic circulation systems. These two modules interact mechanically by means of the pleural pressure, calculated in the mechanical respiratory system, and the intrathoracic blood volume, calculated in the cardiovascular model. The simulation by the proposed model provides results, first, close to experimental data, second, in agreement with the literature results and, finally, highlighting the presence of mechanical cardiorespiratory interactions.

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Liénard systems and potential-Hamiltonian decomposition I – methodology

Cited by 33 publications

References 18 publications

The Isochronal Fibration: Characterization and Implication in Biology

The Isochronal Fibration: Characterization and Implication in Biology

Liénard systems and potential-Hamiltonian decomposition II – algorithm

A model of mechanical interactions between heart and lungs

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