On existence conditions for a two-point oscillating periodic solution in an non-autonomous relay system with a Hurwitz matrix

In this paper, a non-smooth population model with impulsive effects is proposed by combining discontinuity and non-smoothness. According to the qualitative theory of differential equations, the global analysis of the model is discussed. Using the theory of impulsive differential equations, the existence conditions of order one periodic solution are obtained. And the impulsive controllers are designed to make the pest populations stay at the refuge level. Some simulations are carried out to prove the results.

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New Approach to Gene Network Modeling

Modeling and Analysis of Information Systems ◽

10.18255/1818-1015-2019-3-365-404 ◽

2019 ◽

Vol 26 (3) ◽

pp. 365-404

Author(s):

Sergey D. Glyzin ◽

Andgey Yu. Kolesov ◽

Nikolay Kh. Rozov

Keyword(s):

Periodic Solution ◽

Phenomenological Model ◽

Gene Network ◽

Genetic Network ◽

Relay System ◽

New Approach ◽

Relaxation Cycle ◽

The Third ◽

Existence And Stability ◽

Rna Concentration

The article is devoted to the mathematical modeling of artificial genetic networks. A phenomenological model of the simplest genetic network called repressilator is considered. This network contains three elements unidirectionally coupled into a ring. More specifically, the first of them inhibits the synthesis of the second, the second inhibits the synthesis of the third, and the third, which closes the cycle, inhibits the synthesis of the first one. The interaction of the protein concentrations and of mRNA (message RNA) concentration is surprisingly similar to the interaction of six ecological populations — three predators and three preys. This allows us to propose a new phenomenological model, which is represented by a system of unidirectionally coupled ordinary differential equations. We study the existence and stability problem of a relaxation periodic solution that is invariant with respect to cyclic permutations of coordinates. To find the asymptotics of this solution, a special relay system is constructed. It is proved in the paper that the periodic solution of the relay system gives the asymptotic approximation of the orbitally asymptotically stable relaxation cycle of the problem under consideration.

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A unified description method of the harmonic oscillator and its periodic solution existence conditions

Physics Essays ◽

10.4006/0836-1398-28.1.73 ◽

2015 ◽

Vol 28 (1) ◽

pp. 73-77

Author(s):

Gangfeng Yan ◽

Bing Guo

Keyword(s):

Periodic Solution ◽

Harmonic Oscillator ◽

Solution Existence ◽

Existence Conditions ◽

Unified Description

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On nearly-commensurable periods in the restricted problem of three bodies, with calculations of the long-period variations in the interior 2:1 case

Symposium - International Astronomical Union ◽

10.1017/s0074180900105480 ◽

1966 ◽

Vol 25 ◽

pp. 197-222 ◽

Cited By ~ 4

Author(s):

P. J. Message

Keyword(s):

Periodic Solution ◽

Periodic Solutions ◽

Large Amplitude ◽

Restricted Problem ◽

Small Amplitude ◽

Minor Planets ◽

Long Period ◽

Numerical Integrations ◽

Period Variations ◽

The Mean

An analytical discussion of that case of motion in the restricted problem, in which the mean motions of the infinitesimal, and smaller-massed, bodies about the larger one are nearly in the ratio of two small integers displays the existence of a series of periodic solutions which, for commensurabilities of the typep+ 1:p, includes solutions of Poincaré'sdeuxième sortewhen the commensurability is very close, and of thepremière sortewhen it is less close. A linear treatment of the long-period variations of the elements, valid for motions in which the elements remain close to a particular periodic solution of this type, shows the continuity of near-commensurable motion with other motion, and some of the properties of long-period librations of small amplitude.To extend the investigation to other types of motion near commensurability, numerical integrations of the equations for the long-period variations of the elements were carried out for the 2:1 interior case (of which the planet 108 “Hecuba” is an example) to survey those motions in which the eccentricity takes values less than 0·1. An investigation of the effect of the large amplitude perturbations near commensurability on a distribution of minor planets, which is originally uniform over mean motion, shows a “draining off” effect from the vicinity of exact commensurability of a magnitude large enough to account for the observed gap in the distribution at the 2:1 commensurability.

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