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ΠΠ°ΡΠ΅ΠΌΠ°ΡΠΈΡΠ΅ΡΠΊΠ°Ρ ΠΌΠΎΠ΄Π΅Π»Ρ ΡΠΊΡΠΏΠ΅ΡΠΈΠΌΠ΅Π½ΡΠ° ΠΠΈΠΊΠΎΠ»ΡΠΎΠ½Π°
ConsideredΒ is a mathematical model of insectsΒ population dynamics,Β andΒ an attempt is madeΒ to explainΒ classical experimental resultsΒ of Nicholson withΒ its help.Β In theΒ first sectionΒ of the paperΒ Nicholsonβs experiment is describedΒ and dynamicΒ equationsΒ for its modeling are chosen.Β A priori estimatesΒ for model parameters can be made more precise by means of local analysisΒ of theΒ dynamical system,Β that is carriedΒ out in the second section.Β For parameter values found thereΒ the stability loss of theΒ problemΒ equilibriumΒ of theΒ leads to theΒ bifurcation of a stableΒ two-dimensional torus.Β Β Numerical simulationsΒ basedΒ on theΒ estimatesΒ from theΒ second sectionΒ allows to explainΒ theΒ classical Nicholsonβs experiment, whose detailedΒ theoretical substantiation is given in the last section.Β There for an atrractor of theΒ systemΒ theΒ largestΒ LyapunovΒ exponent is computed. TheΒ nature of thisΒ exponent change allows to additionally narrowΒ the area of model parameters search.Β Justification of this experiment was made possible Β onlyΒ dueΒ toΒ theΒ combination of analytical andΒ numericalΒ methodsΒ in studyingΒ equationsΒ of insectsΒ population dynamics.Β Β At theΒ same time,Β theΒ analytical approach madeΒ it possible to perform numericalΒ analysisΒ in a rather narrowΒ region of theΒ parameter space.Β It is notΒ possible to get into this area,Β based only on general considerations.Π Π°ΡΡΠΌΠ°ΡΡΠΈΠ²Π°Π΅ΡΡΡ ΠΌΠ°ΡΠ΅ΠΌΠ°ΡΠΈΡΠ΅ΡΠΊΠ°Ρ ΠΌΠΎΠ΄Π΅Π»ΡΒ Π΄ΠΈΠ½Π°ΠΌΠΈΠΊΠΈΒ ΡΠΈΡΠ»Π΅Π½Π½ΠΎΡΡΠΈΒ Π½Π°ΡΠ΅ΠΊΠΎΠΌΡΡ
Β ΠΈ ΠΏΡΠ΅Π΄ΠΏΡΠΈΠ½ΠΈΠΌΠ°Π΅ΡΡΡ ΠΏΠΎΠΏΡΡΠΊΠ°Β ΠΎΠ±ΡΡΡΠ½Π΅Π½ΠΈΡΒ Ρ Π΅Π΅ ΠΏΠΎΠΌΠΎΡΡΡ ΠΊΠ»Π°ΡΡΠΈΡΠ΅ΡΠΊΠΈΡ
ΡΠΊΡΠΏΠ΅ΡΠΈΠΌΠ΅Π½ΡΠ°Π»ΡΠ½ΡΡ
ΡΠ΅Π·ΡΠ»ΡΡΠ°ΡΠΎΠ² ΠΠΈΠΊΠΎΠ»ΡΠΎΠ½Π°. Π ΠΏΠ΅ΡΠ²ΠΎΠΉ ΡΠ°ΡΡΠΈ ΡΠ°Π±ΠΎΡΡ ΠΎΠΏΠΈΡΡΠ²Π°Π΅ΡΡΡ ΡΠΊΡΠΏΠ΅ΡΠΈΠΌΠ΅Π½Ρ ΠΠΈΠΊΠΎΠ»ΡΠΎΠ½Π° ΠΈ Π²ΡΠ±ΠΈΡΠ°ΡΡΡΡ Π΄ΠΈΠ½Π°ΠΌΠΈΡΠ΅ΡΠΊΠΈΠ΅ ΡΡΠ°Π²Π½Π΅Π½ΠΈΡΒ Π΄Π»Ρ Π΅Π³ΠΎ ΠΌΠΎΠ΄Π΅Π»ΠΈΡΠΎΠ²Π°Π½ΠΈΡ. ΠΠΏΡΠΈΠΎΡΠ½ΡΠ΅ ΠΎΡΠ΅Π½ΠΊΠΈ ΠΏΠ°ΡΠ°ΠΌΠ΅ΡΡΠΎΠ² ΠΌΠΎΠ΄Π΅Π»ΠΈ ΡΠ΄Π°Π΅ΡΡΡΒ ΡΡΠΎΡΠ½ΠΈΡΡ Ρ ΠΏΠΎΠΌΠΎΡΡΡ Π»ΠΎΠΊΠ°Π»ΡΠ½ΠΎΠ³ΠΎΒ Π°Π½Π°Π»ΠΈΠ·Π°Β Π΄ΠΈΠ½Π°ΠΌΠΈΡΠ΅ΡΠΊΠΎΠΉΒ ΡΠΈΡΡΠ΅ΠΌΡ,Β ΠΊΠΎΡΠΎΡΡΠΉΒ Π²ΡΠΏΠΎΠ»Π½Π΅Π½Β Π²ΠΎ Π²ΡΠΎΡΠΎΠΌΒ ΡΠ°Π·Π΄Π΅Π»Π΅.Β Π Π½Π΅ΠΌ Π½Π°ΠΉΠ΄Π΅Π½Ρ Π·Π½Π°ΡΠ΅Π½ΠΈΡΒ ΠΏΠ°ΡΠ°ΠΌΠ΅ΡΡΠΎΠ², ΠΏΡΠΈ ΠΊΠΎΡΠΎΡΡΡ
ΠΏΠΎΡΠ΅ΡΡ ΡΡΡΠΎΠΉΡΠΈΠ²ΠΎΡΡΠΈ ΡΠΎΡΡΠΎΡΠ½ΠΈΠ΅ΠΌ ΡΠ°Π²Π½ΠΎΠ²Π΅ΡΠΈΡ Π·Π°Π΄Π°ΡΠΈ ΠΏΡΠΈΠ²ΠΎΠ΄ΠΈΡ ΠΊ Π±ΠΈΡΡΡΠΊΠ°ΡΠΈΠΈ ΡΡΡΠΎΠΉΡΠΈΠ²ΠΎΠ³ΠΎ Π΄Π²ΡΠΌΠ΅ΡΠ½ΠΎΠ³ΠΎ ΡΠΎΡΠ°. Π§ΠΈΡΠ»Π΅Π½Π½ΡΠΉΒ ΡΡΠ΅Ρ, Π²ΡΠΏΠΎΠ»Π½Π΅Π½Π½ΡΠΉ Π½Π° ΠΎΡΠ½ΠΎΠ²Π΅ ΠΎΡΠ΅Π½ΠΎΠΊ ΠΈΠ· Π²ΡΠΎΡΠΎΠ³ΠΎ ΡΠ°Π·Π΄Π΅Π»Π°, ΠΏΠΎΠ·Π²ΠΎΠ»ΡΠ΅ΡΒ ΠΎΠ±ΡΡΡΠ½ΠΈΡΡΒ ΠΊΠ»Π°ΡΡΠΈΡΠ΅ΡΠΊΠΈΠΉ ΡΠΊΡΠΏΠ΅ΡΠΈΠΌΠ΅Π½ΡΒ ΠΠΈΠΊΠΎΠ»ΡΠΎΠ½Π°,Β ΡΠ°Π·Π²Π΅ΡΠ½ΡΡΠΎΠ΅ ΡΠ΅ΠΎΡΠ΅ΡΠΈΡΠ΅ΡΠΊΠΎΠ΅Β ΠΎΠ±ΠΎΡΠ½ΠΎΠ²Π°Π½ΠΈΠ΅ ΠΊΠΎΡΠΎΡΠΎΠ³ΠΎ Π΄Π°Π½ΠΎ Π² ΠΏΠΎΡΠ»Π΅Π΄Π½Π΅ΠΌ ΡΠ°Π·Π΄Π΅Π»Π΅.Β Π Π½Π΅ΠΌ Π΄Π»Ρ Π°ΡΡΡΠ°ΠΊΡΠΎΡΠ° ΡΠΈΡΡΠ΅ΠΌΡ Π²ΡΡΠΈΡΠ»Π΅Π½ ΡΡΠ°ΡΡΠΈΠΉΒ Π»ΡΠΏΡΠ½ΠΎΠ²ΡΠΊΠΈΠΉΒ ΠΏΠΎΠΊΠ°Π·Π°ΡΠ΅Π»Ρ. Π₯Π°ΡΠ°ΠΊΡΠ΅Ρ ΠΈΠ·ΠΌΠ΅Π½Π΅Π½ΠΈΡΒ ΡΡΠΎΠ³ΠΎ ΠΏΠΎΠΊΠ°Π·Π°ΡΠ΅Π»Ρ ΠΏΡΠΈ ΠΈΠ·ΠΌΠ΅Π½Π΅Π½ΠΈΠΈ ΠΊΠΎΡΡΡΠΈΡΠΈΠ΅Π½ΡΠ° Π»ΠΈΠ½Π΅ΠΉΠ½ΠΎΠ³ΠΎ ΡΠΎΡΡΠ° Π·Π°Π΄Π°ΡΠΈΒ ΠΏΠΎΠ·Π²ΠΎΠ»ΡΠ΅ΡΒ Π΄ΠΎΠΏΠΎΠ»Π½ΠΈΡΠ΅Π»ΡΠ½ΠΎΒ ΡΡΠ·ΠΈΡΡΒ ΠΎΠ±Π»Π°ΡΡΡΒ ΠΏΠΎΠΈΡΠΊΠ° ΠΏΠ°ΡΠ°ΠΌΠ΅ΡΡΠΎΠ² ΠΌΠΎΠ΄Π΅Π»ΠΈ.Β ΠΠ±ΠΎΡΠ½ΠΎΠ²Π°Π½ΠΈΠ΅Β Π΄Π°Π½Π½ΠΎΠ³ΠΎΒ ΡΠΊΡΠΏΠ΅ΡΠΈΠΌΠ΅Π½ΡΠ° ΡΡΠ°Π»ΠΎΒ Π²ΠΎΠ·ΠΌΠΎΠΆΠ½ΡΠΌ Π»ΠΈΡΡΒ Π² ΡΠ΅Π·ΡΠ»ΡΡΠ°ΡΠ΅ ΡΠΎΡΠ΅ΡΠ°Π½ΠΈΡ Π°Π½Π°Π»ΠΈΡΠΈΡΠ΅ΡΠΊΠΈΡ
ΠΈ ΡΠΈΡΠ»Π΅Π½Π½ΡΡ
Β ΠΌΠ΅ΡΠΎΠ΄ΠΎΠ²Β ΠΈΡΡΠ»Π΅Π΄ΠΎΠ²Π°Π½ΠΈΡΒ ΡΡΠ°Π²Π½Π΅Π½ΠΈΠΉΒ Π΄ΠΈΠ½Π°ΠΌΠΈΠΊΠΈΒ ΠΏΠΎΠΏΡΠ»ΡΡΠΈΠΉΒ Π½Π°ΡΠ΅ΠΊΠΎΠΌΡΡ
. ΠΡΠΈ ΡΡΠΎΠΌ Π°Π½Π°Π»ΠΈΡΠΈΡΠ΅ΡΠΊΠΈΠΉ ΠΏΠΎΠ΄Ρ
ΠΎΠ΄ Π΄Π°Π» Π²ΠΎΠ·ΠΌΠΎΠΆΠ½ΠΎΡΡΡ ΠΏΡΠΎΠ²ΠΎΠ΄ΠΈΡΡΒ ΡΠΈΡΠ»Π΅Π½Π½ΡΠΉΒ Π°Π½Π°Π»ΠΈΠ·Β Π² Π΄ΠΎΡΡΠ°ΡΠΎΡΠ½ΠΎΒ ΡΠ·ΠΊΠΎΠΉ ΠΎΠ±Π»Π°ΡΡΠΈ ΠΏΡΠΎΡΡΡΠ°Π½ΡΡΠ²Π° ΠΏΠ°ΡΠ°ΠΌΠ΅ΡΡΠΎΠ². ΠΠΎΠΏΠ°ΡΡΡ Π² ΡΡΡ ΠΎΠ±Π»Π°ΡΡΡ, ΠΈΡΡ
ΠΎΠ΄Ρ Π»ΠΈΡΡ ΠΈΠ· ΠΎΠ±ΡΠΈΡ
ΡΠΎΠΎΠ±ΡΠ°ΠΆΠ΅Π½ΠΈΠΉ, Π½Π΅ ΠΏΡΠ΅Π΄ΡΡΠ°Π²Π»ΡΠ΅ΡΡΡ Π²ΠΎΠ·ΠΌΠΎΠΆΠ½ΡΠΌ
Π Π΅Π»Π°ΠΊΡΠ°ΡΠΈΠΎΠ½Π½ΡΠ΅ ΠΊΠΎΠ»Π΅Π±Π°Π½ΠΈΡ ΡΠ»Π΅ΠΊΡΡΠΈΡΠ΅ΡΠΊΠΈ ΡΠ²ΡΠ·Π°Π½Π½ΡΡ Π½Π΅ΠΉΡΠΎΠΏΠΎΠ΄ΠΎΠ±Π½ΡΡ ΠΎΡΡΠΈΠ»Π»ΡΡΠΎΡΠΎΠ² Ρ Π·Π°ΠΏΠ°Π·Π΄ΡΠ²Π°Π½ΠΈΠ΅ΠΌ
The system of diffusion couplexl nonlinear differential-difference equations with delay modelling the elexctrical interacction of pulse neurons is studiexl. Given the spexxl of electrical processes in the system is high, the limit system, responsible for relaxation ccydes, is construcctexL Along with a synchronous cycle the system permits stable asynchronous ccydes, which asymptotics are presentexLΠ Π°ΡΡΠΌΠ°ΡΡΠΈΠ²Π°Π΅ΡΡΡ ΡΠΈΡΡΠ΅ΠΌΠ° Π΄ΠΈΡΡΡΠ·ΠΈΠΎΠ½Π½ΠΎ ΡΠ²ΡΠ·Π°Π½Π½ΡΡ
Π½Π΅Π»ΠΈΠ½Π΅ΠΉΠ½ΡΡ
Π΄ΠΈΡΡΠ΅ΡΠ΅Π½ΡΠΈΠ°Π»ΡΠ½ΠΎ-ΡΠ°Π·Π½ΠΎΡΡΠ½ΡΡ
ΡΡΠ°Π²Π½Π΅Π½ΠΈΠΉ Ρ Π·Π°ΠΏΠ°Π·Π΄ΡΠ²Π°Π½ΠΈΠ΅ΠΌ, ΠΌΠΎΠ΄Π΅Π»ΠΈΡΡΡΡΠΈΡ
ΡΠ»Π΅ΠΊΡΡΠΈΡΠ΅ΡΠΊΠΎΠ΅ Π²Π·Π°ΠΈΠΌΠΎΠ΄Π΅ΠΉΡΡΠ²ΠΈΠ΅ ΠΈΠΌΠΏΡΠ»ΡΡΠ½ΡΡ
Π½Π΅ΠΉΡΠΎΠ½ΠΎΠ². ΠΡΠΈ ΡΡΠ»ΠΎΠ²ΠΈΠΈ, ΡΡΠΎ ΡΠΊΠΎΡΠΎΡΡΡ ΠΏΡΠΎΡΠ΅Β¬ΠΊΠ°Π½ΠΈΡ ΡΠ»Π΅ΠΊΡΡΠΈΡΠ΅ΡΠΊΠΈΡ
ΠΏΡΠΎΡΠ΅ΡΡΠΎΠ² Π² ΡΠΈΡΡΠ΅ΠΌΠ΅ Π²Π΅Π»ΠΈΠΊΠ°, ΠΏΠΎΡΡΡΠΎΠ΅Π½Π° ΠΏΡΠ΅Π΄Π΅Π»ΡΠ½Π°Ρ ΡΠΈΡΡΠ΅Β¬ΠΌΠ°, ΠΎΡΠ²Π΅ΡΠ°ΡΡΠ°Ρ Π·Π° ΡΠ΅Π»Π°ΠΊΡΠ°ΡΠΈΠΎΠ½Π½ΡΠ΅ ΡΠΈΠΊΠ»Ρ ΡΠΈΡΡΠ΅ΠΌΡ. ΠΠΎΠΊΠ°Π·Π°Π½ΠΎ, ΡΡΠΎ, Π½Π°ΡΡΠ΄Ρ Ρ ΡΠΈΠ½Ρ
ΡΠΎΠ½Π½ΡΠΌ, ΡΠΈΡΡΠ΅ΠΌΠ° ΠΌΠΎΠΆΠ΅Ρ ΠΈΠΌΠ΅ΡΡ ΡΡΡΠΎΠΉΡΠΈΠ²ΡΠ΅ Π½Π΅ΡΠΈΠ½Ρ
ΡΠΎΠ½Π½ΡΠ΅ ΡΠΈΠΊΠ»Ρ, Π½Π°ΠΉΠ΄Π΅Π½Ρ Π°ΡΠΈΠΌΠΏΡΠΎΡΠΈΠΊΠΈ ΡΡΠΈΡ
ΡΠΈΠΊΠ»ΠΎΠ²
Π Π°Π·ΠΌΠ΅ΡΠ½ΠΎΡΡΠ½ΡΠ΅ Ρ Π°ΡΠ°ΠΊΡΠ΅ΡΠΈΡΡΠΈΠΊΠΈ Π΄ΠΈΡΡΡΠ·ΠΈΠΎΠ½Π½ΠΎΠ³ΠΎ Ρ Π°ΠΎΡΠ°
The phenomenon of multimode diffusion chaos is considered. For a number of examples it is shown that the Lyapunov dimension of the attractor of a distributed dynamical system increases as the diffusion coefficient tends to 0.Π Π°ΡΡΠΌΠ°ΡΡΠΈΠ²Π°Π΅ΡΡΡ ΡΠ΅Π½ΠΎΠΌΠ΅Π½ ΠΌΠ½ΠΎΠ³ΠΎΠΌΠΎΠ΄ΠΎΠ²ΠΎΠ³ΠΎ Π΄ΠΈΡΡΡΠ·ΠΈΠΎΠ½Π½ΠΎΠ³ΠΎ Ρ
Π°ΠΎΡΠ°, ΠΎΠ΄Π½ΠΈΠΌ ΠΈΠ· ΠΏΡΠΈΠ·Π½Π°ΠΊΠΎΠ² ΠΊΠΎΡΠΎΡΠΎΠ³ΠΎ ΡΠ²Π»ΡΠ΅ΡΡΡ ΡΠ²Π΅Π»ΠΈΡΠ΅Π½ΠΈΠ΅ Π»ΡΠΏΡΠ½ΠΎΠ²ΡΠΊΠΎΠΉ ΡΠ°Π·ΠΌΠ΅ΡΠ½ΠΎΡΡΠΈ Π°ΡΡΡΠ°ΠΊΡΠΎΡΠ° ΡΠ°ΡΠΏΡΠ΅Π΄Π΅Π»Π΅Π½Π½ΡΡ
ΡΠ²ΠΎΠ»ΡΡΠΈΠΎΠ½Π½ΡΡ
Π΄ΠΈΠ½Π°ΠΌΠΈΡΠ΅ΡΠΊΠΈΡ
ΡΠΈΡΡΠ΅ΠΌ ΠΏΡΠΈ ΡΠΌΠ΅Π½ΡΡΠ΅Π½ΠΈΠΈ ΠΊΠΎΡΡΡΠΈΡΠΈΠ΅Π½ΡΠ° Π΄ΠΈΡΡΡΠ·ΠΈΠΈ. ΠΠ»Ρ ΡΡΠ΄Π° ΠΏΡΠΈΠΌΠ΅ΡΠΎΠ² Π²ΡΠΏΠΎΠ»Π½Π΅Π½ ΠΎΠ±ΡΠΈΡΠ½ΡΠΉ ΡΠΈΡΠ»Π΅Π½Π½ΡΠΉ ΡΠΊΡΠΏΠ΅ΡΠΈΠΌΠ΅Π½Ρ, Π² ΠΊΠΎΡΠΎΡΠΎΠΌ ΠΏΡΠΎΠΈΠ»Π»ΡΡΡΡΠΈΡΠΎΠ²Π°Π½ ΡΡΠΎΡ ΡΡΡΠ΅ΠΊΡ
Π£ΡΠ΅Ρ Π²ΠΎΠ·ΡΠ°ΡΡΠ½ΡΡ Π³ΡΡΠΏΠΏ Π² ΡΡΠ°Π²Π½Π΅Π½ΠΈΠΈ Π₯Π°ΡΡΠΈΠ½ΡΠΎΠ½Π°
The dynamics of a generalized Hutchinson's equation with two delays is investigated. A local analysis of a loss of a stability for the nonzero equilibrium state for the problem has been made. Phase reorganizations have been analyzed with numerical methods with the help of the obtained asymptotic formulas. The bifurcation curves that correspond to the principal bifurcations, which take place in the system, have been built on the parameters' plane.ΠΠ·ΡΡΠ°Π΅ΡΡΡ Π΄ΠΈΠ½Π°ΠΌΠΈΠΊΠ° ΠΎΠ±ΠΎΠ±ΡΠ΅Π½Π½ΠΎΠ³ΠΎ ΡΡΠ°Π²Π½Π΅Π½ΠΈΡ Π₯Π°ΡΡΠΈΠ½ΡΠΎΠ½Π° Ρ Π΄Π²ΡΠΌΡ Π·Π°ΠΏΠ°Π·Π΄ΡΠ²Π°Π½ΠΈΡΠΌΠΈ. ΠΡΠΎΠ²Π΅Π΄Π΅Π½ Π»ΠΎΠΊΠ°Π»ΡΠ½ΡΠΉ Π°Π½Π°Π»ΠΈΠ· ΠΏΠΎΡΠ΅ΡΠΈ ΡΡΡΠΎΠΉΡΠΈΠ²ΠΎΡΡΠΈ Π½Π΅Π½ΡΠ»Π΅Π²ΠΎΠ³ΠΎ ΡΠΎΡΡΠΎΡΠ½ΠΈΡ ΡΠ°Π²Π½ΠΎΠ²Π΅ΡΠΈΡ Π·Π°Π΄Π°ΡΠΈ. Π‘ ΡΡΠ΅ΡΠΎΠΌ ΠΏΠΎΠ»ΡΡΠ΅Π½Π½ΡΡ
Π°ΡΠΈΠΌΠΏΡΠΎΡΠΈΡΠ΅ΡΠΊΠΈΡ
ΡΠΎΡΠΌΡΠ» ΡΠΈΡΠ»Π΅Π½Π½ΠΎ ΠΏΡΠΎΠ°Π½Π°Π»ΠΈΠ·ΠΈΡΠΎΠ²Π°Π½Ρ ΡΠ°Π·ΠΎΠ²ΡΠ΅ ΠΏΠ΅ΡΠ΅ΡΡΡΠΎΠΉΠΊΠΈ, ΠΏΡΠΎΠΈΡΡ
ΠΎΠ΄ΡΡΠΈΠ΅ Ρ ΠΈΠ·ΡΡΠ°Π΅ΠΌΡΠΌ ΡΡΠ°Π²Π½Π΅Π½ΠΈΠ΅ΠΌ. ΠΠ° ΠΏΠ»ΠΎΡΠΊΠΎΡΡΠΈ ΠΏΠ°ΡΠ°ΠΌΠ΅ΡΡΠΎΠ² ΠΏΠΎΡΡΡΠΎΠ΅Π½Ρ Π±ΠΈΡΡΡΠΊΠ°ΡΠΈΠΎΠ½Π½ΡΠ΅ ΠΊΡΠΈΠ²ΡΠ΅, ΡΠΎΠΎΡΠ²Π΅ΡΡΡΠ²ΡΡΡΠΈΠ΅ ΠΎΡΠ½ΠΎΠ²Π½ΡΠΌ Π±ΠΈΡΡΡΠΊΠ°ΡΠΈΡΠΌ, ΠΏΡΠΎΠΈΡΡ
ΠΎΠ΄ΡΡΠΈΠΌ Π² ΡΠΈΡΡΠ΅ΠΌΠ΅
Π Π°Π·Π½ΠΎΡΡΠ½ΡΠ΅ Π°ΠΏΠΏΡΠΎΠΊΡΠΈΠΌΠ°ΡΠΈΠΈ ΡΡΠ°Π²Π½Π΅Π½ΠΈΡ Β«ΡΠ΅Π°ΠΊΡΠΈΡ - Π΄ΠΈΡΡΡΠ·ΠΈΡΒ» Π½Π° ΠΎΡΡΠ΅Π·ΠΊΠ΅
The system of phase differences for a chain of diffuse weakly coupled oscillators on a stable integral manifold is constructed and analysed. It is shown by means of numerical methods that as the number of oscillators in the chain increases, the Lyapunov dimention growth is close to linear. The extensive computations performed for difference model of Ginsburg-Landau equation illustrate this result and determine the applicability limits for asymptotic methods.ΠΠ»Ρ ΡΠ΅ΠΏΠΎΡΠΊΠΈ Π΄ΠΈΡΡΡΠ·ΠΈΠΎΠ½Π½ΠΎ ΡΠ»Π°Π±ΠΎ ΡΠ²ΡΠ·Π°Π½Π½ΡΡ
ΠΊΠΎΠ»Π΅Π±Π°ΡΠ΅Π»ΡΠ½ΡΡ
ΡΠΈΡΡΠ΅ΠΌ Π½Π° ΡΡΡΠΎΠΉΡΠΈΠ²ΠΎΠΌ ΠΈΠ½ΡΠ΅Π³ΡΠ°Π»ΡΠ½ΠΎΠΌ ΠΌΠ½ΠΎΠ³ΠΎΠΎΠ±ΡΠ°Π·ΠΈΠΈ ΠΏΠΎΡΡΡΠΎΠ΅Π½Π° ΠΈ ΠΏΡΠΎΠ°Π½Π°Π»ΠΈΠ·ΠΈΡΠΎΠ²Π°Π½Π° ΡΠΈΡΡΠ΅ΠΌΠ° ΡΠ°Π·Π½ΠΎΡΡΠ΅ΠΉ ΡΠ°Π· ΠΎΡΡΠΈΠ»Π»ΡΡΠΎΡΠΎΠ². Π ΡΠ»ΡΡΠ°Π΅, ΠΊΠΎΠ³Π΄Π° ΡΠΈΡΠ»ΠΎ ΠΎΡΡΠΈΠ»Π»ΡΡΠΎΡΠΎΠ² Π² ΡΠ΅ΠΏΠΎΡΠΊΠ΅ ΡΠ°ΡΡΠ΅Ρ, ΡΠΈΡΠ»Π΅Π½Π½ΡΠΌΠΈ ΠΌΠ΅ΡΠΎΠ΄Π°ΠΌΠΈ ΠΏΠΎΠΊΠ°Π·Π°Π½ΠΎ, ΡΡΠΎ Π»ΡΠΏΡΠ½ΠΎΠ²ΡΠΊΠ°Ρ ΡΠ°Π·ΠΌΠ΅ΡΠ½ΠΎΡΡΡ Π°ΡΡΡΠ°ΠΊΡΠΎΡΠ° ΡΠ²Π΅Π»ΠΈΡΠΈΠ²Π°Π΅ΡΡΡ ΠΏΠΎ Π±Π»ΠΈΠ·ΠΊΠΎΠΌΡ ΠΊ Π»ΠΈΠ½Π΅ΠΉΠ½ΠΎΠΌΡ Π·Π°ΠΊΠΎΠ½Ρ. ΠΡΠΎΠΈΠ·Π²Π΅Π΄Π΅Π½ ΠΎΠ±ΡΠΈΡΠ½ΡΠΉ ΡΠΈΡΠ»Π΅Π½Π½ΡΠΉ ΡΠΊΡΠΏΠ΅ΡΠΈΠΌΠ΅Π½Ρ Π΄Π»Ρ ΡΠ°Π·Π½ΠΎΡΡΠ½ΠΎΠΉ ΠΌΠΎΠ΄Π΅Π»ΠΈ ΡΡΠ°Π²Π½Π΅Π½ΠΈΡ ΠΠΈΠ½Π·Π±ΡΡΠ³Π° - ΠΠ°Π½Π΄Π°Ρ, Π² ΠΊΠΎΡΠΎΡΠΎΠΌ ΠΏΡΠΎΠΈΠ»Π»ΡΡΡΡΠΈΡΠΎΠ²Π°Π½ ΡΡΠΎΡ ΡΠ΅Π·ΡΠ»ΡΡΠ°Ρ ΠΈ ΠΎΠΏΡΠ΅Π΄Π΅Π»Π΅Π½Ρ Π³ΡΠ°Π½ΠΈΡΡ ΠΏΡΠΈΠΌΠ΅Π½ΠΈΠΌΠΎΡΡΠΈ Π°ΡΠΈΠΌΠΏΡΠΎΡΠΈΡΠ΅ΡΠΊΠΈΡ
ΠΌΠ΅ΡΠΎΠ΄ΠΎΠ²
Features of the Algorithmic Implementation of Difference Analogues of the Logistic Equation with Delay
The logistic equation with delay or Hutchinsonβs equation is one of the fundamental equations of population dynamics and is widely used in problems of mathematical ecology. We consider a family of mappings built for this equation based on central separated differences. Such difference schemes are usually used in the numerical simulation of this problem. The presented mappings themselves can serve as models of population dynamics; therefore, their study is of considerable interest. We compare the properties of the trajectories of these mappings and the original equation with delay. It is shown that the behavior of the solutions of the mappings constructed on the basis of the central separated differences does not preserve, even with a sufficiently small value of the time step, the basic dynamic properties of the logistic equation with delay. In particular, this map does not have a stable invariant curve bifurcating under the oscillatory loss of stability of a nonzero equilibrium state. This curve corresponds in such mappings to the stable limit cycle of the original continuous equation. Thus, it is shown that such a difference scheme cannot be used for numerical modeling of the logistic equation with delay
Π Π΅Π»Π°ΠΊΡΠ°ΡΠΈΠΎΠ½Π½ΡΠ΅ ΡΠΈΠΊΠ»Ρ Π² ΠΎΠ±ΠΎΠ±ΡΠ΅Π½Π½ΠΎΠΉ Π½Π΅ΠΉΡΠΎΠ½Π½ΠΎΠΉ ΠΌΠΎΠ΄Π΅Π»ΠΈ Ρ Π΄Π²ΡΠΌΡ Π·Π°ΠΏΠ°Π·Π΄ΡΠ²Π°Π½ΠΈΡΠΌΠΈ
A method of modeling the phenomenon of bursting behavior in neural systems based on delay equations is proposed. A singularly perturbed scalar nonlinear differentialdifference equation of Volterra type is a mathematical model of a neuron and a separate pulse containing one function without delay and two functions with different lags. It is established that this equation, for a suitable choice of parameters, has a stable periodic motion with any preassigned number of bursts in the time interval of the period length. To prove this assertion we first go to a relay-type equation and then determine the asymptotic solutions of a singularly perturbed equation. On the basis of this asymptotics the Poincare operator is constructed. The resulting operator carries a closed bounded convex set of initial conditions into itself, which suggests that it has at least one fixed point. The Frechet derivative evaluation of the succession operator, made in the paper, allows us to prove the uniqueness and stability of the resulting relax of the periodic solution.ΠΡΠ΅Π΄Π»ΠΎΠΆΠ΅Π½ ΡΠΏΠΎΡΠΎΠ± ΠΌΠΎΠ΄Π΅Π»ΠΈΡΠΎΠ²Π°Π½ΠΈΡ ΡΠ΅Π½ΠΎΠΌΠ΅Π½Π° βbursting behaviorβ Π² Π½Π΅ΠΉΡΠΎΠ½Π½ΡΡ
ΡΠΈΡΡΠ΅ΠΌΠ°Ρ
, ΠΎΡΠ½ΠΎΠ²Π°Π½Π½ΡΠΉ Π½Π° ΠΈΡΠΏΠΎΠ»ΡΠ·ΠΎΠ²Π°Π½ΠΈΠΈ ΡΡΠ°Π²Π½Π΅Π½ΠΈΠΉ Ρ Π·Π°ΠΏΠ°Π·Π΄ΡΠ²Π°Π½ΠΈΠ΅ΠΌ. Π Π°ΡΡΠΌΠ°ΡΡΠΈΠ²Π°Π΅ΡΡΡ ΡΠΈΠ½Π³ΡΠ»ΡΡΠ½ΠΎ Π²ΠΎΠ·ΠΌΡΡΠ΅Π½Π½ΠΎΠ΅ ΡΠΊΠ°Π»ΡΡΠ½ΠΎΠ΅ Π½Π΅Π»ΠΈΠ½Π΅ΠΉΠ½ΠΎΠ΅ Π΄ΠΈΡΡΠ΅ΡΠ΅Π½ΡΠΈΠ°Π»ΡΠ½ΠΎ-ΡΠ°Π·Π½ΠΎΡΡΠ½ΠΎΠ΅ ΡΡΠ°Π²Π½Π΅Π½ΠΈΠ΅ Π²ΠΎΠ»ΡΡΠ΅ΡΡΠΎΠ²ΡΠΊΠΎΠ³ΠΎ ΡΠΈΠΏΠ°, ΡΠ²Π»ΡΡΡΠ΅Π΅ΡΡ ΠΌΠ°ΡΠ΅ΠΌΠ°ΡΠΈΡΠ΅ΡΠΊΠΎΠΉ ΠΌΠΎΠ΄Π΅Π»ΡΡ ΠΎΡΠ΄Π΅Π»ΡΠ½ΠΎΠ³ΠΎ ΠΈΠΌΠΏΡΠ»ΡΡΠ½ΠΎΠ³ΠΎ Π½Π΅ΠΉΡΠΎΠ½Π° ΠΈ ΡΠΎΠ΄Π΅ΡΠΆΠ°ΡΠ΅Π΅ ΠΎΠ΄Π½Ρ ΡΡΠ½ΠΊΡΠΈΡ Π±Π΅Π· Π·Π°ΠΏΠ°Π·Π΄ΡΠ²Π°Π½ΠΈΡ ΠΈ Π΄Π²Π΅ ΡΡΠ½ΠΊΡΠΈΠΈ Ρ ΡΠ°Π·Π»ΠΈΡΠ½ΡΠΌΠΈ Π·Π°ΠΏΠ°Π·Π΄ΡΠ²Π°Π½ΠΈΡΠΌΠΈ. Π£ΡΡΠ°Π½ΠΎΠ²Π»Π΅Π½ΠΎ, ΡΡΠΎ Ρ ΡΡΠΎΠ³ΠΎ ΡΡΠ°Π²Π½Π΅Π½ΠΈΡ ΠΏΡΠΈ ΠΏΠΎΠ΄Ρ
ΠΎΠ΄ΡΡΠ΅ΠΌ Π²ΡΠ±ΠΎΡΠ΅ ΠΏΠ°ΡΠ°ΠΌΠ΅ΡΡΠΎΠ² ΡΡΡΠ΅ΡΡΠ²ΡΠ΅Ρ ΡΡΡΠΎΠΉΡΠΈΠ²ΠΎΠ΅ ΠΏΠ΅ΡΠΈΠΎΠ΄ΠΈΡΠ΅ΡΠΊΠΎΠ΅ Π΄Π²ΠΈΠΆΠ΅Π½ΠΈΠ΅ Ρ Π»ΡΠ±ΡΠΌ Π½Π°ΠΏΠ΅ΡΠ΅Π΄ Π·Π°Π΄Π°Π½Π½ΡΠΌ ΠΊΠΎΠ»ΠΈΡΠ΅ΡΡΠ²ΠΎΠΌ Π²ΡΠΏΠ»Π΅ΡΠΊΠΎΠ² Π½Π° ΠΎΡΡΠ΅Π·ΠΊΠ΅ Π²ΡΠ΅ΠΌΠ΅Π½ΠΈ Π΄Π»ΠΈΠ½Ρ ΠΏΠ΅ΡΠΈΠΎΠ΄Π°. ΠΠ»Ρ Π΄ΠΎΠΊΠ°Π·Π°ΡΠ΅Π»ΡΡΡΠ²Π° Π΄Π°Π½Π½ΠΎΠ³ΠΎ ΡΡΠ²Π΅ΡΠΆΠ΄Π΅Π½ΠΈΡ ΡΠ½Π°ΡΠ°Π»Π° Π²ΡΠΏΠΎΠ»Π½ΡΠ΅ΡΡΡ ΠΏΠ΅ΡΠ΅Ρ
ΠΎΠ΄ ΠΊ ΡΡΠ°Π²Π½Π΅Π½ΠΈΡ ΡΠ΅Π»Π΅ΠΉΠ½ΠΎΠ³ΠΎ ΡΠΈΠΏΠ°, Π·Π°ΡΠ΅ΠΌ ΠΎΠΏΡΠ΅Π΄Π΅Π»ΡΠ΅ΡΡΡ Π°ΡΠΈΠΌΠΏΡΠΎΡΠΈΠΊΠ° ΡΠ΅ΡΠ΅Π½ΠΈΡ ΡΠΈΠ½Π³ΡΠ»ΡΡΠ½ΠΎ Π²ΠΎΠ·ΠΌΡΡΠ΅Π½Π½ΠΎΠ³ΠΎ ΡΡΠ°Π²Π½Π΅Π½ΠΈΡ ΠΈ Π½Π° ΠΎΡΠ½ΠΎΠ²Π΅ ΡΡΠΎΠΉ Π°ΡΠΈΠΌΠΏΡΠΎΡΠΈΠΊΠΈ ΡΡΡΠΎΠΈΡΡΡ ΠΎΠΏΠ΅ΡΠ°ΡΠΎΡ ΠΏΠΎΡΠ»Π΅Π΄ΠΎΠ²Π°Π½ΠΈΡ ΠΡΠ°Π½ΠΊΠ°ΡΠ΅. ΠΠΎΠ»ΡΡΠ΅Π½Π½ΡΠΉ ΠΎΠΏΠ΅ΡΠ°ΡΠΎΡ ΠΏΠ΅ΡΠ΅Π²ΠΎΠ΄ΠΈΡ Π·Π°ΠΌΠΊΠ½ΡΡΠΎΠ΅, ΠΎΠ³ΡΠ°Π½ΠΈΡΠ΅Π½Π½ΠΎΠ΅ Π²ΡΠΏΡΠΊΠ»ΠΎΠ΅ ΠΌΠ½ΠΎΠΆΠ΅ΡΡΠ²ΠΎ Π½Π°ΡΠ°Π»ΡΠ½ΡΡ
ΡΡΠ»ΠΎΠ²ΠΈΠΉ Π² ΡΠ΅Π±Ρ, ΡΡΠΎ ΠΏΠΎΠ·Π²ΠΎΠ»ΡΠ΅Ρ ΡΡΠ²Π΅ΡΠΆΠ΄Π°ΡΡ, ΡΡΠΎ ΠΎΠ½ ΠΈΠΌΠ΅Π΅Ρ Ρ
ΠΎΡΡ Π±Ρ ΠΎΠ΄Π½Ρ Π½Π΅ΠΏΠΎΠ΄Π²ΠΈΠΆΠ½ΡΡ ΡΠΎΡΠΊΡ. ΠΡΠΏΠΎΠ»Π½Π΅Π½Π½Π°Ρ Π² ΡΠ°Π±ΠΎΡΠ΅ ΠΎΡΠ΅Π½ΠΊΠ° ΠΏΡΠΎΠΈΠ·Π²ΠΎΠ΄Π½ΠΎΠΉ Π€ΡΠ΅ΡΠ΅ ΠΎΠΏΠ΅ΡΠ°ΡΠΎΡΠ° ΠΏΠΎΡΠ»Π΅Π΄ΠΎΠ²Π°Π½ΠΈΡ ΠΏΠΎΠ·Π²ΠΎΠ»ΡΠ΅Ρ Π΄ΠΎΠΊΠ°Π·Π°ΡΡ Π΅Π΄ΠΈΠ½ΡΡΠ²Π΅Π½Π½ΠΎΡΡΡ ΠΈ ΡΡΡΠΎΠΉΡΠΈΠ²ΠΎΡΡΡ ΠΏΠΎΠ»ΡΡΠ΅Π½Π½ΠΎΠ³ΠΎ ΡΠ΅Π»Π°ΠΊΡΠ°ΡΠΈΠΎΠ½Π½ΠΎΠ³ΠΎ ΠΏΠ΅ΡΠΈΠΎΠ΄ΠΈΡΠ΅ΡΠΊΠΎΠ³ΠΎ ΡΠ΅ΡΠ΅Π½ΠΈΡ
ΠΠ΅ΡΠΏΠΎΡΡΠ΄ΠΎΡΠ΅Π½Π½ΡΠ΅ ΠΊΠΎΠ»Π΅Π±Π°Π½ΠΈΡ Π² Π½Π΅ΠΉΡΠΎΡΠ΅ΡΠΈ ΠΈΠ· ΡΡΠ΅Ρ ΠΎΡΡΠΈΠ»Π»ΡΡΠΎΡΠΎΠ² Ρ Π·Π°ΠΏΠ°Π·Π΄ΡΠ²Π°ΡΡΠ΅ΠΉ Π²Π΅ΡΠ°ΡΠ΅Π»ΡΠ½ΠΎΠΉ ΡΠ²ΡΠ·ΡΡ
A model of neural association of three pulsed neurons with a delayed broadcast connection is considered. It is assumed that the parameters of the problem are chosen near the critical point of stability loss by the homogeneous equilibrium state of the system. Because of the broadcast connection the equation corresponding to one of the oscillators can be detached in the system. The two remaining impulse neurons interact with each other and, in addition, there is a periodic external action, determined by the broadcast neuron. Under these conditions, the normal form of this system is constructed for the values of parameters close to the critical ones on a stable invariant integral manifold. This normal form is reduced to a four-dimensional system with two variables responsible for the oscillation amplitudes, and the other two, defined as the difference between the phase variables of these oscillators with the phase variable of the broadcast oscillator. The obtained normal form has an invariant manifold on which the amplitude and phase variables of the oscillators coincide. The dynamics of the problem on this manifold is described. An important result was obtained on the basis of numerical analysis of the normal form. It turned out that periodic and chaotic oscillatory solutions can occur when the coupling between the oscillators is weakened. Moreover, a cascade of bifurcations associated with the same type of phase rearrangements was discovered, where a self-symmetric stable cycle alternately loses symmetry with the appearance of two symmetrical cycles. A cascade of bifurcations of doubling occurs with each of these cycles with the appearance of symmetric chaotic regimes. With further reduction of the coupling parameter, these symmetric chaotic regimes are combined into a self-symmetric one, which is then rebuilt into a self-symmetric cycle of a more complex form compared to the cycle obtained at the previous step. Then the whole process is repeated. Lyapunov exponents were calculated to study chaotic attractors of the system.Π Π°ΡΡΠΌΠ°ΡΡΠΈΠ²Π°Π΅ΡΡΡ ΠΌΠΎΠ΄Π΅Π»Ρ Π½Π΅ΠΉΡΠΎΠ½Π½ΠΎΠΉ Π°ΡΡΠΎΡΠΈΠ°ΡΠΈΠΈ ΠΈΠ· ΡΡΠ΅Ρ
ΠΈΠΌΠΏΡΠ»ΡΡΠ½ΡΡ
Π½Π΅ΠΉΡΠΎΠ½ΠΎΠ² Ρ Π²Π΅ΡΠ°ΡΠ΅Π»ΡΠ½ΠΎΠΉ Π·Π°ΠΏΠ°Π·Π΄ΡΠ²Π°ΡΡΠ΅ΠΉ ΡΠ²ΡΠ·ΡΡ ΠΌΠ΅ΠΆΠ΄Ρ Π½ΠΈΠΌΠΈ. Π£ΡΠΈΡΡΠ²Π°Ρ, ΡΡΠΎ ΡΠ²ΡΠ·Ρ Π²Π΅ΡΠ°ΡΠ΅Π»ΡΠ½Π°Ρ, Π² ΡΠΈΡΡΠ΅ΠΌΠ΅ ΠΎΡΡΠ΅ΠΏΠ»ΡΠ΅ΡΡΡ ΡΡΠ°Π²Π½Π΅Π½ΠΈΠ΅, ΡΠΎΠΎΡΠ²Π΅ΡΡΡΠ²ΡΡΡΠ΅Π΅ ΠΎΠ΄Π½ΠΎΠΌΡ ΠΈΠ· ΠΎΡΡΠΈΠ»Π»ΡΡΠΎΡΠΎΠ². ΠΠ²Π° ΠΎΡΡΠ°Π²ΡΠΈΡ
ΡΡ ΠΈΠΌΠΏΡΠ»ΡΡΠ½ΡΡ
Π½Π΅ΠΉΡΠΎΠ½Π° Π²Π·Π°ΠΈΠΌΠΎΠ΄Π΅ΠΉΡΡΠ²ΡΡΡ Π΄ΡΡΠ³ Ρ Π΄ΡΡΠ³ΠΎΠΌ, ΠΈ, ΠΊΡΠΎΠΌΠ΅ ΡΠΎΠ³ΠΎ, ΠΈΠΌΠ΅Π΅ΡΡΡ ΠΏΠ΅ΡΠΈΠΎΠ΄ΠΈΡΠ΅ΡΠΊΠΎΠ΅ Π²Π½Π΅ΡΠ½Π΅Π΅ Π²ΠΎΠ·Π΄Π΅ΠΉΡΡΠ²ΠΈΠ΅, ΠΎΠΏΡΠ΅Π΄Π΅Π»ΡΠ΅ΠΌΠΎΠ΅ Π²Π΅ΡΠ°ΡΠ΅Π»ΡΠ½ΡΠΌ Π½Π΅ΠΉΡΠΎΠ½ΠΎΠΌ. Π ΡΡΠΈΡ
ΡΡΠ»ΠΎΠ²ΠΈΡΡ
, ΠΏΡΠΈ Π·Π½Π°ΡΠ΅Π½ΠΈΡΡ
ΠΏΠ°ΡΠ°ΠΌΠ΅ΡΡΠΎΠ², Π±Π»ΠΈΠ·ΠΊΠΈΡ
ΠΊ ΠΊΡΠΈΡΠΈΡΠ΅ΡΠΊΠΈΠΌ, Π½Π° ΡΡΡΠΎΠΉΡΠΈΠ²ΠΎΠΌ ΠΈΠ½Π²Π°ΡΠΈΠ°Π½ΡΠ½ΠΎΠΌ ΠΈΠ½ΡΠ΅Π³ΡΠ°Π»ΡΠ½ΠΎΠΌ ΠΌΠ½ΠΎΠ³ΠΎΠΎΠ±ΡΠ°Π·ΠΈΠΈ ΠΏΠΎΡΡΡΠΎΠ΅Π½Π° Π½ΠΎΡΠΌΠ°Π»ΡΠ½Π°Ρ ΡΠΎΡΠΌΠ° Π΄Π°Π½Π½ΠΎΠΉ ΡΠΈΡΡΠ΅ΠΌΡ. ΠΡΠ° Π½ΠΎΡΠΌΠ°Π»ΡΠ½Π°Ρ ΡΠΎΡΠΌΠ° ΡΠ²ΠΎΠ΄ΠΈΡΡΡ ΠΊ ΡΠ΅ΡΡΡΠ΅Ρ
ΠΌΠ΅ΡΠ½ΠΎΠΉ ΡΠΈΡΡΠ΅ΠΌΠ΅, Π΄Π²Π΅ ΠΏΠ΅ΡΠ΅ΠΌΠ΅Π½Π½ΡΡ
ΠΊΠΎΡΠΎΡΠΎΠΉ ΠΎΡΠ²Π΅ΡΠ°ΡΡ Π·Π° Π°ΠΌΠΏΠ»ΠΈΡΡΠ΄Ρ ΠΊΠΎΠ»Π΅Π±Π°Π½ΠΈΠΉ ΠΎΡΡΠΈΠ»Π»ΡΡΠΎΡΠΎΠ², Π° Π΄Π²Π΅ Π΄ΡΡΠ³ΠΈΠ΅ ΠΎΠΏΡΠ΅Π΄Π΅Π»ΡΡΡΡΡ ΡΠ°Π·Π½ΠΎΡΡΡΡ ΡΠ°Π·ΠΎΠ²ΡΡ
ΠΏΠ΅ΡΠ΅ΠΌΠ΅Π½Π½ΡΡ
ΡΡΠΈΡ
ΠΎΡΡΠΈΠ»Π»ΡΡΠΎΡΠΎΠ² Ρ ΡΠ°Π·ΠΎΠ²ΠΎΠΉ ΠΏΠ΅ΡΠ΅ΠΌΠ΅Π½Π½ΠΎΠΉ Π²Π΅ΡΠ°ΡΠ΅Π»ΡΠ½ΠΎΠ³ΠΎ ΠΎΡΡΠΈΠ»Π»ΡΡΠΎΡΠ°. ΠΠΎΠ»ΡΡΠ΅Π½Π½Π°Ρ Π½ΠΎΡΠΌΠ°Π»ΡΠ½Π°Ρ ΡΠΎΡΠΌΠ° ΠΈΠΌΠ΅Π΅Ρ ΠΈΠ½Π²Π°ΡΠΈΠ°Π½ΡΠ½ΠΎΠ΅ ΠΌΠ½ΠΎΠ³ΠΎΠΎΠ±ΡΠ°Π·ΠΈΠ΅, Π½Π° ΠΊΠΎΡΠΎΡΠΎΠΌ Π°ΠΌΠΏΠ»ΠΈΡΡΠ΄Π½ΡΠ΅ ΠΈ ΡΠ°Π·ΠΎΠ²ΡΠ΅ ΠΏΠ΅ΡΠ΅ΠΌΠ΅Π½Π½ΡΠ΅ ΠΎΡΡΠΈΠ»Π»ΡΡΠΎΡΠΎΠ² ΡΠΎΠ²ΠΏΠ°Π΄Π°ΡΡ. ΠΠΏΠΈΡΠ°Π½Π° Π΄ΠΈΠ½Π°ΠΌΠΈΠΊΠ° Π·Π°Π΄Π°ΡΠΈ Π½Π° ΡΡΠΎΠΌ ΠΌΠ½ΠΎΠ³ΠΎΠΎΠ±ΡΠ°Π·ΠΈΠΈ. ΠΠ°ΠΆΠ½ΡΠΉ ΡΠ΅Π·ΡΠ»ΡΡΠ°Ρ ΡΠ΄Π°Π»ΠΎΡΡ ΠΏΠΎΠ»ΡΡΠΈΡΡ Π½Π° ΠΎΡΠ½ΠΎΠ²Π΅ ΡΠΈΡΠ»Π΅Π½Π½ΠΎΠ³ΠΎ Π°Π½Π°Π»ΠΈΠ·Π° Π½ΠΎΡΠΌΠ°Π»ΡΠ½ΠΎΠΉ ΡΠΎΡΠΌΡ. ΠΠΊΠ°Π·Π°Π»ΠΎΡΡ, ΡΡΠΎ ΠΏΡΠΈ ΠΎΡΠ»Π°Π±Π»Π΅Π½ΠΈΠΈ ΡΠ²ΡΠ·ΠΈ ΠΌΠ΅ΠΆΠ΄Ρ ΠΎΡΡΠΈΠ»Π»ΡΡΠΎΡΠ°ΠΌΠΈ ΠΌΠΎΠ³ΡΡ Π²ΠΎΠ·Π½ΠΈΠΊΠ°ΡΡ ΠΏΠ΅ΡΠΈΠΎΠ΄ΠΈΡΠ΅ΡΠΊΠΈΠ΅ ΠΈ Ρ
Π°ΠΎΡΠΈΡΠ΅ΡΠΊΠΈΠ΅ ΠΊΠΎΠ»Π΅Π±Π°ΡΠ΅Π»ΡΠ½ΡΠ΅ ΡΠ΅ΡΠ΅Π½ΠΈΡ. ΠΠΎΠ»Π΅Π΅ ΡΠΎΠ³ΠΎ, Π±ΡΠ» ΠΎΠ±Π½Π°ΡΡΠΆΠ΅Π½ ΠΊΠ°ΡΠΊΠ°Π΄ Π±ΠΈΡΡΡΠΊΠ°ΡΠΈΠΉ, ΡΠ²ΡΠ·Π°Π½Π½ΡΠΉ Ρ ΠΎΠ΄Π½ΠΎΡΠΈΠΏΠ½ΡΠΌΠΈ ΡΠ°Π·ΠΎΠ²ΡΠΌΠΈ ΠΏΠ΅ΡΠ΅ΡΡΡΠΎΠΉΠΊΠ°ΠΌΠΈ, Π² ΠΊΠΎΡΠΎΡΠΎΠΌ ΠΏΠΎΠΎΡΠ΅ΡΠ΅Π΄Π½ΠΎ ΡΠ°ΠΌΠΎΡΠΈΠΌΠΌΠ΅ΡΡΠΈΡΠ½ΡΠΉ ΡΡΡΠΎΠΉΡΠΈΠ²ΡΠΉ ΡΠΈΠΊΠ» ΡΠ΅ΡΡΠ΅Ρ ΡΠΈΠΌΠΌΠ΅ΡΡΠΈΡ Ρ Π²ΠΎΠ·Π½ΠΈΠΊΠ½ΠΎΠ²Π΅Π½ΠΈΠ΅ΠΌ Π΄Π²ΡΡ
ΡΠΈΠΌΠΌΠ΅ΡΡΠΈΡΠ½ΡΡ
Π΄ΡΡΠ³ Π΄ΡΡΠ³Ρ ΡΠΈΠΊΠ»ΠΎΠ²; Ρ ΠΊΠ°ΠΆΠ΄ΡΠΌ ΠΈΠ· ΡΡΠΈΡ
ΡΠΈΠΊΠ»ΠΎΠ² ΠΏΡΠΎΠΈΡΡ
ΠΎΠ΄ΠΈΡ ΠΊΠ°ΡΠΊΠ°Π΄ Π±ΠΈΡΡΡΠΊΠ°ΡΠΈΠΉ ΡΠ΄Π²ΠΎΠ΅Π½ΠΈΡ Ρ ΠΏΠΎΡΠ²Π»Π΅Π½ΠΈΠ΅ΠΌ ΡΠΈΠΌΠΌΠ΅ΡΡΠΈΡΠ½ΡΡ
Ρ
Π°ΠΎΡΠΈΡΠ΅ΡΠΊΠΈΡ
ΡΠ΅ΠΆΠΈΠΌΠΎΠ². ΠΡΠΈ ΡΠΈΠΌΠΌΠ΅ΡΡΠΈΡΠ½ΡΠ΅ Ρ
Π°ΠΎΡΠΈΡΠ΅ΡΠΊΠΈΠ΅ ΡΠ΅ΠΆΠΈΠΌΡ ΠΏΡΠΈ Π΄Π°Π»ΡΠ½Π΅ΠΉΡΠ΅ΠΌ ΡΠΌΠ΅Π½ΡΡΠ΅Π½ΠΈΠΈ ΠΏΠ°ΡΠ°ΠΌΠ΅ΡΡΠ° ΡΠ²ΡΠ·ΠΈ ΠΎΠ±ΡΠ΅Π΄ΠΈΠ½ΡΡΡΡΡ Π² ΡΠ°ΠΌΠΎΡΠΈΠΌΠΌΠ΅ΡΡΠΈΡΠ½ΡΠΉ, ΠΊΠΎΡΠΎΡΡΠΉ Π·Π°ΡΠ΅ΠΌ ΠΏΠ΅ΡΠ΅ΡΡΡΠ°ΠΈΠ²Π°Π΅ΡΡΡ Π² ΡΠ°ΠΌΠΎΡΠΈΠΌΠΌΠ΅ΡΡΠΈΡΠ½ΡΠΉ ΡΠΈΠΊΠ» Π±ΠΎΠ»Π΅Π΅ ΡΠ»ΠΎΠΆΠ½ΠΎΠ³ΠΎ Π²ΠΈΠ΄Π° ΠΏΠΎ ΡΡΠ°Π²Π½Π΅Π½ΠΈΡ Ρ ΠΏΠΎΠ»ΡΡΠ΅Π½Π½ΡΠΌ Π½Π° ΠΏΡΠ΅Π΄ΡΠ΄ΡΡΠ΅ΠΌ ΡΠ°Π³Π΅. ΠΠ°Π»Π΅Π΅ Π²Π΅ΡΡ ΠΏΡΠΎΡΠ΅ΡΡ ΠΏΠΎΠ²ΡΠΎΡΡΠ΅ΡΡΡ. ΠΠ»Ρ ΠΈΠ·ΡΡΠ΅Π½ΠΈΡ Ρ
Π°ΠΎΡΠΈΡΠ΅ΡΠΊΠΈΡ
Π°ΡΡΡΠ°ΠΊΡΠΎΡΠΎΠ² ΡΠΈΡΡΠ΅ΠΌΡ Π²ΡΡΠΈΡΠ»ΡΠ»ΠΈΡΡ Π»ΡΠΏΡΠ½ΠΎΠ²ΡΠΊΠΈΠ΅ ΠΏΠΎΠΊΠ°Π·Π°ΡΠ΅Π»ΠΈ
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