Conference publications


XXIII conference

Cascades of bifurcation in the two-parameter ecological system

Gurina T.A.

Мoscow Aviation Institute (National Reseach University), Faculty "Applied Mathematics and Physics", Volokolamskoe shosse, 4, Moscow, 125993, Russia, Phone: (915)-281-23-87, E-mail:

1 pp. (accepted)

The model Hastings - Powell's ecological system of the "prey-predator-tор-predator" describes a system of differential equations with three parameters. As the bifurcation parameters are considered two parameters of the system, other parameters are fixed.

For special points that are in the region of positive values of the variables, a partition of the plane was built on an area of two parameters for the type of rough singular point of the linearized system. When crossing the boundary of a saddle-focus with positive real parts of a pair of complex conjugate roots going Andronov-Hopf bifurcation birthday stable limit cycle, followed by a cascade of period-doubling bifurcations cycle and subharmonic cascade Sharkovskii ending cycle period of the birth of three. A further change of the parameters appear in the system cycles homoclinic bifurcation cascade leading to the formation of a strange attractor.

With the transformation of computing and evidence shows the existence of a homoclinic orbit of a saddle-focus, the destruction of which is the principal homoclinic bifurcation cascade and defined range of parameters in which it exists. Bifurcation diagrams, charts Lyapunov exponents of saddle graphics, fractal dimension of the strange attractor. Work is executed with the use of analytical and numerical calculations “Maple”.


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