Proceedings of the Royal Society of Edinburgh Section A Mathematics · 1999 · 100 citations · 22 references
Natural SciencesMathematicsSystems EngineeringDifferential EquationsDynamical SystemsSystem DynamicsBifurcation TheoryDelay ModelingNonlinear ControlNonlinear DynamicsStability TheorySystems Of Differential EquationsBifurcationDelay-dependent StabilityDelays In SystemsExistence ConditionsHopf BifurcationHopf Bifurcation With DelayNeural Networks With DelayPeriodic Solutions Of Differential Equations
In this paper, we consider a planar system with two delays: ẋ 1 ( t ) = − a 0 x 1 ( t ) + a 1 F 1 ( x 1 ( t − τ 1 ), x 2 ( τ − t 2 )). ẋ 2 ( t ) = − b 0 x 2 ( t ) + b 1 F 2 ( x 1( t − τ 1 ), x 2 ( t − τxs 2 )). Firstly, linearized stability and local Hopf bifurcations are studied. Then, existence conditions for non-constant periodic solutions are derived using degree theory methods. Finally, a simple neural network model with two delays is analysed as an example.
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