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Title of Journal: Nonlinear Dyn

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Abbravation: Nonlinear Dynamics

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Springer Netherlands

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1573-269X

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Stability convergence limit cycles and chaos in

Authors: Sreelakshmi Manjunath Anusha Podapati Gaurav Raina
Publish Date: 2016/11/28
Volume: 87, Issue: 4, Pages: 2577-2595
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Abstract

In this paper we study three models of population dynamics i the classical delay logistic equation ii its variant which incorporates a harvesting rate and iii the Perez–Malta–Coutinho PMC equation For all these models we first conduct a local stability analysis around their respective equilibria In particular we outline the necessary and sufficient condition for local stability In the case of the PMC equation we also outline a sufficient condition for local stability which may help guide design considerations We also characterise the rate of convergence about the locally stable equilibria for all these models We then conduct a detailed local bifurcation analysis We first show by using a suitably motivated bifurcation parameter that the models undergo a Hopf bifurcation when the necessary and sufficient condition gets violated Then we use Poincaré normal forms and centre manifold theory to study the dynamics of the systems just beyond the region of local stability We outline an analytical basis to establish the type of the Hopf and determine the stability of the limit cycles In some cases we are able to derive explicit analytic expressions for the amplitude and period of the bifurcating limit cycles We also highlight that in the PMC model variations in one of the model parameters can readily induce chaotic dynamics Finally we construct an equivalent electronic circuit for the PMC model and demonstrate the existence of multiple bifurcations in a hardwarebased circuit implementationHaving established that the system undergoes a Hopf bifurcation we now present the analysis that enables us to explore the bifurcation properties of the system In particular the following analysis enables us to characterise the type of the Hopf bifurcation and determine the orbital stability of the emergent limit cycles

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