Abstract
Bifurcation theory (center manifold and Ljapunov–Schmidt reduction, normal form theory, universal unfolding, calculation of bifurcation diagrams) has become an important and very useful means in the solution of nonlinear stability problems in many branches of engineering. The present study deals with qualitative behavior of a two-dimensional discrete-time system for interaction between prey and predator. The discrete-time model has more chaotic and rich dynamical behavior as compare to its continuous counterpart. We investigate the qualitative behavior of a discrete-time Lotka-Volterra model with linear functional response for prey. The local asymptotic behavior of equilibria is discussed for discrete-time Lotka-Volterra model. Furthermore, with the help of bifurcation theory and center manifold theorem, explicit parametric conditions for directions and existence of flip and Hopf bifurcations are investigated. Moreover, two chaos control methods, that is, OGY feedback control and hybrid control strategy, are implemented. Numerical simulations are provided to illustrate theoretical discussion and their effectiveness.
Highlights
AND PRELIMINARIESFor the many different deterministic non-linear dynamic systems, the discovery of irregular vibrations in addition to periodic and almost periodic vibrations is one of the most significant achievements of modern science
CONCLUDING REMARKS In this paper, piecewise constant argument is implemented in order to obtain a discrete-time predator-prey model with linear functional response
The proposed model is governed by two-dimensional planar system of difference equations in exponential form, which has more rich dynamics and chaotic behavior as compare to its continuous counterpart
Summary
For the many different deterministic non-linear dynamic systems (physical, mechanical, technical, chemical, ecological, economic, and civil and structural engineering), the discovery of irregular vibrations in addition to periodic and almost periodic vibrations is one of the most significant achievements of modern science. Keeping in view the herd behavior for prey population, Salman et al [17] implemented a functional response of square root type in order to discuss bifurcation and chaos control for a discrete prey-predator model. Crowley–Martin functional response was implemented by Ren et al [19] in order to investigate Hopf bifurcation, flip bifurcation, and chaos for a discrete prey-predator model. The coming investigation reveals that piecewise constant arguments method is more appropriate to discuss rich dynamics, chaotic behavior, bifurcation analysis and chaos control for discrete counterpart of system (2). In order to discuss parametric conditions for existence and direction of flip bifurcation at fixed point P = It follows that characteristic polynomial (5) has distinct real roots, say λ1 and λ2. Ac−b c are stable, and if l2 < 0, these orbits are unstable
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