Chaos in perturbed Lotka-Volterra systems (Q2711364)

Here, the following slowly varying time-periodically perturbed Lotka-Volterra system is considered NEWLINE\[NEWLINE\begin{aligned} {dx\over dt}= & x(c-bx-2dy+az)+ \varepsilon \bigl[\lambda_1 x+(\lambda_2+ \lambda_3x) \sin\omega t\bigr],\\ {dy \over dt}= & y(-c+2bx+dy-az)+\varepsilon \bigl[\lambda_1 y+(\lambda_2+ \lambda_3 y)\sin\omega t\bigr],\\ {dz\over dt}= & \varepsilon z(\lambda_4+ \lambda_5 \sin \omega t-\lambda_6 x+\lambda_7 y-\lambda_8z), \end{aligned}NEWLINE\]NEWLINE where \(0\leq \varepsilon \ll 1\) denotes the perturbation parameter, \(\omega>0\) is the frequency of the perturbation, \(a,b,c,d>0\) and \(\lambda_j\), \(j=1,\dots,8\), are real parameters. Using Melnikov's method, sufficient conditions are obtained for the above system to have a transverse heteroclinic cycle and hence to posses chaotic behavior in the sense of Smale. Then a special case involving a reduction to a two-dimensional Lotka-Volterra system is examined, leading finally to an application with a model for the self-organisation of macromolecules.