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We consider a Hamiltonian system of coupled Van der Pol-Duffing(VdPD)oscillators with balanced loss and gain. The system is analyzed perturbativelyby using Renormalization Group(RG) techniques as well as Multiple ScaleAnalysis(MSA). Both the methods produce identical results in the leading orderof the perturbation. The RG flow equation is exactly solvable and the slowvariation of amplitudes and phases in time can be computed analytically. Thesystem is analyzed numerically and shown to admit periodic solutions in regionsof parameter-space, confirming the results of the linear stability analysis andperturbation methods. The complex dynamical behavior of the system is studiedin detail by using time-series, Poincar$\acute{e}$-sections, power-spectra,auto-correlation function and bifurcation diagrams. The Lyapunov exponents arecomputed numerically. The numerical analysis reveals chaotic behaviour in thesystem beyond a critical value of the parameter that couples the two VdPDoscillators through linear coupling, thereby providing yet another example ofHamiltonian chaos in a system with balanced loss and gain. Further, we modifythe nonlinear terms of the model to make it a non-Hamiltonian system of coupledVdPD oscillators with balanced loss and gain. The non-Hamiltonian system isanalyzed perturbativly as well as numerically and shown to posses regularperiodic as well as chaotic solutions. It is seen that the${\cal{PT}}$-symmetry is not an essential requirement for the existence ofregular periodic solutions in both the Hamiltonian as well as non-Hamiltoniansystems.

Complex dynamical properties of coupled Van der Pol-Duffing oscillators with balanced loss and gain