Bifurcations and Arnold Tongues of a Multiplier-Accelerator Model

In this paper, we investigate the bifurcations of a multiplier-acceler-ator model with nonlinear investment function in an anti-cyclical fiscal policy rule. Firstly, we give the conditions that the model produces supercritical flip bifurcation and subcritical one respectively. Secondly, we prove that the model undergoes a generalized flip bifurcation and present a parameter region such that the model possesses two 2-periodic orbits. Thirdly, it is proved that the model undergoes supercritical Neimark-Sacker bifurcation and produces an attracting invariant circle surrounding a fixed point. Fourthly, we present the Arnold tongues such that the model has periodic orbits on the invariant circle produced from the Neimark-Sacker bifurcation. Finally, to verify the correctness of our results, we numerically simulate a attracting 2-periodic orbit, an stable invariant circle, an Arnold tongue with rotation number 1/7 and an attracting 7-periodic orbit on the invariant circle.

Characterizing the Dynamics of the Watt Governor System Under Harmonic Perturbation and Gaussian Noise

We investigate the disturbance on the dynamics of a Watt governor system model due to the addition of a harmonic perturbation and a Gaussian noise, by analyzing the numerical results using two distinct methods for the nonlinear dynamics characterization: (i) the well-known Lyapunov spectrum, and (ii) the 0-1 test for chaos. The results clearly show that for tiny harmonic perturbations only the smallest stable periodic structures (SPSs) immersed in chaotic domains are destroyed, whereas for intermediate harmonic perturbation amplitudes there is the emergence of quasiperiodic motion, with the existence of typical Arnold tongues and, the consequent distortion of the SPSs embedded in the chaotic region. For large enough harmonic perturbations, the SPSs immersed in chaotic domains are suppressed and the dynamics becomes essentially chaotic. Regarding the noise perturbations, it is able to suppress periodic motion even if tiny noise intensities are considered, as analyzed by a periodic attractor subject to different noise intensities. The threshold of noise amplitude for chaos generation in periodic structures is reported by both methods. Additionally, we investigate the robustness of the 0-1 test for chaos characterization in both noiseless and noise cases, and for the first time, we compare the Lyapunov exponents and 0-1 test methods in the parameter-planes. Our findings are generic due to their remarkable agreement with results previously reported for dynamical systems in other contexts.

Fold-Pitchfork Bifurcation, Arnold Tongues and Multiple Chaotic Attractors in a Minimal Network of Three Sigmoidal Neurons

Bifurcations and chaos in a network of three identical sigmoidal neurons are examined. The network consists of a two-neuron oscillator of the Wilson–Cowan type and an additional third neuron, which has a simpler structure than chaotic neural networks in the previous studies. A codimension-two fold-pitchfork bifurcation connecting two periodic solutions exists, which is accompanied by the Neimark–Sacker bifurcation. A stable quasiperiodic solution is generated and Arnold’s tongues emanate from the locus of the Neimark–Sacker bifurcation in a two-dimensional parameter space. The merging, splitting and crossing of the Arnold tongues are observed. Further, multiple chaotic attractors are generated through cascades of period-doubling bifurcations of periodic solutions in the Arnold tongues. The chaotic attractors grow and are destroyed through crises. Transient chaos and crisis-induced intermittency due to the crises are also observed. These quasiperiodic solutions and chaotic attractors are robust to small asymmetry in the output function of neurons.