Narrow Results

The model introduced by Goodwin [1967] in "A Growth Cycle" represents a milestone in the nonlinear modeling of economic dynamics. On the basis of a few simple assumptions, the Goodwin Model (GM) is formulated exactly as the well-known Lotka–Volterra system, in terms of the two variables "wage share" and "employment rate". A number of extensions have been proposed with the aim to make the model more robust, in particular, to obtain structural stability, lacking in GM original formulation. We propose a new extension that: (a) removes the limiting hypothesis of "Harrod-neutral" technical progress: (b) on the line of Lotka–Volterra models with adaptation, introduces the concept of "memory", which plays a relevant role in the dynamics of economic systems. As a consequence, an additional equation appears, the validity of the model is substantially extended and a rich phenomenology is obtained, in particular, transition to chaotic behavior via period-doubling bifurcations.

We discuss an attractor neural network in which only a fraction ρ of nodes is simultaneously updated. In addition, the network has a heterogeneous distribution of connection weights and, depending on the current degree of order, connections are changed at random by a factor Φ on short-time scales. The resulting dynamic attractors may become unstable in a certain range of Φ thus ensuing chaotic itineracy which highly depends on ρ. For intermediate values of ρ, we observe that the number of attractors visited increases with ρ, and that the trajectory may change from regular to chaotic and vice versa as ρ is modified. Statistical analysis of time series shows a power-law spectra under conditions in which the attractors' space is most efficiently explored.

We consider a new mathematical model of a CO₂ laser with a fast saturable absorber. The system exhibits isolas of periodic solutions, along with their bifurcations, as the main parameter, the incoherent pump of the laser, is changed. The characteristic feature of these lasers is their spiking behavior; the spikes, passive Q-switching pulses, correspond to stable periodic solutions on the isolas. We also study the changes in the solution structure as a second parameter varies, namely, the one that is responsible for the extent of nonlinear losses. In particular, we determine what happens to the isolas and to the stability properties of the periodic solutions along them.

In this paper, the feasibility of observing chaotic behavior in the model of a human arm is discussed. Two-Link Arm driven by Six Muscles (TLASM) which is a well-known model of planar human arm reaching movements in the horizontal plane is investigated. Reinforcement learning (RL) that is considered as a model for Dopamine-based learning in the brain is used to control the TLSAM. Finally, the existence of chaos phenomena in the TLASM model controlled with RL is researched using tools like bifurcation maps, Lyapunov exponents, phase-plane trajectories, and spectral analysis using FFT. Results yield that chaos phenomena may occur in the overall system by changing some internal parameters of muscles that have a physiological explanation.

We give a systematic comparison of a molecular model for a CO₂ laser with a fast saturable absorber and a reduced version of this model. Overall, we find that there is good agreement between these models. We use numerical continuation algorithms to analyze the bifurcation structure of the equations, and complement the results by numerical simulations to model possible experiments. Our study predicts the existence of isolas of periodic passive Q-switching self-pulsations and a rich structure of Q-intervals of stability for these periodic orbits, where Q represents the incoherent pump of the laser. These intervals correspond to the observed phenomenon known as period-adding cascades. Computed loci of codimension-1 bifurcations show that a small shift of a secondary parameter in the reduced model with respect to that of the complete model substantially improves their quantitative agreement. This parameter is associated with the action of the Stark effect in the absorber. We also discuss a necessary condition for chaotic windows to arise as Q changes.

We investigate the dynamics and the transitions to spatiotemporal chaos observed in a partial differential equation known as Nikolaevskiy equation in the regime of small domains while applying periodic boundary conditions. In contrast to generic chaotic solutions in large domains called soft-mode turbulence, the Nikolaevskiy model exhibits a rich variety of different chaotic and nonchaotic dynamics if the considered domain size is constrained to only a few characteristic wavelengths. Extending the work by Tanaka, we provide (i) an in-depth numerical analysis including maps of solution types for several parameter subspaces and (ii) results from the numerical continuation of selected types of regular dynamics. Doing this, we detect and classify the highly elaborate scenario of different transitions from regular dynamics to chaos that occur if the system size is varied. Due to the model’s simplicity, we expect those results to be adaptable in similar partial differential equations.

The layer of water at temperatures close to the point of the density maximum is considered. The density–temperature dependence for water can be approximated by quadratic function in the vicinity of the point density maximum. If the point of density maximum is inside the layer then the stable and unstable regions are formed. We studied transition of convective regimes from static to stochastic in two-dimensional horizontal plane layer in case of equal heights of stable and unstable layers in the conductive state. The stability of two-dimensional steady and periodic patterns was investigated to choose a horizontal scale for the study of transitional processes. Regions of hysteresis were obtained. The most realizable scenario of transition to chaos is through bifurcation of the steady solution to the periodic one, then there happens the secondary subcritical Hopf bifurcation to quasiperiodic torus, but initially the motion is doubly periodic. With the increase of supercriticality, intermittency arises with strong turbulent bursts on the background of base quasiperiodic motion. Then, there is a window of quasiperiodic solutions inside intermittent solutions which are different from initial quasiperiodic motion.

Transition from steady state to intermittent chaos in the cubical lid-driven cavity flow is investigated numerically. Fully 3D global stability analysis has revealed that the flow experiences a Hopf bifurcation at Re_c = 1910. As for the 2D-periodic LDC flow, the unstable mode is related to a centrifugal instability of the primary vortex core. Once unstable, DNS shows that the flow is driven toward a limit cycle before eventually exhibiting intermittent chaotic dynamics.