Quasiperiodic-Chaotic Neural Networks and Short-Term Analog Memory

A model of quasiperiodic-chaotic neural networks is proposed on the basis of chaotic neural networks. A quasiperiodic-chaotic neuron exhibits quasiperiodic dynamics that an original chaotic neuron does not have. Quasiperiodic and chaotic solutions are exclusively isolated in the parameter space. The chaotic domain can be identified by the presence of a folding structure of an invariant closed curve. Using the property that the influence of perturbation is conserved in the quasiperiodic solution, we demonstrate short-term visual memory in which real numbers are acceptable for representing colors. The quasiperiodic solution is sensitive to dynamical noise when images are restored. However, the quasiperiodic synchronization among neurons can reduce the influence of noise. Short-term analog memory using quasiperiodicity is important in that it can directly store analog quantities. The quasiperiodic-chaotic neural networks are shown to work as large-scale analog storage arrays. This type of analog memory has potential applications to analog computation such as deep learning.

Quasiperiodicity and Chaos Through Hopf–Hopf Bifurcation in Minimal Ring Neural Oscillators Due to a Single Shortcut

Quasiperiodicity and chaos in a ring of unidirectionally coupled sigmoidal neurons (a ring neural oscillator) caused by a single shortcut is examined. A codimension-two Hopf–Hopf bifurcation for two periodic solutions exists in a ring of six neurons without self-couplings and in a ring of four neurons with self-couplings in the presence of a shortcut at specific locations. The locus of the Neimark–Sacker bifurcation of the periodic solution emanates from the Hopf–Hopf bifurcation point and a stable quasiperiodic solution is generated. Arnold’s tongues emanate from the locus of the Neimark–Sacker bifurcation, and multiple chaotic oscillations are generated through period-doubling cascades of periodic solutions in the Arnold’s tongues. Further, such chaotic irregular oscillations due to a single shortcut are also observed in propagating oscillations in a ring of Bonhoeffer–van der Pol (BVP) neurons coupled unidirectionally by slow synapses.

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.

Quasiperiodic galloping of a wind-excited tower near secondary resonances of order 2

Quasiperiodic galloping of a wind-excited tower under unsteady wind is investigated analytically near secondary (sub/superharmonic) resonances of order 2 considering a single degree-of-freedom model. The case where the unsteady wind develops multiharmonic excitations consisting of the two first harmonic terms is examined. We perform two successive multiple scale methods to obtain analytical expressions of a quasiperiodic solution and its modulation envelope near the secondary resonances. The influence of unsteady wind on the quasiperiodic galloping and on the frequency of its modulation is examined for different cases of wind excitation. The results show that the quasiperiodic galloping onset and its modulation envelope can be influenced, depending on the activated resonance and the harmonic component induced by the unsteady wind. It is also shown that the frequency of the quasiperiodic galloping is higher near the 2-superharmonic resonance in all cases of wind excitation.

TRANSITION TO CHAOS IN A SHELL MODEL OF TURBULENCE WITH THE CHANGE OF THE VISCOSITY

In this paper, a shell model for the energy cascade in three-dimensional turbulence at varying the viscosity is studied. When the viscosity parameter ν is large enough, the phase orbit of each shell in the system is a stable fixed point. As ν diminishes down to a critical value, the point becomes unstable and a stable periodic orbit appears via a supercritical Hopf bifurcation. For the quasiperiodic solution of the system, each shell has a own dynamic behavior. At ν = 6.88×10-5, the system initiates a complex scenario of bifurcations, where the phase orbits of un and un+3 begin to form a intersected section. Farther decreasing ν, we observe a transition from quasiperiodic solution to chaos through a sequence of Neimark–Sacker bifurcations. The behaviors can be vividly shown in the change of the average energy function E with ν. At last, the transition to chaos in the ν–δ parameter space is also obtained.

Instability and Chaos of a Flexible Rotor Ball Bearing System: An Investigation on the Influence of Rotating Imbalance and Bearing Clearance

In this paper, a horizontal flexible rotor supported on two deep groove ball bearings is theoretically investigated for instability and chaos. The system is biperiodically excited. The two sources of excitation are rotating imbalance and self excitation due to varying compliance effect of ball bearing. A generalized Timoshenko beam finite element (FE) formulation, which can be used for both flexible and rigid rotor systems with equal effectiveness, is developed. The novel scheme proposed in the literature to analyze quasiperiodic response is coupled with the existing nonautonomous shooting method and is thus modified; the shooting method is used to obtain a steady state quasiperiodic solution. The eigenvalues of monodromy matrix provide information about stability and nature of bifurcation of the quasiperiodic solution. The maximum value of the Lyapunov exponent is used for quantitative measure of chaos in the dynamic response. The effect of three parameters, viz., rotating unbalance, bearing clearance, and rotor flexibility, on an unstable and chaotic behavior of a horizontal flexible rotor is studied. Interactive effects between the three parameters are examined in detail in respect of rotor system instability and chaos, and finally the range of parameters is established for the same.

On quasiperiodic oscillations of a nonlinear dynamic system of Liapunov type with time lag

The paper is concerned with the investigation of the quasiperiodic oscillations of a nonlinear dynamic system of Liapunov type with time lag. The following results are obtained:- The necessary and sufficient conditions for the existence of the quasiperiodic solution describing the oscillating processes.- The approximate quasiperiodic solution in the power series.- The quasiperiodic oscillations of a nonlinear dynamic system of Duffing type with the quasiperiodic perturbations.