A. Mareno, L. Q. English
In this paper we consider a system of strongly coupled logistic maps involving two parameters. We classify and investigate the stability of its fixed points. A local bifurcation analysis of the system using Center Manifold is undertaken and then supported by numerical computations.This reveals the existence of reverse flip and Neimark-Sacker bifurcations.
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Bouncing droplets on a vibrating fluid bath can exhibit wave-particle behavior, such as being propelled by interacting with its own wave field. These droplets seem to walk across the bath, and thus are dubbed walkers. Experiments have shown that walkers can exhibit exotic dynamical behavior indicative of chaos. While the integro-differential models developed for these systems agree well with the experiments, they are difficult to analyze mathematically. In recent years, simpler discrete dynamical models have been derived and studied numerically. The numerical simulations of these models show e…
We investigate a population dynamics model that exhibits a Neimark Sacker bifurcation with a period that is naturally close to 4. Beyond the bifurcation, the period becomes soon locked at 4 due to a strong resonance, and a second attractor of period 2 emerges, which coexists with the first attractor over a considerable parameter range. A linear stability analysis and a numerical investigation of the second attractor reveal that the bifurcations producing the second attractor occur naturally in this type of system.
We report using Clarke's concept of generalised differential and a modification of Floquet theory to non-smooth oscillations, the occurrence of discontinuity induced Hopf bifurcations and Neimark-Sacker bifurcations leading to quasiperiodic attractors in a memristive Murali-Lakshmanan-Chua (memristive MLC) circuit. The above bifurcations arise because of the fact that a memristive MLC circuit is basically a nonsmooth system by virtue of having a memristive element as its nonlinearity. The switching and modulating properties of the memristor which we have considered endow the circuit with two d…
Recently, it has been demonstrated that asymptotic states of open quantum system can undergo qualitative changes resembling pitchfork, saddle-node, and period doubling classical bifurcations. Here, making use of the periodically modulated open quantum dimer model, we report and investigate a quantum Neimark-Sacker bifurcation. Its classical counterpart is the birth of a torus (an invariant curve in the Poincaré section) due to instability of a limit cycle (fixed point of the Poincaré map). The quantum system exhibits a transition from unimodal to bagel shaped stroboscopic distributions, as for…
Dynamical properties of a small heterogeneous chain network of neurons in discrete time
提出由 Chialvo 与 Rulkov 映射构成的离散时间双向耦合异质神经元链式网络，并通过分岔、同步度量和样本熵揭示其丰富动力学。
Dynamical analysis of r-Chialvo neuron map with cosine memristive
本文通过引入余弦基忆阻器构建新型二维离散神经元映射，研究电磁调制的动力学特性。该映射表现出多样化的放电模式、稳定性行为和混沌吸引子，揭示了多种分岔结构及多稳态吸引子的共存现象。
Visualizing Attractors of the Three-Dimensional Generalized Hénon Map
本文研究三维广义 Hénon 映射，通过参数分岔分析、旋转数与 Lyapunov 指数刻画不变圆及多种混沌吸引子的形成机制。
Hysteresis phenomena and multistability play crucial roles in the dynamics of coupled oscillators, which are now interpreted from the point of view of codimension-two bifurcations. On the Ott-Antonsen's manifold, complete bifurcation sets of delay-coupled Kuramoto model are derived regarding coupling strength and delay as bifurcation parameters. It is rigorously proved that the system must undergo Bautin bifurcations for some critical values, thus there always exists saddle-node bifurcation of periodic solutions inducing hysteresis loop. With the aid of center manifold reduction method and the…
This paper presents an investigation of the dynamics of two coupled non-identical FitzHugh-Nagumo neurons with delayed synaptic connection. We consider coupling strength and time delay as bifurcation parameters, and try to classify all possible dynamics which is fairly rich. The neural system exhibits a unique rest point or three ones for the different values of coupling strength by employing the pitchfork bifurcation of non-trivial rest point. The asymptotic stability and possible Hopf bifurcations of the trivial rest point are studied by analyzing the corresponding characteristic equation. H…
Bifurcations of mode-locked periodic orbits in three-dimensional maps
研究三维及更高维映射中锁模周期轨道的两类“加倍”分岔，验证其类型由第三特征值符号决定，并揭示 Neimark-Sacker 分岔与非同步鞍–结连接结构。
A system of two identical SQUIDs (superconducting quantum interference devices) symmetrically coupled through their mutual inductance and driven by a sinusoidal field is investigated numerically with respect to dynamical properties such as its multibranched resonance curve, its bifurcation structure, as well as its synchronization behavior. The SQUID dimer is found to exhibit a hysteretic resonance curve with a bubble connected to it through Neimark-Sacker (torus) bifurcations, along with coexisting chaotic branches in their vicinity. Interestingly, the transition of the SQUID dimer to chaos o…
Torus and hyperchaos in 3D Lotka-Volterra map
本研究探讨了三维 Lotka-Volterra 映射中三频准周期环面与超混沌的出现，通过计算流形和分岔分析揭示了通向准周期结构与超混沌的多种路径及转变机制。
We study the dynamics of the five-parameter quadratic family of volume-preserving diffeomorphisms of R^3. This family is the unfolded normal form for a bifurcation of a fixed point with a triple-one multiplier and also is the general form of a quadratic three-dimensional map with a quadratic inverse. Much of the nontrivial dynamics of this map occurs when its two fixed points are saddle-foci with intersecting two-dimensional stable and unstable manifolds that bound a spherical ``vortex-bubble''. We show that this occurs near a saddle-center-Neimark-Sacker (SCNS) bifurcation that also creates, …
We investigate the origin of various convective patterns using bifurcation diagrams that are constructed using direct numerical simulations. We perform two-dimensional pseudospectral simulations for a Prandtl number 6.8 fluid that is confined in a box with aspect ratio $Γ= 2\sqrt{2}$. Steady convective rolls are born from the conduction state through a pitchfork bifurcation at $r=1$, where $r$ is the reduced Rayleigh number. These fixed points bifurcate successively to time-periodic and quasiperiodic rolls through Hopf and Neimark-Sacker bifurcations at $r \simeq 80$ and $r \simeq 500 $ respec…
Distributed delays modeled by 'weak generic kernels' are introduced in the well-known coupled Landau-Stuart system, as well as a chaotic van der Pol-Rayleigh system with parametric forcing. The systems are close via the 'linear chain trick'. Linear stability analysis of the systems and conditions for Hopf bifurcation which initiates oscillations are investigated, including deriving the normal form at bifurcation, and deducing the stability of the resulting limit cycle attractor. The value of the delay parameter $a=a_{Hopf}$ at Hopf bifurcation picks out the onset of Amplitude Death(AD) in all …
Predator Extinction arose from Chaos of the Prey: the Chaotic Behavior of a Homomorphic Two-Dimensional Logistic Map in the Form of Lotka-Volterra Equations
本文提出一种保持Lotka-Volterra方程特征的二维同态Logistic映射，发现捕食者种群远大于猎物时可引发猎物混沌并导致捕食者快速灭绝，并据此提出抑制病毒害虫和实现超导态两种应用。
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