scholarly journals Bifurcation delay - the case of the sequence: Stable focus - unstable focus - unstable node

2009 ◽  
Vol 2 (4) ◽  
pp. 911-929 ◽  
Author(s):  
Eric Benoît ◽  
Keyword(s):  
Stable Focus ◽  
Unstable Node ◽  
Bifurcation Delay ◽  
Unstable Focus

Flow topology in compressible turbulent boundary layer

2012 ◽  
Vol 703 ◽  
pp. 255-278 ◽  
Author(s):  
Li Wang ◽  
Xi-Yun Lu
Keyword(s):  
Outer Layer ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Flow Topology ◽  
Gradient Tensor ◽  
Navier Stokes Equations ◽  
Velocity Gradient Tensor ◽  
Compressibility Effect ◽  
Unstable Node ◽  
Unstable Focus

AbstractThe flow topologies of compressible turbulent boundary layers at Mach 2 are investigated by means of direct numerical simulation (DNS) of the compressible Navier–Stokes equations, and statistical analysis of the invariants of the velocity gradient tensor. We identify a preference for an unstable focus/compressing topology in the inner layer and an unstable node/saddle/saddle (UN/S/S) topology in the outer layer. The dissipation and dissipation production originate mainly from this UN/S/S topology. The enstrophy depends mainly on an unstable focus/stretching (UFS) topology, and the enstrophy production relies on a UN/S/S topology in the inner layer and on a UFS topology in the outer layer. The compressibility effect on the statistical properties of the topologies is investigated in terms of the ‘incompressible’, compressed and expanding regions. It is found that the locally compressed region tends to be more stable and the locally expanding region tends to be more dissipative. The compressibility is mainly related to unstable focus/compressing and stable focus/stretching topologies. Moreover, the features of the average dissipation, enstrophy, dissipation production and enstrophy production of the various topologies are clarified in the locally compressed and expanding regions.

Topology of fine-scale motions in turbulent channel flow

1996 ◽  
Vol 310 ◽  
pp. 269-292 ◽  
Author(s):  
Hugh M. Blackburn ◽  
Nagi N. Mansour ◽  
Brian J. Cantwell
Keyword(s):  
Strain Rate ◽  
Channel Flow ◽  
Velocity Gradient ◽  
Outer Region ◽  
Turbulent Channel Flow ◽  
Principal Strain ◽  
Wall Region ◽  
Fine Scale ◽  
Stable Focus ◽  
Topological Features

An investigation of topological features of the velocity gradient field of turbulent channel flow has been carried out using results from a direct numerical simulation for which the Reynolds number based on the channel half-width and the centreline velocity was 7860. Plots of the joint probability density functions of the invariants of the rate of strain and velocity gradient tensors indicated that away from the wall region, the fine-scale motions in the flow have many characteristics in common with a variety of other turbulent and transitional flows: the intermediate principal strain rate tended to be positive at sites of high viscous dissipation of kinetic energy, while the invariants of the velocity gradient tensor showed that a preference existed for stable focus/stretching and unstable node/saddle/saddle topologies. Visualization of regions in the flow with stable focus/stretching topologies revealed arrays of discrete downstream-leaning flow structures which originated near the wall and penetrated into the outer region of the flow. In all regions of the flow, there was a strong preference for the vorticity to be aligned with the intermediate principal strain rate direction, with the effect increasing near the walls in response to boundary conditions.

Canard-induced mixed mode oscillations as a mechanism for the Bonhoeffer-van der Pol circuit under parametric perturbation

Circuit World ◽  
2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Yue Yu ◽  
Cong Zhang ◽  
Zhenyu Chen ◽  
Zhengdi Zhang
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Orbit ◽  
Large Amplitude ◽  
Mixed Mode ◽  
Periodic Perturbation ◽  
Van Der Pol ◽  
Content Type ◽  
Stable Focus ◽  
Mixed Mode Oscillations ◽  
Transition Mechanism

Purpose This paper aims to investigate the singular Hopf bifurcation and mixed mode oscillations (MMOs) in the perturbed Bonhoeffer-van der Pol (BVP) circuit. There is a singular periodic orbit constructed by the switching between the stable focus and large amplitude relaxation cycles. Using a generalized fast/slow analysis, the authors show the generation mechanism of two distinct kinds of MMOs. Design/methodology/approach The parametric modulation can be used to generate complicated dynamics. The BVP circuit is constructed as an example for second-order differential equation with periodic perturbation. Then the authors draw the bifurcation parameter diagram in terms of a containing two attractive regions, i.e. the stable relaxation cycle and the stable focus. The transition mechanism and characteristic features are investigated intensively by one-fast/two-slow analysis combined with bifurcation theory. Findings Periodic perturbation can suppress nonlinear circuit dynamic to a singular periodic orbit. The combination of these small oscillations with the large amplitude oscillations that occur due to canard cycles yields such MMOs. The results connect the theory of the singular Hopf bifurcation enabling easier calculations of where the oscillations occur. Originality/value By treating the perturbation as the second slow variable, the authors obtain that the MMOs are due to the canards in a supercritical case or in a subcritical case. This study can reveal the transition mechanism for multi-time scale characteristics in perturbed circuit. The information gained from such results can be extended to periodically perturbed circuits.

scholarly journals Soil nutrient cycles as a nonlinear dynamical system

2004 ◽  
Vol 11 (5/6) ◽  
pp. 589-598 ◽  
Author(s):  
S. Manzoni ◽  
A. Porporato ◽  
P. D'Odorico ◽  
F. Laio ◽  
I. Rodriguez-Iturbe
Keyword(s):  
Dynamical System ◽  
Initial Conditions ◽  
Nonlinear Dynamical System ◽  
Soil Nutrient ◽  
Steady State Solution ◽  
Basins Of Attraction ◽  
Climatic Conditions ◽  
Stable Node ◽  
Nutrient Cycles ◽  
Stable Focus

Abstract. An analytical model for the soil carbon and nitrogen cycles is studied from the dynamical system point of view. Its main nonlinearities and feedbacks are analyzed by considering the steady state solution under deterministic hydro-climatic conditions. It is shown that, changing hydro-climatic conditions, the system undergoes dynamical bifurcations, shifting from a stable focus to a stable node and back to a stable focus when going from dry, to well-watered, and then to saturated conditions, respectively. An alternative degenerate solution is also found in cases when the system can not sustain decomposition under steady external conditions. Different basins of attraction for "normal" and "degenerate" solutions are investigated as a function of the system initial conditions. Although preliminary and limited to the specific form of the model, the present analysis points out the importance of nonlinear dynamics in the soil nutrient cycles and their possible complex response to hydro-climatic forcing.

GWN-INDUCED BURSTING, SPIKING, AND RANDOM SUBTHRESHOLD IMPULSING OSCILLATION BEFORE HOPF BIFURCATIONS IN THE CHAY MODEL

2004 ◽  
Vol 14 (12) ◽  
pp. 4143-4159 ◽  
Author(s):  
ZHUOQIN YANG ◽  
QISHAO LU ◽  
HUAGUANG GU ◽  
WEI REN
Keyword(s):  
Hysteresis Loop ◽  
Gaussian White Noise ◽  
Global Bifurcation ◽  
Intrinsic Noise ◽  
Stable Node ◽  
Bifurcation Curve ◽  
Fast System ◽  
Integer Multiple ◽  
Rest State ◽  
Stable Focus

Gaussian white noise (GWN), as an intrinsic noise source, can give rise to various firing activities at the rest state before a supercritical or subcritical Hopf bifurcation (supH or subH) in the Chay system without or with external current input, when VK, VC, λn and I are considered as changeable control parameters. These firing activities are closely related to the global bifurcation mechanism of the whole system and the fast/slow dynamical subsystems, and can be tackled by means of bifurcation analysis. GWN can induce some typical bursting phenomena in the stochastic Chay system. Firstly, integer multiple "fold/homoclinic" or "circle/homoclinic" bursting due to GWN, with only one spike per burst, can arise from rest states before both subH and supH (with respect to the parameter VK), and their respective trajectories have the same shape and property. However, less spikes appear and their peaks are lower before supH, comparing with those before subH. Secondly, a "fold/fold" point–point hysteresis loop bursting due to GWN is generated before supH (with respect to the parameter VC) on the upper branch of a "Z"-shaped bifurcation curve between two fold bifurcations of the fast system. Thirdly, at a rest state before subH (with respect to the additional current I) situated on the lower branch of a "S"-shaped bifurcation curve between two fold bifurcations of the fast system, a GWN-induced firing pattern appears and is classified as "Hopf/homoclinic" bursting via "fold/homoclinic" point–point hysteresis loop. GWN-induced firing activities other than bursting can also be observed in the stochastic Chay system. For example, sometimes GWN-induced continuous spiking without any particular shape may arise at a rest state before supH (with respect to the parameter VK) for certain values of parameters. Moreover, under the situation that a stable node and a stable focus coexist before subH (with respect to the parameter I) and the attractive region of the stable node is larger than that of the stable focus, GWN only provoke random subthreshold impulsing oscillation near the stable node.

scholarly journals Finite-Time Formation Control without Collisions for Multiagent Systems with Communication Graphs Composed of Cyclic Paths

2015 ◽  
Vol 2015 ◽  
pp. 1-17 ◽  
Author(s):  
J. F. Flores-Resendiz ◽  
E. Aranda-Bricaire ◽  
J. González-Sierra ◽  
J. Santiaguillo-Salinas
Keyword(s):  
Multiagent Systems ◽  
Finite Time ◽  
Formation Control ◽  
Vector Fields ◽  
Control Law ◽  
Closed Loop System ◽  
Focus Structure ◽  
Communication Graphs ◽  
Cyclic Pursuit ◽  
Unstable Focus

This paper addresses the formation control problem without collisions for multiagent systems. A general solution is proposed for the case of any number of agents moving on a plane subject to communication graph composed of cyclic paths. The control law is designed attending separately the convergence to the desired formation and the noncollision problems. First, a normalized version of the directed cyclic pursuit algorithm is proposed. After this, the algorithm is generalized to a more general class of topologies, including all the balanced formation graphs. Once the finite-time convergence problem is solved we focus on the noncollision complementary requirement adding a repulsive vector field to the previous control law. The repulsive vector fields display an unstable focus structure suitably scaled and centered at the position of the rest of agents in a certain radius. The proposed control law ensures that the agents reach the desired geometric pattern in finite time and that they stay at a distance greater than or equal to some prescribed lower bound for all times. Moreover, the closed-loop system does not exhibit undesired equilibria. Numerical simulations and real-time experiments illustrate the good performance of the proposed solution.

Bipolar Pulse-Induced Coexisting Firing Patterns in Two-Dimensional Hindmarsh–Rose Neuron Model

2019 ◽  
Vol 29 (01) ◽  
pp. 1950006 ◽  
Author(s):  
Han Bao ◽  
Aihuang Hu ◽  
Wenbo Liu
Keyword(s):  
Membrane Potential ◽  
Neuron Model ◽  
Periodic Variation ◽  
Constant Current ◽  
Two Dimensional ◽  
Bipolar Pulse ◽  
Firing Patterns ◽  
Current Stimulus ◽  
Unstable Node ◽  
Stimulus Effect

In this paper, a bipolar pulse (BP) current is taken to mimic a periodic stimulus effect on the membrane potential in the axon of a neuron. By introducing the BP current to substitute the externally applied constant current, a BP-forced two-dimensional Hindmarsh–Rose (HR) neuron model is proposed. Based on the proposed neuron model, the BP-switched equilibrium point and its stability evolution with the periodic variation in time are explored. Furthermore, coexisting asymmetric attractors (or coexisting firing patterns) with bistability are revealed by phase plane orbits, time sequences, and attraction basins, as well as the BP-induced coexisting asymmetric attractors’ behaviors are then elaborated through bifurcation analysis. The research results exhibit that, with the increase of the time, the stabilities of the neuron model are continually switched between an unstable node-focus and a stable point, resulting in the coexisting behaviors of numerous asymmetric attractors under the specified initials. Consequently, the newly introduced BP current stimulus, instead of the original constant current stimulus, allows the two-dimensional HR neuron model to possess complex dynamical behaviors for the membrane potential. Additionally, a hardware breadboard is fabricated and circuit experiments are carried out to validate the numerical simulations.

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