scholarly journals On the Dynamics of Coupled Parametrically Forced Oscillators

2008 ◽  
Author(s):  
Jeff Moehlis
Keyword(s):  
Dynamical System ◽  
Hopf Bifurcation ◽  
Periodic Orbit ◽  
Weak Coupling ◽  
Weak Coupling Limit ◽  
Stable Periodic Orbit ◽  
Weakly Coupled ◽  
Forced Oscillators ◽  
Stable Periodic ◽  
Autonomous Dynamical System

It is well known that an autonomous dynamical system can have a stable periodic orbit, arising for example through a Hopf bifurcation. When a collection of such oscillators is coupled together, the system can display a number of phase-locked solutions which can be understood in the weak coupling limit by using a phase model. It is also well known that a stable periodic orbit can be found for a parametrically forced dynamical system, with the phase of the periodic orbit being locked to the forcing. Here we discuss the periodic solutions which occur for a collection of such parametrically forced oscillators that are weakly coupled together.

Nonresonant Double Hopf Bifurcation in Toxic Phytoplankton–Zooplankton Model with Delay

2017 ◽  
Vol 27 (02) ◽  
pp. 1750028 ◽  
Author(s):  
Rui Yuan ◽  
Weihua Jiang ◽  
Yong Wang
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Periodic Orbit ◽  
Digestion Time ◽  
Normal Form Theory ◽  
Stable Periodic Orbit ◽  
Double Hopf Bifurcation ◽  
Toxic Phytoplankton ◽  
Stable Periodic ◽  
The Stability

This paper investigates a toxic phytoplankton–zooplankton model with Michaelis–Menten type phytoplankton harvesting. The model has rich dynamical behaviors. It undergoes transcritical, saddle-node, fold, Hopf, fold-Hopf and double Hopf bifurcation, when the parameters change and go through some of the critical values, the dynamical properties of the system will change also, such as the stability, equilibrium points and the periodic orbit. We first study the stability of the equilibria, and analyze the critical conditions for the above bifurcations at each equilibrium. In addition, the stability and direction of local Hopf bifurcations, and the completion bifurcation set by calculating the universal unfoldings near the double Hopf bifurcation point are given by the normal form theory and center manifold theorem. We obtained that the stable coexistent equilibrium point and stable periodic orbit alternate regularly when the digestion time delay is within some finite value. That is, we derived the pattern for the occurrence, and disappearance of a stable periodic orbit. Furthermore, we calculated the approximation expression of the critical bifurcation curve using the digestion time delay and the harvesting rate as parameters, and determined a large range in terms of the harvesting rate for the phytoplankton and zooplankton to coexist in a long term.

Dynamics of Morse-Smale urn processes

1995 ◽  
Vol 15 (6) ◽  
pp. 1005-1030 ◽  
Author(s):  
Michel Benaïm ◽  
Morris W. Hirsch
Keyword(s):  
Dynamical System ◽  
Sample Path ◽  
Asymptotically Stable ◽  
Martingale Differences ◽  
Stable Periodic Orbit ◽  
Sample Paths ◽  
Stable Periodic ◽  
Image Position ◽  
Deterministic Dynamical System ◽  
Uniformly Bounded

AbstractWe consider stochastic processes {xn}n≥0 of the formwhere F: ℝm → ℝm is C2, {λi}i≥1 is a sequence of positive numbers decreasing to 0 and {Ui}i≥1 is a sequence of uniformly bounded ℝm-valued random variables forming suitable martingale differences. We show that when the vector field F is Morse-Smale, almost surely every sample path approaches an asymptotically stable periodic orbit of the deterministic dynamical system dy/dt = F(y). In the case of certain generalized urn processes we show that for each such orbit Γ, the probability of sample paths approaching Γ is positive. This gives the generic behavior of three-color urn models.

scholarly journals Gaussian-optical approach to stable periodic orbit resonances of partially chaotic dielectric micro-cavities

2002 ◽  
Vol 10 (16) ◽  
pp. 752 ◽  
Author(s):  
Hakan Tureci ◽  
H. Schwefel ◽  
A. Stone ◽  
E. Narimanov
Keyword(s):  
Periodic Orbit ◽  
Stable Periodic Orbit ◽  
Stable Periodic

scholarly journals Bifurcation Gaps in Asymmetric and High-Dimensional Hypercycles

2018 ◽  
Vol 28 (01) ◽  
pp. 1830001
Author(s):  
Júlia Puig ◽  
Gerard Farré ◽  
Antoni Guillamon ◽  
Ernest Fontich ◽  
Josep Sardanyés
Keyword(s):  
Periodic Orbit ◽  
Fixed Points ◽  
Quality Factor ◽  
Equilibrium Points ◽  
High Dimensional ◽  
Catalytic Systems ◽  
Stable Periodic Orbit ◽  
Replication Quality ◽  
Stable Periodic ◽  
Nonzero Equilibrium

Hypercycles are catalytic systems with cyclic architecture. These systems have been suggested to play a key role in the maintenance and increase of information in prebiotic replicators. It is known that for a large enough number of hypercycle species ([Formula: see text]) the coexistence of all hypercycle members is governed by a stable periodic orbit. Previous research has characterized saddle-node (s-n) bifurcations involving abrupt transitions from stable hypercycles to extinction of all hypercycle members, or, alternatively, involving the outcompetition of the hypercycle by so-called mutant sequences or parasites. Recently, the presence of a bifurcation gap between a s-n bifurcation of periodic orbits and a s-n of fixed points has been described for symmetric five-member hypercycles. This gap was found between the value of the replication quality factor [Formula: see text] from which the periodic orbit vanishes ([Formula: see text]) and the value where two unstable (nonzero) equilibrium points collide ([Formula: see text]). Here, we explore the persistence of this gap considering asymmetries in replication rates in five-member hypercycles as well as considering symmetric, larger hypercycles. Our results indicate that both the asymmetry in Malthusian replication constants and the increase in hypercycle members enlarge the size of this gap. The implications of this phenomenon are discussed in the context of delayed transitions associated to the so-called saddle remnants.

A Four-Leaf Chaotic Attractor of a Three-Dimensional Dynamical System

2015 ◽  
Vol 25 (02) ◽  
pp. 1530003 ◽  
Author(s):  
Tomoyuki Miyaji ◽  
Hisashi Okamoto ◽  
Alex D. D. Craik
Keyword(s):  
Dynamical System ◽  
Three Dimensional ◽  
Period Doubling ◽  
Numerical Verification ◽  
Unstable Periodic Orbits ◽  
Verification Method ◽  
Stable Periodic ◽  
Approach Infinity ◽  
Dimensional Dynamical System ◽  
Autonomous Dynamical System

A three-dimensional autonomous dynamical system proposed by Pehlivan is untypical in simultaneously possessing both unbounded and chaotic solutions. Here, this topic is studied in some depth, both numerically and analytically. We find, by standard methods, that four-leaf chaotic orbits result from a period-doubling cascade; we identify unstable fixed points and both stable and unstable periodic orbits; and we examine how initial data determines whether orbits approach infinity or a stable periodic orbit. Further, we describe and apply a strict numerical verification method that rigorously proves the existence of sequences of period doublings.

Almost Super Stable Periodic Orbit in an Electric Impact Oscillator

2014 ◽  
Vol 1 ◽  
pp. 832-835
Author(s):  
Hiroyuki Asahara ◽  
Jun Hosokawa ◽  
Kazuyuki Aihara ◽  
Soumitro Banerjee ◽  
Takuji Kousaka
Keyword(s):  
Periodic Orbit ◽  
Impact Oscillator ◽  
Stable Periodic Orbit ◽  
Stable Periodic

Jacobi analysis of a segmented disc dynamo system

2020 ◽  
Vol 17 (14) ◽  
pp. 2050205
Author(s):  
Aimin Liu ◽  
Biyu Chen ◽  
Yuming Wei
Keyword(s):  
Angular Velocity ◽  
Periodic Orbit ◽  
Finsler Geometry ◽  
Small Perturbations ◽  
Stable Periodic Orbit ◽  
Onset Of Chaos ◽  
Internal Parameters ◽  
Geometric Objects ◽  
Stable Periodic ◽  
Deviation Vector

In this paper, Jacobi stability of a segmented disc dynamo system is geometrically investigated from viewpoint of Kosambi–Cartan–Chern (KCC) theory in Finsler geometry. First, the geometric objects associated to the reformulated system are explicitly obtained. Second, the Jacobi stability of equilibria and a periodic orbit are discussed in the light of deviation curvature tensor. It is shown that all the equilibria are always Jacobi unstable for any parameters, a Lyapunov stable periodic orbit falls into both Jacobi stable regions and Jacobi unstable regions. The considered system is not robust to small perturbations of the equilibria, and some fragments of the periodic orbit are included in fragile region, indicating that the system is extremely sensitive to internal parameters and environment. Finally, the dynamics of the deviation vector and its curvature near all the equilibria are presented to interpret the onset of chaos in the dynamo system. In a physical sense, magnetic fluxes and angular velocity can show irregular oscillations under some certain cases, these oscillations may reveal the irregularity of magnetic field’s evolution and reversals.

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