Routes to Chaos in the Peroxidase−Oxidase Reaction:  Period-Doubling and Period-Adding

1997 ◽  
Vol 101 (25) ◽  
pp. 5075-5083 ◽  
Author(s):  
Marcus J. B. Hauser ◽  
Lars F. Olsen ◽  
Tatiana V. Bronnikova ◽  
William M. Schaffer
Keyword(s):  
Period Doubling ◽  
Routes To Chaos ◽  
Oxidase Reaction

scholarly journals Complexity of a Peroxidase-Oxidase Reaction Model

2021 ◽  
Author(s):  
Jason Gallas ◽  
Marcus Hauser ◽  
Lars Folke Olsen
Keyword(s):  
Chemical Reaction ◽  
Mixed Mode ◽  
Period Doubling ◽  
Reaction Model ◽  
Chaotic Behaviour ◽  
Mixed Mode Oscillations ◽  
Oscillating Reaction ◽  
Oxidase Reaction

The peroxidase-oxidase oscillating reaction was the first (bio)chemical reaction to show chaotic behaviour. The reaction is rich in bifurcation scenarios, from period-doubling to peak-adding mixed mode oscillations. Here, we study...

scholarly journals CLASSIFICATION OF BIFURCATIONS AND ROUTES TO CHAOS IN A VARIANT OF MURALI–LAKSHMANAN–CHUA CIRCUIT

2002 ◽  
Vol 12 (04) ◽  
pp. 783-813 ◽  
Author(s):  
K. THAMILMARAN ◽  
M. LAKSHMANAN
Keyword(s):  
Period Doubling ◽  
Routes To Chaos ◽  
Rich Variety ◽  
Chua Circuit ◽  
Bifurcation Phenomena ◽  
Simulation Results ◽  
The Rich ◽  
Doubling Sequence ◽  
Two Parameter

We present a detailed investigation of the rich variety of bifurcations and chaos associated with a very simple nonlinear parallel nonautonomous LCR circuit with Chua's diode as its only nonlinear element as briefly reported recently [Thamilmaran et al., 2000]. It is proposed as a variant of the simplest nonlinear nonautonomous circuit introduced by Murali, Lakshmanan and Chua (MLC) [Murali et al., 1994]. In our study we have constructed two-parameter phase diagrams in the forcing amplitude-frequency plane, both numerically and experimentally. We point out that under the influence of a periodic excitation a rich variety of bifurcation phenomena, including the familiar period-doubling sequence, intermittent and quasiperiodic routes to chaos as well as period-adding sequences, occur. In addition, we have also observed that the periods of many windows satisfy the familiar Farey sequence. Further, reverse bifurcations, antimonotonicity, remerging chaotic band attractors, and so on, also occur in this system. Numerical simulation results using Poincaré section, Lyapunov exponents, bifurcation diagrams and phase trajectories are found to be in agreement with experimental observations. The chaotic dynamics of this circuit is observed experimentally and confirmed both by numerical and analytical studies as well PSPICE simulation results. The results are also compared with the dynamics of the original MLC circuit with reference to the two-parameter space to show the richness of the present circuit.

scholarly journals Complex Dynamics on the Routes to Chaos in a Discrete Predator-Prey System with Crowley-Martin Type Functional Response

2018 ◽  
Vol 2018 ◽  
pp. 1-18 ◽  
Author(s):  
Huayong Zhang ◽  
Shengnan Ma ◽  
Tousheng Huang ◽  
Xuebing Cong ◽  
Zichun Gao ◽  
...  
Keyword(s):  
Functional Response ◽  
Complex Dynamics ◽  
Three Dimensional ◽  
Period Doubling ◽  
Flip Bifurcation ◽  
Predator Prey ◽  
Center Manifold Theorem ◽  
Routes To Chaos ◽  
Prey System ◽  
The One

We present in this paper an investigation on a discrete predator-prey system with Crowley-Martin type functional response to know its complex dynamics on the routes to chaos which are induced by bifurcations. Via application of the center manifold theorem and bifurcation theorems, occurrence conditions for flip bifurcation and Neimark-Sacker bifurcation are determined, respectively. Numerical simulations are performed, on the one hand, verifying the theoretical results and, on the other hand, revealing new interesting dynamical behaviors of the discrete predator-prey system, including period-doubling cascades, period-2, period-3, period-4, period-5, period-6, period-7, period-8, period-9, period-11, period-13, period-15, period-16, period-20, period-22, period-24, period-30, and period-34 orbits, invariant cycles, chaotic attractors, sub-flip bifurcation, sub-(inverse) Neimark-Sacker bifurcation, chaotic interior crisis, chaotic band, sudden disappearance of chaotic dynamics and abrupt emergence of chaos, and intermittent periodic behaviors. Moreover, three-dimensional bifurcation diagrams are utilized to study the transition between flip bifurcation and Neimark-Sacker bifurcation, and a critical case between the two bifurcations is found. This critical bifurcation case is a combination of flip bifurcation and Neimark-Sacker bifurcation, showing the nonlinear characteristics of both bifurcations.

Fractal bifurcation sets, renormalization strange sets and their universal invariants

1987 ◽  
Vol 413 (1844) ◽  
pp. 45-61 ◽  
Keyword(s):  
Hausdorff Dimension ◽  
Fractal Structure ◽  
Period Doubling ◽  
Invariant Sets ◽  
Transition Structure ◽  
Routes To Chaos

The transition structure of the most common routes to chaos are organized by fractal bifurcation sets. Examples include the quasiperiodic transitions to chaos and the period-doubling structure found in Arnol’d tongues. In this paper I discuss the universality of such fractal bifurcation sets and their relation to strange invariant sets of renormalization transformations. An important result is that fractal bifurcation sets from within the same universality chaos are lipeomorphic. This implies that they have the same fractal structure and, in particular, the same Hausdorff dimension and scaling spectra. Some other invariants are introduced.

scholarly journals Effect of linewidth enhancement factor (LEF) on routes to chaos in optically injected semiconductor lasers

2021 ◽  
Vol 51 (4) ◽  
Author(s):  
N.M. Al-Hosiny
Keyword(s):  
Semiconductor Laser ◽  
Semiconductor Lasers ◽  
Enhancement Factor ◽  
Period Doubling ◽  
Bifurcation Diagrams ◽  
Linewidth Enhancement Factor ◽  
Routes To Chaos ◽  
Linewidth Enhancement

The effect of the linewidth enhancement factor (LEF) or α-factor on two common routes to chaos (mainly period-doubling and quasi-periodic routes) in optically injected semiconductor laser is theoretically investigated using bifurcation diagrams. The value of the LEF is slightly modified to examine the sensitivity of routes to chaos to any variation in the LEF. Despite the fact that LEF enhances chaos in the system, both routes are found to be highly insensitive to the variation in the LEF.

A POSSIBLE NEW ROUTE TO CHAOS VIA SEMICONDUCTOR SUPERLATTICES

2007 ◽  
Vol 21 (23n24) ◽  
pp. 3967-3974
Author(s):  
X. R. WANG ◽  
Z. Z. SUN ◽  
ZHENYU ZHANG
Keyword(s):  
Limit Cycles ◽  
Logistic Map ◽  
Period Doubling ◽  
Current Understanding ◽  
Semiconductor Superlattices ◽  
Frequency Locking ◽  
Routes To Chaos ◽  
Route To Chaos ◽  
Geometric Picture ◽  
Torus Bifurcation

Our current understanding of routes to chaos is mainly based on torus bifurcation where new periods are generated, the period-doubling mechanism revealed in the logistic map, and intermittency where periodic and burst motion appear alternatively. We present a possible new route to chaos based on our geometric picture of the frequency-locking of limit-cycles in semiconductor superlattices. In the period-double route and/or its variations, the period increases exponentially with bifurcation order, whereas the period in the new route increases linearly with the order of bifurcations.

Period-doubling bifurcations and chaos in a detailed model of the peroxidase-oxidase reaction

1995 ◽  
Vol 99 (23) ◽  
pp. 9309-9312 ◽  
Author(s):  
T. V. Bronnikova ◽  
V. R. Fed'kina ◽  
W. M. Schaffer ◽  
L. F. Olsen
Keyword(s):  
Period Doubling ◽  
Detailed Model ◽  
Oxidase Reaction ◽  
Period Doubling Bifurcations
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