scholarly journals Stable Polymorphisms in a Two-Locus Gene-for-Gene System

2005 ◽  
Vol 95 (7) ◽  
pp. 728-736 ◽  
Author(s):  
J. Segarra
Keyword(s):  
Initial Conditions ◽  
Stable Limit Cycle ◽  
Infinite Family ◽  
Host Population ◽  
Stable Limit ◽  
Final State ◽  
Closed Curves ◽  
Interior Equilibrium ◽  
Wide Range ◽  
Gene For Gene

A two-locus gene-for-gene model is presented to analyze coevolutionary dynamics in interactions between host plants and their pathogens. Using both analytical and simulation approximations, we show that the behavior of the model is very simple with one locus. In the reciprocal genetic feedback version, there is a smooth outward spiral toward the boundaries. In the delayed feedback version, there is an infinite family of closed curves corresponding to different initial conditions. Both versions of the model are stabilized by the addition of recurrent mutation. Either a stable interior equilibrium or a stable limit cycle appears. But with the two-locus model, different coevolutionary outcomes are predicted according to the parameter values. For a wide range of small and medium values of virulence and resistance costs, complex fluctuations arise. The number of virulence alleles per isolate and the number of resistance alleles per plant cycle indefinitely. If the costs of both virulence and resistance are above a threshold, the final state of the coevolutionary dynamics is a stable single-resistance static polymorphism in the host and avirulence in the parasite. An equivalent threshold to maintain a disease-free host population was obtained analytically for a multilocus system. These expressions can be used to determine the number of single-resistance host genotypes that would have to be present in a mixture to prevent the spread of any virulent race of pathogen. The model demonstrates that it is preferable to use mixtures of single-resistant genotypes rather than using multiple resistance alleles in the same cultivar.

Coexistence of Multiple Attractors, Metastable Chaos and Bursting Oscillations in a Multiscroll Memristive Chaotic Circuit

2017 ◽  
Vol 27 (05) ◽  
pp. 1750067 ◽  
Author(s):  
N. Henry Alombah ◽  
Hilaire Fotsin ◽  
Kengne Romanic
Keyword(s):  
Limit Cycle ◽  
Initial Conditions ◽  
Basin Of Attraction ◽  
Stable Limit Cycle ◽  
Phase Portraits ◽  
Stable Limit ◽  
Nonlinear Phenomenon ◽  
Chaotic Circuit ◽  
Bursting Oscillations ◽  
Wide Range

In this paper, some complex nonlinear behaviors in a four-dimensional multiscroll autonomous memristor based chaotic system are investigated. This system is derived from the three-dimensional autonomous charge-controlled Muthuswamy–Chua simplest chaotic circuit. The system can generate four different coexisting attractors for a fixed set of parameters and different initial conditions. This phenomenon is relatively rare given that we have four different attractors namely: an equilibrium point, a stable limit cycle, a 16-peak limit cycle and a strange attractor that coexist in the system within a wide range of parameters. The nonlinear phenomenon of transient chaos is studied and revealed numerically in Matlab and Pspice environments. The complex transient dynamics of this memristive system under different initial states shows that the transient time depends strongly on the initial conditions. Moreover, this model displays spiking and bursting oscillations. The bursting behavior is classified according to the dynamics of separated slow and fast subsystems. It is shown to be of the fold-Hopf type. These complex dynamical behaviors of this system are investigated by means of numerical simulations and via Pspice circuit simulations. The use of bifurcation diagrams, Lyapunov exponents diagrams, power spectrums, phase portraits, time series, isospike diagram, basin of attraction, clearly shows these complex phenomena.

scholarly journals Statistical characteristics of irreversible predictability time in regional ocean models

2005 ◽  
Vol 12 (1) ◽  
pp. 129-138 ◽  
Author(s):  
P. C. Chu ◽  
L. M. Ivanov
Keyword(s):  
Boundary Conditions ◽  
Initial Conditions ◽  
Stable Limit Cycle ◽  
Ocean Model ◽  
Statistical Characteristics ◽  
Open Boundary ◽  
Stable Limit ◽  
Open Boundary Conditions ◽  
Stochastic Perturbations ◽  
Non Gaussian

Abstract. Probabilistic aspects of regional ocean model predictability is analyzed using the probability density function (PDF) of the irreversible predictability time (IPT) (called τ-PDF) computed from an unconstrained ensemble of stochastic perturbations in initial conditions, winds, and open boundary conditions. Two-attractors (a chaotic attractor and a small-amplitude stable limit cycle) are found in the wind-driven circulation. Relationship between attractor's residence time and IPT determines the τ-PDF for the short (up to several weeks) and intermediate (up to two months) predictions. The τ-PDF is usually non-Gaussian but not multi-modal for red-noise perturbations in initial conditions and perturbations in the wind and open boundary conditions. Bifurcation of τ-PDF occurs as the tolerance level varies. Generally, extremely successful predictions (corresponding to the τ-PDF's tail toward large IPT domain) are not outliers and share the same statistics as a whole ensemble of predictions.

scholarly journals Asymptotic analysis of internal relaxation oscillations in a conceptual climate model

2020 ◽  
Vol 85 (3) ◽  
pp. 467-494
Author(s):  
Łukasz Płociniczak
Keyword(s):  
Asymptotic Analysis ◽  
Climate Model ◽  
Stable Limit Cycle ◽  
Stable Limit ◽  
External Forcing ◽  
Relaxation Oscillations ◽  
Robust Model ◽  
Wide Range ◽  
Paleoclimatic Data ◽  
The Stability

Abstract We construct a dynamical system based on the Källén–Crafoord–Ghil conceptual climate model which includes the ice–albedo and precipitation–temperature feedbacks. Further, we classify the stability of various critical points of the system and identify a parameter which change generates a Hopf bifurcation. This gives rise to a stable limit cycle around a physically interesting critical point. Moreover, it follows from the general theory that the periodic orbit exhibits relaxation-oscillations that are a characteristic feature of the Pleistocene ice ages. We provide an asymptotic analysis of their behaviour and derive a formula for the period along with several estimates. They, in turn, are in a decent agreement with paleoclimatic data and are independent of any parametrization used. Whence, our simple but robust model shows that a climate may exhibit internal relaxation oscillations without any external forcing and for a wide range of parameters.

scholarly journals Estimation of oscillation velocities of oscillator network

2018 ◽  
Vol 32 ◽  
pp. 182-189
Author(s):  
Volodymyr Shcherbak ◽  
Iryna Dmytryshyn
Keyword(s):  
Dynamical System ◽  
Nonlinear Oscillations ◽  
Initial Conditions ◽  
Nonlinear Dynamical System ◽  
Stable Limit Cycle ◽  
Nonlinear Observer ◽  
Stable Limit ◽  
Nonlinear Dynamical ◽  
Biomechanical Models ◽  
Neural Ensembles

The study of the collective behavior of multiscale dynamic processes is currently one of the most urgent problems of nonlinear dynamics. Such systems arise on modelling of many cyclical biological or physical processes. It is of fundamental importance for understanding the basic laws of synchronous dynamics of distributed active subsystems with oscillations, such as neural ensembles, biomechanical models of cardiac or locomotor activity, models of turbulent media, etc. Since the nonlinear oscillations that are observed in such systems have a stable limit cycle , which does not depend on the initial conditions, then a system of interconnected nonlinear oscillators is usually used as a model of multiscale processes. The equations of Lienar type are often used as the main dynamic model of each of these oscillators. In a number of practical control problems of such interconnected oscillators it is necessary to determine the oscillation velocities by known data. This problem is considered as observation problem for nonlinear dynamical system. A new method – a synthesis of invariant relations is used to design a nonlinear observer. The method allows us to represent unknowns as a function of known quantities. The scheme of the construction of invariant relations consists in the expansion of the original dynamical system by equations of some controlled subsystem (integrator). Control in the additional system is used for the synthesis of some relations that are invariant for the extended system and have the attraction property for all of its trajectories. Such relations are considered in observation problems as additional equations for unknown state vector of initial oscillators ensemble. To design the observer, first we introduce a observer for unique oscillator of Lienar type and prove its exponential convergence. This observer is then extended on several coupled Lienar type oscillators. The performance of the proposed method is investigated by numerical simulations.

Analysis of Self-Excited Longitudinal Vibration of a Moving Tape

1993 ◽  
Author(s):  
Jonathan A. Wickert
Keyword(s):  
Initial Conditions ◽  
Longitudinal Vibration ◽  
Stable Limit Cycle ◽  
Longitudinal Motion ◽  
Stable Limit ◽  
Contact Parameter ◽  
All Solutions ◽  
Stationary Observer ◽  
Axially Moving ◽  
Almost All

Abstract Longitudinal vibration of a moving magnetic tape can be excited by frictional contact with a recording head, or with a guide that defines the path of the tape. The friction force depends on the velocity, relative to a stationary observer, of the tape element that is instantaneously located at the point of contact. The response of a moving tape under such nonlinear dissipation is determined using new methods for the vibration analysis of axially-moving materials. In a particular transport speed range, longitudinal motion of the tape is self-excited through negative (unstable) damping for small amplitude vibration and positive (stable) damping for large amplitudes. Independent of the initial conditions, almost all solutions are attracted to a stable limit cycle, the amplitude of which is sensitive to the transport speed and to the axial location of contact. Parameter combinations that reduce, or eliminate altogether, the self-excited motion are identified. Some of the conclusions differ from those of previous analyses that neglect the effects of convection on the tape’s velocity and acceleration.

scholarly journals Dynamics and Optimal Taxation Control in a Bioeconomic Model with Stage Structure and Gestation Delay

2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
Author(s):  
Yi Zhang ◽  
Qingling Zhang ◽  
Fenglan Bai
Keyword(s):  
Optimal Taxation ◽  
Stable Limit Cycle ◽  
Stage Structure ◽  
Model System ◽  
Optimal Harvesting ◽  
Stable Limit ◽  
Gestation Delay ◽  
Interior Equilibrium ◽  
Control Instrument ◽  
Predator Model

A prey-predator model with gestation delay, stage structure for predator, and selective harvesting effort on mature predator is proposed, where taxation is considered as a control instrument to protect the population resource in prey-predator biosystem from overexploitation. It shows that interior equilibrium is locally asymptotically stable when the gestation delay is zero, and there is no periodic orbit within the interior of the first quadrant of state space around the interior equilibrium. An optimal harvesting policy can be obtained by virtue of Pontryagin's Maximum Principle without considering gestation delay; on the other hand, the interior equilibrium of model system loses as gestation delay increases through critical certain threshold, a phenomenon of Hopf bifurcation occurs, and a stable limit cycle corresponding to the periodic solution of model system is also observed. Finally, numerical simulations are carried out to show consistency with theoretical analysis.

ALLEE EFFECT, EMIGRATION AND IMMIGRATION IN A CLASS OF PREDATOR-PREY MODELS

2008 ◽  
Vol 03 (01n02) ◽  
pp. 195-215 ◽  
Author(s):  
EDUARDO GONZÁLEZ-OLIVARES ◽  
JAIME MENA-LORCA ◽  
HÉCTOR MENESES-ALCAY ◽  
BETSABÉ GONZÁLEZ-YAÑEZ ◽  
JOSÉ D. FLORES
Keyword(s):  
Allee Effect ◽  
Initial Conditions ◽  
Stable Limit Cycle ◽  
Threshold Level ◽  
Prey Population ◽  
Stable Limit ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Stable Attractor ◽  
Predator Prey Models

In this work we analyze a predator-prey model proposed by A. Kent et al. in Ecol. Model.162, 233 (2003), in which two aspect of the model are considered: an effect of emigration or immigration on prey population to constant rate and a prey threshold level for predators. We prove that the system when the immigration effect is introduced in the model has a dynamics that is similar to the Rosenzweig-MacArthur model. Also, when emigration is considered in the model, we show that the behavior of the system is strongly dependent on this phenomenon, this due to the fact that trajectories are highly sensitive to the initial conditions, in similar way as when Allee effect is assumed on prey. Furthermore, we determine constraints in the parameters space for which two stable attractor exist, indicating that the extinction of both population is possible in addition with the coexistence of oscillating of populations size in a unique stable limit cycle. We also show that the consideration of a threshold level of prey population for the predator is not essential in the dynamics of the model.

OREGONATOR-BASED SIMULATION OF THE BELOUSOV–ZHABOTINSKII REACTION

2007 ◽  
Vol 17 (12) ◽  
pp. 4337-4353 ◽  
Author(s):  
MO-HONG CHOU ◽  
HSIU-CHUAN WEI ◽  
YU-TUAN LIN
Keyword(s):  
Initial Conditions ◽  
Global Analysis ◽  
Three Dimensional ◽  
Power Spectra ◽  
Stable Limit Cycle ◽  
Fractal Dimensions ◽  
Slow Manifold ◽  
Phase Lag ◽  
Stable Limit ◽  
Belousov Zhabotinskii Reaction

Some observations are made on the Belousov–Zhabotinskii reaction simulated via the Field–Noyes model, also referred to as the Oregonator, and its modification. The simulation is performed with the aid of a cell-to-cell mapping for global analysis. Regarding the standard Oregonator, a two-dimension-like region in the three-dimensional phase space is detected showing the sensitive dependence of short-term ODE integrations on initial conditions. Trajectories with initial conditions closely located in this region may experience a phase lag if they eventually approach the same stable limit cycle connected with a subcritical Hopf bifurcation. When a flow term is added to the Oregonator, chaos can be brought about to mimic the experimental finding by suitably pleating the slow manifold. Coexistent attractors now may have a chaotic member and a fractal separatrix detected by the global analysis. The above mentioned sensitive region is found to play a significant role in shaping the pleating in order for chaos to happen in a manner analogous to the "screw-type" proposed by [Rössler, 1977] as one of the two prototypes for three-variable systems. Some relevant calculations of Lyapunov exponents, fractal dimensions and power spectra are also included.

Stability of a relay dynamic system with non-linear speed sensor and delay under constant disturbance

2020 ◽  
Author(s):  
R.P. Simonyants ◽  
B.R. Khudaybergenov
Keyword(s):  
Dynamic System ◽  
Limit Cycle ◽  
Initial Conditions ◽  
Stable Limit Cycle ◽  
Approximate Determination ◽  
Cycle Phase ◽  
Stable Limit ◽  
Speed Sensor ◽  
Unstable Limit Cycle ◽  
The Stability

The paper considers the joint effect of the control delay and speed sensor output signal limiting on the stability of the relay dynamic system under the constant disturbance. It is shown that in this case a new property is detected in the system – the appearance of the unstable limit cycle. Phase trajectories are drawn to a stable limit cycle only from the area of initial conditions where their boundaries are determined by the trajectory of an unstable limit cycle. Using the method of Poincare mappings, the parameters of fixed points defining the unstable limit cycle as the boundary of the stability region are found. A simplified method for approximate determination of simple limit cycles and stability in the “large” is proposed based on the property of dynamic symmetry of the system. The method allows the study of the problem under consideration to be limited to applying shift and symmetry mappings to the switching lines.

Linear and Nonlinear Stability Analysis of a Fixed Caliper Brake During Forward and Backward Driving

2019 ◽  
Vol 141 (3) ◽  
Author(s):  
Oliver Stump ◽  
Ronaldo Nunes ◽  
Karl Häsler ◽  
Wolfgang Seemann
Keyword(s):  
Limit Cycle ◽  
Initial Conditions ◽  
Robustness Analysis ◽  
Stable Limit Cycle ◽  
Brake System ◽  
Complex Eigenvalue ◽  
Stable Limit ◽  
Nonlinear Stability Analysis ◽  
Linear And Nonlinear ◽  
Small Models

The parking maneuver of a passenger car is known by bench and vehicle testers to sometimes produce brake squeal, even though the brake system is otherwise quiet. This phenomenon is examined in this work. Pressure foil measurements at the pad caliper contact and acceleration measurements are done on a real break system in order to better understand the mechanisms of the forward–backward driving maneuver. The contact area at the caliper is under a large change during a forward and backward driving maneuver. The measurements motivate linear and nonlinear simulations. A proposal has been made to include the linear effects of parking into the standard robustness analysis with the complex eigenvalues calculation. A time integration of the full nonlinear system shows a possible stable limit cycle, when the brake pad moves from the leading to the trailing side, like in a parking maneuver. This growth of amplitude is not anticipated from the complex eigenvalue analysis (CEA), because no instable eigenvalue is found in the linearized equation of motion at that working point. This subcritical flutter-type behavior is known for small models in the literature and is examined in this paper with a more realistic brake system. It is found that the resulting error of the linearization cannot be neglected. Furthermore, different initial conditions are analyzed to narrow the zone of attraction of the stable limit cycle and the decrease of the critical friction value due to this kind of bifurcation behavior.

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