Showing papers on "Bifurcation diagram published in 2013"
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30 Jul 2013TL;DR: The Dynamic Bifurcation Theory of Functional Differential Equations (DBDE) as discussed by the authors is an extension of FDEs with center manifold reduction and Lyapunov-Schmidt reduction.
Abstract: Introduction to Dynamic Bifurcation Theory.- Introduction to Functional Differential Equations.-Center Manifold Reduction.- Normal form theory.- Lyapunov-Schmidt Reduction.- Degree theory.- Bifurcation in Symmetric FDEs .
120 citations
TL;DR: In this paper, the authors investigated the process of pattern formation in a two dimensional domain for a reaction-diffusion system with nonlinear diffusion terms and the competitive Lotka-Volterra kinetics.
Abstract: In this work we investigate the process of pattern formation in a two dimensional domain for a reaction–diffusion system with nonlinear diffusion terms and the competitive Lotka–Volterra kinetics. The linear stability analysis shows that cross-diffusion, through Turing bifurcation, is the key mechanism for the formation of spatial patterns. We show that the bifurcation can be regular, degenerate non-resonant and resonant. We use multiple scales expansions to derive the amplitude equations appropriate for each case and show that the system supports patterns like rolls, squares, mixed-mode patterns, supersquares, and hexagonal patterns.
104 citations
TL;DR: In this article, the effect of constant-yield predator harvesting on the dynamics of a Leslie-Gower type predator-prey model was studied. And the authors showed that the model has a Bogdanov-Takens singularity (cusp case) or a weak focus of multiplicity two for some parameter values, respectively.
Abstract: In this paper we study the effect of constant-yield predator harvesting on the dynamics
of a Leslie-Gower type predator-prey model. It is shown that the model has a Bogdanov-Takens singularity (cusp case)
of codimension 3 or a weak focus of multiplicity two for some parameter values, respectively. Saddle-node
bifurcation, repelling and attracting Bogdanov-Takens bifurcations, supercritical and subcritical
Hopf bifurcations, and degenerate Hopf bifurcation are shown as the values of parameters vary. Hence, there are
different parameter values for which the model has a homoclinic loop or two limit cycles.
It is also proven that there exists a critical harvesting value such that the predator specie goes extinct
for all admissible initial densities of both species when the harvest rate is greater than the critical value.
These results indicate that the dynamical behavior of the model is very sensitive to the constant-yield predator
harvesting and the initial densities of both species and it requires careful management in the applied conservation
and renewable resource contexts. Numerical simulations, including the repelling and attracting Bogdanov-Takens
bifurcation diagrams and corresponding phase portraits, two limit cycles, the coexistence of a stable homoclinic
loop and an unstable limit cycle, and a stable limit cycle
enclosing an unstable multiple focus with multiplicity one, are presented which not only support the theoretical
analysis but also indicate the existence of Bogdanov-Takens bifurcation (cusp case) of codimension 3. These results
reveal far richer and much more complex dynamics compared to the model without harvesting
or with only constant-yield prey harvesting.
97 citations
TL;DR: In this article, the authors investigate stochastic bifurcation for a tumor-immune system in the presence of a symmetric non-Gaussian Levy noise and find that the dynamics induced by Gaussian and nonGaussian noises are quite different.
Abstract: In this paper, we investigate stochastic bifurcation for a tumor–immune system in the presence of a symmetric non-Gaussian Levy noise. Stationary probability density functions will be numerically obtained to define stochastic bifurcation via the criteria of its qualitative change, and bifurcation diagram at parameter plane is presented to illustrate the bifurcation analysis versus noise intensity and stability index. The effects of both noise intensity and stability index on the average tumor population are also analyzed by simulation calculation. We find that stochastic dynamics induced by Gaussian and non-Gaussian Levy noises are quite different.
72 citations
TL;DR: The improved digital secure communication scheme is achieved based on hybrid synchronization in coupled fractional-order complex Chen system, that means anti-synchronization in real part of state variables and projective synchronization in imaginary part, respectively.
Abstract: In this paper, a novel dynamic system, the fractional-order complex Chen system, is presented for the first time. Dynamic behaviors of system are studied analytically and numerically. Different routes to chaos are shown, and diverse kinds of motions are identified and exhibited by means of bifurcation diagram, portrait phase and the largest Lyapunov exponent. Secondly, an application to digital secure communication based on the novel system is proposed, in which security is enhanced by continually switching different orders of derivative in an irregular pattern. Furthermore, making full use of the advantage of high-capacity transmission of complex system, the improved digital secure communication scheme is achieved based on hybrid synchronization in coupled fractional-order complex Chen system, that means anti-synchronization in real part of state variables and projective synchronization in imaginary part, respectively. The corresponding numerical simulations demonstrate the effectiveness and feasibility of the proposed schemes.
68 citations
TL;DR: Bifurcations of spatially nonhomogeneous periodic orbits and steady state solutions are rigorously proved for a reaction-diffusion system modeling Schnakenberg chemical reaction in this article.
Abstract: Bifurcations of spatially nonhomogeneous periodic orbits and steady state solutions are rigorously proved for a reaction–diffusion system modeling Schnakenberg chemical reaction The existence of these patterned solutions shows the richness of the spatiotemporal dynamics such as oscillatory behavior and spatial patterns
68 citations
TL;DR: In this article, two approaches to incorporate the effects of rotation and curvature in scalar eddy viscosity models are explored: modified coefficients approach and bifurcation approach.
66 citations
TL;DR: Three different generalized logistic maps are introduced with arbitrary powers which can be reduced to the conventional logistic map, and the added parameter (arbitrary power) increases the degree of freedom of each map and gives a versatile response that can fit many applications.
62 citations
TL;DR: In this paper, the authors used rigorous numerics to compute several global smooth branches of steady states for a system of three reaction-diffusion PDEs introduced by Iida et al.
Abstract: In this paper, we use rigorous numerics to compute several global smooth branches of steady states for a system of three reaction-diffusion PDEs introduced by Iida et al. [J. Math. Biol. 53(4):617---641, 2006] to study the effect of cross-diffusion in competitive interactions. An explicit and mathematically rigorous construction of a global bifurcation diagram is done, except in small neighborhoods of the bifurcations. The proposed method, even though influenced by the work of van den Berg et al. [Math. Comput. 79(271):1565---1584, 2010], introduces new analytic estimates, a new gluing-free approach for the construction of global smooth branches and provides a detailed analysis of the choice of the parameters to be made in order to maximize the chances of performing successfully the computational proofs.
59 citations
TL;DR: In this article, an example of a Rijke tube model with an explicit time delay is presented, and a linear stability analysis of the model is performed to identify parameter values at the onset of linear instability via a Hopf bifurcation.
Abstract: This paper analyses subcritical transition to instability, also known as triggering in thermoacoustic systems, with an example of a Rijke tube model with an explicit time delay. Linear stability analysis of the thermoacoustic system is performed to identify parameter values at the onset of linear instability via a Hopf bifurcation. We then use the method of multiple scales to recast the model of a general thermoacoustic system near the Hopf point into the Stuart–Landau equation. From the Stuart–Landau equation, the relation between the nonlinearity in the model and the criticality of the ensuing bifurcation is derived. The specific example of a model for a horizontal Rijke tube is shown to lose stability through a subcritical Hopf bifurcation as a consequence of the nonlinearity in the model for the unsteady heat release rate. Analytical estimates are obtained for the triggering amplitudes close to the critical values of the bifurcation parameter corresponding to loss of linear stability. The unstable limit cycles born from the subcritical Hopf bifurcation undergo a fold bifurcation to become stable and create a region of bistability or hysteresis. Estimates are obtained for the region of bistability by locating the fold points from a fully nonlinear analysis using the method of harmonic balance. These analytical estimates help to identify parameter regions where triggering is possible. Results obtained from analytical methods compare reasonably well with results obtained from both experiments and numerical continuation.
57 citations
TL;DR: In this article, a three dimensional eco-epidemiological model consisting of susceptible prey, infected prey and predator is proposed and analyzed, where the parameter delay is introduced in the model system for considering the time taken by a susceptible prey to become infected.
Abstract: A three dimensional ecoepidemiological model consisting of susceptible prey, infected prey and predator is proposed and analysed in the present work. The parameter delay is introduced in the model system for considering the time taken by a susceptible prey to become infected. Mathematically we analyze the dynamics of the system such as, boundedness of the solutions, existence of non-negative equilibria, local and global stability of interior equilibrium point. Next we choose delay as a bifurcation parameter to examine the existence of the Hopf bifurcation of the system around its interior equilibrium. Moreover we use the normal form method and center manifold theorem to investigate the direction of the Hopf bifurcation and stability of the bifurcating limit cycle. Some numerical simulations are carried out to support the analytical results.
TL;DR: In this article, the existence of a global bifurcation branch of 2π -periodic, smooth, traveling-wave solutions of the Whitham equation is proved.
Abstract: We prove the existence of a global bifurcation branch of 2π -periodic, smooth, traveling-wave solutions of the Whitham equation. It is shown that any subset of solutions in the global branch contains a sequence which converges uniformly to some solution of Holder class C α , α < 1/2. Bifurcation formulas are given, as well as some properties along the global bifurcation branch. In addition, a spectral scheme for computing approximations to those waves is put forward, and several numerical results along the global bifurcation branch are presented, including the presence of a turning point and a ‘highest’, cusped wave. Both analytic and numerical results are compared to traveling-wave solutions of the KdV equation.
TL;DR: The existence and global bifurcation for periodic solutions of a class of differential variational inequalities are studied by using the topological degree theory for multivalued maps and the method of guiding functions.
Abstract: In this paper, by using the topological degree theory for multivalued maps and the method of guiding functions, the existence and global bifurcation for periodic solutions of a class of differential variational inequalities are studied.
TL;DR: The bifurcation analysis of a predator–prey system of Holling and Leslie type with constant-yield prey harvesting is carried out and it is shown that the model has a Bogdanov–Takens singularity of codimension at least 4 for some parameter values.
Abstract: The bifurcation analysis of a predator–prey system of Holling and Leslie type with constant-yield prey harvesting is carried out in this paper It is shown that the model has a Bogdanov–Takens singularity (cusp case) of codimension at least 4 for some parameter values Various kinds of bifurcations, such as saddle-node bifurcation, Hopf bifurcation, repelling and attracting Bogdanov–Takens bifurcations of codimensions 2 and 3, are also shown in the model as parameters vary Hence, there are different parameter values for which the model has a limit cycle, a homoclinic loop, two limit cycles, or a limit cycle coexisting with a homoclinic loop These results present far richer dynamics compared to the model with no harvesting Numerical simulations, including the repelling and attracting Bogdanov–Takens bifurcation diagrams and corresponding phase portraits, and the existence of two limit cycles or an unstable limit cycle enclosing a stable multiple focus with multiplicity one, are also given to support the theoretical analysis
TL;DR: A memristive Murali–Lakshmanan–Chua (MLC) circuit is built by replacing the nonlinear element of an ordinary MLC circuit with a three-segment piecewise-linear active flux controlled memristor, which introduces two discontinuity boundaries or switching manifolds in the circuit topology.
Abstract: In this paper, a memristive Murali-Lakshmanan-Chua (MLC) circuit is built by replacing the nonlinear element of an ordinary MLC circuit, namely the Chua's diode, with a three segment piecewise linear active flux controlled memristor. The bistability nature of the memristor introduces two discontinuty boundaries or switching manifolds in the circuit topology. As a result, the circuit becomes a piecewise smooth system of second order. Grazing bifurcations, which are essentially a form of discontinuity induced non-smooth bifurcations, occur at these boundaries and govern the dynamics of the circuit. While the interaction of the memristor aided self oscillations of the circuit and the external sinusoidal forcing result in the phenomenon of beats occurring in the circuit, grazing bifurcations endow them with chaotic and hyper chaotic nature. In addition the circuit admits a codimension-5 bifurcation and transient hyper chaos. Grazing bifurcations as well as other behaviors have been analyzed numerically using time series plots, phase portraits, bifurcation diagram, power spectra and Lyapunov spectrum, as well as the recent 0-1 K test for chaos, obtained after constructing a proper Zero Time Discontinuity Map (ZDM) and Poincare Discontinuity Map (PDM) analytically. Multisim simulations using a model of piecewise linear memristor have also been used to confirm some of the behaviors.
TL;DR: The asymptotical method is used to analyze phenomenological bifurcation and it is found that the neuronal activity of spiking and bursting chaos remains for finite values of the noise intensity.
Abstract: We analyze the bifurcations occurring in the 3D Hindmarsh-Rose neuronal model with and without random signal. When under a sufficient stimulus, the neuron activity takes place; we observe various types of bifurcations that lead to chaotic transitions. Beside the equilibrium solutions and their stability, we also investigate the deterministic bifurcation. It appears that the neuronal activity consists of chaotic transitions between two periodic phases called bursting and spiking solutions. The stochastic bifurcation, defined as a sudden change in character of a stochastic attractor when the bifurcation parameter of the system passes through a critical value, or under certain condition as the collision of a stochastic attractor with a stochastic saddle, occurs when a random Gaussian signal is added. Our study reveals two kinds of stochastic bifurcation: the phenomenological bifurcation (P-bifurcations) and the dynamical bifurcation (D-bifurcations). The asymptotical method is used to analyze phenomenological bifurcation. We find that the neuronal activity of spiking and bursting chaos remains for finite values of the noise intensity.
TL;DR: In this paper, the fractional-order sliding-mode controller is designed to control a fractional order hyperchaotic system and the minimum orders for chaos and hyperchaos to exist in such systems are 2.89 and 3.66.
Abstract: In this paper we numerically investigate the fractional-order sliding-mode control for a novel fractional-order hyperchaotic system. Firstly, the dynamic analysis approaches of the hyperchaotic system involving phase portraits, Lyapunov exponents, bifurcation diagram, Lyapunov dimension, and Poincare maps are investigated. Then the fractional-order generalizations of the chaotic and hyperchaotic systems are studied briefly. The minimum orders we found for chaos and hyperchaos to exist in such systems are 2.89 and 3.66, respectively. Finally, the fractional-order sliding-mode controller is designed to control the fractional-order hyperchaotic system. Numerical experimental examples are shown to verify the theoretical results.
TL;DR: In this paper, the authors investigate qualitative changes of probability distributions (stochastic bifurcations, coherence resonance, and stochastic synchronization) in nonlinear dynamical systems near a sub-critical Hopf Bifurcation.
Abstract: We analyze noise-induced phenomena in nonlinear dynamical systems near a subcritical Hopf bifurcation. We investigate qualitative changes of probability distributions (stochastic bifurcations), coherence resonance, and stochastic synchronization. These effects are studied in dynamical systems for which a subcritical Hopf bifurcation occurs. We perform analytical calculations, numerical simulations and experiments on an electronic circuit. For the generalized Van der Pol model we uncover the similarities between the behavior of a self-sustained oscillator characterized by a subcritical Hopf bifurcation and an excitable system. The analogy is manifested through coherence resonance and stochastic synchronization. In particular, we show both experimentally and numerically that stochastic oscillations that appear due to noise in a system with hard excitation, can be partially synchronized even outside the oscillatory regime of the deterministic system.
TL;DR: This paper theoretically demonstrate and numerically evidence that the emergence of geometric power series in the vicinity of simple bifurcation points is a generic behavior, and proposes to use this hallmark as a bIfurcation indicator to locate and compute very efficiently any simple b ifurcation point.
TL;DR: In this article, a memristive Murali-Lakshmanan-Chua (MLC) circuit is built by replacing the nonlinear element of an ordinary MLC circuit, namely the Chua's diode, with a three-segment piecewise-linear active flux controlled memristor.
Abstract: In this paper, a memristive Murali–Lakshmanan–Chua (MLC) circuit is built by replacing the nonlinear element of an ordinary MLC circuit, namely the Chua's diode, with a three-segment piecewise-linear active flux controlled memristor. The bistability nature of the memristor introduces two discontinuity boundaries or switching manifolds in the circuit topology. As a result, the circuit becomes a piecewise-smooth system of second order. Grazing bifurcations, which are essentially a form of discontinuity-induced nonsmooth bifurcations, occur at these boundaries and govern the dynamics of the circuit. While the interaction of the memristor-aided self oscillations of the circuit and the external sinusoidal forcing result in the phenomenon of beats occurring in the circuit, grazing bifurcations endow them with chaotic and hyperchaotic nature. In addition, the circuit admits a codimension-5 bifurcation and transient hyperchaos. Grazing bifurcations as well as other behaviors have been analyzed numerically using time series plots, phase portraits, bifurcation diagram, power spectra and Lyapunov spectrum, as well as the recent 0–1 K test for chaos, obtained after constructing a proper Zero Time Discontinuity Map (ZDM) and Poincare Discontinuity Map (PDM) analytically. Multisim simulations using a model of piecewise linear memristor have also been used to confirm some of the behaviors.
TL;DR: In this article, the authors generalized the economic growth model with endogenous labor shift under a dual economy proposed by Cai (Applied Mathematics Letters 21, 774-779 (2008)) by introducing a time delay in the physical capital.
Abstract: The economic growth model with endogenous labor shift under a dual economy proposed by Cai (Applied Mathematics Letters 21, 774-779 (2008)) is generalized in this paper by introducing a time delay in the physical capital. By choosing the delay as a bifurcation parameter, it is proved that the delayed model has unique nonzero equilibrium and a Hopf bifurcation is proven to exist as the delay crosses a critical value. Moreover the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are investigated in this paper by applying the center manifold theorem and the normal form theory.
TL;DR: The rich and complicated dynamics exhibit that the model is very sensitive to parameter perturbations, which has important implications for disease control of endangered species.
Abstract: In this paper we completely study bifurcations of an epidemic model with five parameters introduced by Hilker et al. (Am Nat 173:72–88, 2009), which describes the joint interplay of a strong Allee effect and infectious diseases in a single population. Existence of multiple positive equilibria and all kinds of bifurcation are examined as well as related dynamical behavior. It is shown that the model undergoes a series of bifurcations such as saddle-node bifurcation, pitchfork bifurcation, Bogdanov–Takens bifurcation, degenerate Hopf bifurcation of codimension two and degenerate elliptic type Bogdanov–Takens bifurcation of codimension three. Respective bifurcation surfaces in five-dimensional parameter spaces and related dynamical behavior are obtained. These theoretical conclusions confirm their numerical simulations and conjectures by Hilker et al., and reveal some new bifurcation phenomena which are not observed in Hilker et al. (Am Nat 173:72–88, 2009). The rich and complicated dynamics exhibit that the model is very sensitive to parameter perturbations, which has important implications for disease control of endangered species.
TL;DR: A two-dimensional delay differential system with two delays in which the time delays are used as the bifurcation parameter is considered and the dynamical behaviors of a model describing the interaction between tumor cells and effector cells of the immune system are studied.
Abstract: In this paper, we consider a two-dimensional delay differential system with two delays. By analyzing the distribution of eigenvalues, linear stability of the equilibria and existence of Hopf, Bautin, and Hopf--Hopf bifurcations are obtained in which the time delays are used as the bifurcation parameter. General formula for the direction, period, and stability of the bifurcated periodic solutions are given for codimension one and codimension two bifurcations, including Hopf bifurcation, Bautin bifurcation, and Hopf--Hopf bifurcation. As an application, we study the dynamical behaviors of a model describing the interaction between tumor cells and effector cells of the immune system. Numerical examples and simulations are presented to illustrate the obtained results.
TL;DR: In this paper, the authors considered the differential-algebraic predator-prey model with predator harvesting and two delays and analyzed the stability and Hopf bifurcation of the proposed system.
Abstract: In this paper, we consider the differential-algebraic predator–prey model with predator harvesting and two delays. By using the new normal form of differential-algebraic systems, center manifold theorem and bifurcation theory, we analyze the stability and the Hopf bifurcation of the proposed system. In addition, the new effective analytical method enriches the toolbox for the qualitative analysis of the delayed differential-algebraic systems. Finally, numerical simulations are given to show the consistency with theoretical analysis obtained here.
TL;DR: In this article, the authors investigate a non-invasive, locally stabilizing control scheme necessary for an experimental bifurcation analysis and propose a sequence of experiments that allow one to choose optimal control gains, filter parameters and settings for a continuation method without a priori study of a model.
TL;DR: In this paper, the existence and structure of the positive solutions of a simple superlinear indefinite semilinear elliptic model under nonhomogeneous boundary conditions was analyzed under different parameter values.
Abstract: This paper
analyzes the existence and structure of the positive solutions of a
very simple superlinear indefinite semilinear elliptic prototype
model under non-homogeneous boundary conditions, measured by $M\leq
\infty$. Rather strikingly, there are ranges of values of the
parameters involved in its setting for which the model admits an
arbitrarily large number of positive solutions, as a result of their
fast oscillatory behavior, for sufficiently large $M$. Further,
using the amplitude of the superlinear term as the main bifurcation
parameter, we can ascertain the global bifurcation diagram of the
positive solutions. This seems to be the first work where these
multiplicity results have been documented.
TL;DR: In this article, the authors cast neural field models with transmission delays as abstract delay differential equations (DDE) and derive a characteristic equation for a Hopf bifurcation.
Abstract: Neural fieldmodels with transmission delays may be cast as abstract delay differential equations (DDE). The theory of dual semigroups (also called sun-star calculus) provides a natural framework for the analysis of a broad class of delay equations, among which DDE. In particular, it may be used advantageously for the investigation of stability and bifurcation of steady states. After introducing the neural field model in its basic functional analytic setting and discussing its spectral properties, we elaborate extensively an example and derive a characteristic equation. Under certain conditions the associated equilibrium may destabilise in a Hopf bifurcation. Furthermore, two Hopf curves may intersect in a double Hopf point in a two-dimensional parameter space. We provide general formulas for the corresponding critical normal form coefficients, evaluate these numerically and interpret the results.
TL;DR: In this article, the authors consider 2D flows with a homoclinic figure-eight to a dissipative saddle and derive the bifurcation diagram using topological techniques.
Abstract: We consider 2D flows with a homoclinic figure-eight to a dissipative saddle. We study the rich dynamics that such a system exhibits under a periodic forcing. First, we derive the bifurcation diagram using topological techniques. In particular, there is a homoclinic zone in the parameter space with a non-smooth boundary. We provide a complete explanation of this phenomenon relating it to primary quadratic homoclinic tangency curves which end up at some cubic tangency (cusp) points. We also describe the possible attractors that exist (and may coexist) in the system. A main goal of this work is to show how the previous qualitative description can be complemented with quantitative global information. To this end, we introduce a return map model which can be seen as the simplest one which is 'universal' in some sense. We carry out several numerical experiments on the model, to check that all the objects predicted to exist by the theory are found in the model, and also to investigate new properties of the system.
TL;DR: In this article, a delayed neural network model with unidirectional coupling is considered and zero-Hopf bifurcation is studied by using the center manifold reduction and the normal form method for retarded functional differential equation.
Abstract: In this paper, a delayed neural network model with unidirectional coupling is considered. Zero–Hopf bifurcation is studied by using the center manifold reduction and the normal form method for retarded functional differential equation. We get the versal unfolding of the norm form at the zero–Hopf singularity and show that the model can exhibit pitchfork, Hopf bifurcation, and double Hopf bifurcation is also found to occur in this model. Some numerical simulations are given to support the analytic results.
TL;DR: In this article, a tri-neuron BAM neural network model with multiple delays is considered and the connection topology of the network plays a fundamental role in classifying the rich dynamics and bifurcation phenomena.
Abstract: In this paper, a tri-neuron BAM neural network model with multiple delays is considered We show that the connection topology of the network plays a fundamental role in classifying the rich dynamics and bifurcation phenomena There is a wide range of different dynamical behaviors which can be produced by varying the coupling strength By choosing the connected weights c
21 and c
31 (the connection weights through the neurons from J-layer to I-layer) as bifurcation parameters, the critical values where a Bogdanov–Takens bifurcation occurs are derived Then, by computing the normal forms for the system, the bifurcation diagrams are obtained Furthermore, some interesting phenomena, such as saddle-node bifurcation, pitchfork bifurcation, homoclinic bifurcation, heteroclinic bifurcation and double limit cycle bifurcation are found by choosing the different connection strengths Some numerical simulations are given to support the analytic results