Local Bifurcation Analysis Research Articles

Complex dynamics of a three species prey-predator model with intraguild predation

‘Omnivory’, a widespread mechanism in interacting populations, is defined as feeding on more than one trophic levels. In this work we have considered a three species model consisting of prey, intermediate predator that predates upon prey and top predator with intraguild predation. It is well known that intraguild predation destabilizes the food webs consisting of three and more trophic levels, and induces chaotic oscillation for Holling type-I functional response when the intensity of the intraguild predation becomes low. Here we have considered Holling type-II functional response between intermediate predator and intraguild predator and other functional responses are assumed to be linear. First we investigate local stability and bifurcation analysis of all axial and boundary equilibrium points. We observe two types of coexistence of three species - a steady-state coexistence and an oscillatory coexistence. Although it is difficult to find the analytical expressions for stability and bifurcation of the interior equilibrium point, we have verified numerically that a stable limit cycle bifurcates from the interior equilibrium point and ultimately chaos occurs via successive period doubling bifurcations. Bifurcation diagrams have been drawn with respect to all the system parameters to explore the complete dynamics of the model. Consideration of saturating functional response instead of law of mass action can suppress the chaotic oscillation leading to stable coexistence of three species at their steady-state.

Local and global Hopf bifurcation analysis in a neutral-type neuron system with two delays

In recent years, neutral-type differential-difference equations have been applied extensively in the field of engineering, and their dynamical behaviors are more complex than that of the delay differential-difference equations. In this paper, the equations used to describe a neutral-type neural network system of differential difference equation with two delays are studied (i.e. neutral-type differential equations). Firstly, by selecting [Formula: see text], [Formula: see text] respectively as a parameter, we provide an analysis about the local stability of the zero equilibrium point of the equations, and sufficient conditions of asymptotic stability for the system are derived. Secondly, by using the theory of normal form and applying the theorem of center manifold introduced by Hassard et al., the Hopf bifurcation is found and some formulas for deciding the stability of periodic solutions and the direction of Hopf bifurcation are given. Moreover, by applying the theorem of global Hopf bifurcation, the existence of global periodic solution of the system is studied. Finally, an example is given, and some computer numerical simulations are taken to demonstrate and certify the correctness of the presented results.

Treating cancerous cells with viruses: insights from a minimal model for oncolytic virotherapy

In recent years, interest in the capability of virus particles as a treatment for cancer has increased. In this work, we present a mathematical model embodying the interaction between tumour cells and virus particles engineered to infect and destroy cancerous tissue. To quantify the effectiveness of oncolytic virotherapy, we conduct a local stability analysis and bifurcation analysis of our model. In the absence of tumour growth or viral decay, the model predicts that oncolytic virotherapy will successfully eliminate the tumour cell population for a large proportion of initial conditions. In comparison, for growing tumours and decaying viral particles there are no stable equilibria in the model; however, oscillations emerge for certain regions in our parameter space. We investigate how the period and amplitude of oscillations depend on tumour growth and viral decay. We find that higher tumour replication rates result in longer periods between oscillations and lower amplitudes for uninfected tumour cells. From our analysis, we conclude that oncolytic viruses can reduce growing tumours into a stable oscillatory state, but are insufficient to completely eradicate them. We propose that it is only with the addition of other anti-cancer agents that tumour eradication may be achieved by oncolytic virus.

Epidemic dynamics on information-driven adaptive networks

Research on the interplay between the dynamics on the network and the dynamics of the network has attracted much attention in recent years. In this work, we propose an information-driven adaptive model, where disease and disease information can evolve simultaneously. For the information-driven adaptive process, susceptible (infected) individuals who have abilities to recognize the disease would break the links of their infected (susceptible) neighbors to prevent the epidemic from further spreading. Simulation results and numerical analyses based on the pairwise approach indicate that the information-driven adaptive process can not only slow down the speed of epidemic spreading, but can also diminish the epidemic prevalence at the final state significantly. In addition, the disease spreading and information diffusion pattern on the lattice as well as on a real-world network give visual representations about how the disease is trapped into an isolated field with the information-driven adaptive process. Furthermore, we perform the local bifurcation analysis on four types of dynamical regions, including healthy, a continuous dynamic behavior, bistable and endemic, to understand the evolution of the observed dynamical behaviors. This work may shed some lights on understanding how information affects human activities on responding to epidemic spreading.

Local and global Hopf bifurcation analysis on simplified bidirectional associative memory neural networks with multiple delays

In this paper, a class of simplified bidirectional associative memory (BAM) neural networks with multiple delays are considered. By analyzing the associated characteristic transcendental equation, their linear stability is investigated and Hopf bifurcation is demonstrated. By applying Nyquist criterion, the length of delay which preserves the stability of the zero equilibrium is estimated. Some explicit results are derived for stability and direction of the bifurcating periodic orbit by using the normal form theory and center manifold arguments. Global existence of periodic orbits is also established by using a global Hopf bifurcation theorem for functional differential equations (FDE) and a Bendixson’s criterion for high-dimensional ordinary differential equations (ODE) due to Li and Muldowney. Finally, numerical simulations supporting the theoretical analysis are carried out.

Stability and bifurcation analysis of micro-electromechanical nonlinear coupling system with delay

In this paper, we study dynamics in delayed micro-electromechanical nonlinear coupling system, with particular attention focused on Hopf and Hopf-pitchfork bifurcations. Based on the distribution of eigenvalues, we prove that a sequence of Hopf and Hopf-pitchfork bifurcations occur at the trivial equilibrium as the delay increases and obtain the critical values of two types of bifurcations. Next, by applying the multiple time scales method, the normal forms near the Hopf and Hopf-pitchfork bifurcations critical points are derived. Finally, bifurcation analysis and numerical simulations are presented to demonstrate the application of the theoretical results. We show the regions near above bifurcation critical points in which the micro-electromechanical nonlinear coupling system exists stable fixed point or stable periodic solution. Detailed numerical analysis using MATLAB extends the local bifurcation analysis to a global picture, and stable windows are observed as we change control parameters. Namely, the stable fixed point and stable periodic solution can exist in large regions of unfolding parameters as the unfolding parameters increase away from the critical value.

Treating Cancerous Cells with Viruses: Insights from Aminimal Model for Oncolytic Virotherapy

In recent years, interest in the capability of virus particles as a treatment for cancer has increased. In this work, we present a mathematical model embodying the interaction between tumour cells and virus particles engineered to infect and destroy cancerous tissue. To quantify the effectiveness of oncolytic virotherapy, we conduct a local stability analysis and bifurcation analysis of our model. In the absence of tumour growth or viral decay, the model predicts that oncolytic virotherapy will successfully eliminate the tumour cell population for a large proportion of initial conditions. In comparison, for growing tumours and decaying viral particles there are no stable equilibria in the model; however, oscillations emerge for certain regions in our parameter space. We investigate how the period and amplitude of oscillations depend on tumour growth and viral decay. We find that higher tumour replication rates result in longer periods between oscillations and lower amplitudes for uninfected tumour cells. From our analysis, we conclude that oncolytic viruses can reduce growing tumours into a stable oscillatory state, but are insufficient to completely eradicate them. We propose that it is only with the addition of other anti-cancer agents that tumour eradication may be achieved by oncolytic virus.

Stability and Bifurcation Analysis of Cellular Neural Networks with Discrete and Distributed Delays

This work is devoted to study the application of contraction mapping theorem to analyze a class of n-dimensional neural network model consisting of multiple delays with self-feedback and studying the dynamic behaviour of the system and obtain stability condition under the effect of time delay. Some novel and sufficient conditions are derived to ensure the existence and uniqueness of the solution and the global stability of the considered system at the same time under delay effect. Global asymptotic stability is discussed by constructing suitable Lyapunov functional. To check the dynamics of system, Local Hopf bifurcation analysis has also been done when the number of neurons reduces to two followed by finding the direction and stability of Hopf bifurcation. In the end, the analytical findings are validated using two numerical examples.

Transitions between asynchronous and synchronous states: a theory of correlations in small neural circuits

The study of correlations in neural circuits of different size, from the small size of cortical microcolumns to the large-scale organization of distributed networks studied with functional imaging, is a topic of central importance to systems neuroscience. However, a theory that explains how the parameters of mesoscopic networks composed of a few tens of neurons affect the underlying correlation structure is still missing. Here we consider a theory that can be applied to networks of arbitrary size with multiple populations of homogeneous fully-connected neurons, and we focus its analysis to a case of two populations of small size. We combine the analysis of local bifurcations of the dynamics of these networks with the analytical calculation of their cross-correlations. We study the correlation structure in different regimes, showing that a variation of the external stimuli causes the network to switch from asynchronous states, characterized by weak correlation and low variability, to synchronous states characterized by strong correlations and wide temporal fluctuations. We show that asynchronous states are generated by strong stimuli, while synchronous states occur through critical slowing down when the stimulus moves the network close to a local bifurcation. In particular, strongly positive correlations occur at the saddle-node and Andronov-Hopf bifurcations of the network, while strongly negative correlations occur when the network undergoes a spontaneous symmetry-breaking at the branching-point bifurcations. These results show how the correlation structure of firing-rate network models is strongly modulated by the external stimuli, even keeping the anatomical connections fixed. These results also suggest an effective mechanism through which biological networks may dynamically modulate the encoding and integration of sensory information.

Spiky Steady States of a Chemotaxis System with Singular Sensitivity

This paper is concerned with the steady state problem of a chemotaxis model with singular sensitivity function in a one dimensional spatial domain. Using the chemotactic coefficient $$\chi $$ as the bifurcation parameter, we perform local and global bifurcation analysis for the model. It is shown that positive monotone steady states exist as long as $$\chi $$ is larger than the first bifurcation value $$\bar{\chi }_1.$$ We further obtain asymptotic profiles of these steady states, as $$\chi $$ becomes large. In particular, our results show that the cell density function forms a spike, which models the important physical phenomenon of cell aggregation.

Local Bifurcation Analysis of a Fractional-Order Dynamic Model of Genetic Regulatory Networks with Delays

In this paper, we propose a delayed fractional-order gene regulatory network model. Firstly, the sum of delays is chosen as the bifurcation parameter, and the conditions of the existence for Hopf bifurcations are achieved through analyzing its characteristic equation. Secondly, it is shown that the fractional order can be effectively manipulated to control the dynamics of such network, and the stability domain can be changed with different fractional orders. The fractional-order genetic network can generate a Hopf bifurcation (oscillation appears) as the sum of delays passes through some critical values. Therefore, we can achieve some desirable dynamical behaviors by choosing the appropriate fractional order. Finally, numerical simulations are carried out to illustrate the validity of our theoretical analysis.

Secondary bifurcation of a shearable rod with nonlinear spring supports

Abstract The postbuckling behavior of a compressed elastic rod is studied. The rod is supported by two identical hardening spring supports at clamped ends. The constitutive equations of the rod take into account the effect of shear. For local bifurcation analysis the Liapunov-Schmidt method is used. It is shown that for the critical value of auxiliary parameter the lowest eigenvalue becomes double. This fact and double Z 2 symmetry lead to the occurrence of secondary bifurcations for the values of auxiliary parameter near the critical one. For primary and secondary branches asymptotic expansions are constructed. Depending on the values of parameters of nonlinear spring supports and shear rigidity five different groups of bifurcation diagrams are found.

On the bifurcation of Marotto’s map and its application in image encryption

The aim of this paper is to address the codimension-one bifurcation of Marotto’s map and its utility in image encryption. First of all, local stability analysis and local bifurcation analysis of fixed points of the considered map are investigated in details. According to the classical bifurcation theory and the center manifold theorem, the map exhibits various bifurcation types such as transcritical, flip and Neimark–Sacker bifurcations. Second of all, the map is proven to be chaotic in the sense of Marotto. Since image encryption based on chaotic maps is very promising for cryptography, Marotto’s map, compound chaos, and spatiotemporal chaos are combined to encrypt and decrypt images. Numerical simulations agree with the analytical framework for the complex dynamics of the map. Furthermore, different test images are used to demonstrate the effectiveness of the method implemented for encryption.

Local bifurcation analysis of a delayed fractional-order dynamic model of dual congestion control algorithms

In this paper, we propose a delayed fractional-order congestion control model which is more accurate than the original integer-order model when depicting the dual congestion control algorithms. The presence of fractional orders requires the use of suitable criteria which usually make the analytical work so harder. Based on the stability theorems on delayed fractionalorder differential equations, we study the issue of the stability and bifurcations for such a model by choosing the communication delay as the bifurcation parameter. By analyzing the associated characteristic equation, some explicit conditions for the local stability of the equilibrium are given for the delayed fractionalorder model of congestion control algorithms. Moreover, the Hopf bifurcation conditions for general delayed fractional-order systems are proposed. The existence of Hopf bifurcations at the equilibrium is established. The critical values of the delay are identified, where the Hopf bifurcations occur and a family of oscillations bifurcate from the equilibrium. Same as the delay, the fractional order normally plays an important role in the dynamics of delayed fractional-order systems. It is found that the critical value of Hopf bifurcations is crucially dependent on the fractional order. Finally, numerical simulations are carried out to illustrate the main results.

Walking dynamics of the passive compass-gait model under OGY-based state-feedback control: Analysis of local bifurcations via the hybrid Poincaré map

In our previous work, we have analyzed the passive dynamic walking of the compass-gait biped model under the OGY-based state-feedback control using the impulsive hybrid nonlinear dynamics. Such study was carried out through bifurcation diagrams. It was shown that the controlled bipedal gait exhibits attractive nonlinear phenomena such as the cyclic-fold (saddle-node) bifurcation, the period-doubling (flip) bifurcation and chaos. Moreover, we revealed that, using the controlled continuous-time dynamics, we encountered a problem in finding, identifying and hence following branches of (un)stable solutions in order to characterize local bifurcations. The present paper solves such problem and then provides a further investigation of the controlled bipedal walking dynamics using the developed analytical expression of the controlled hybrid Poincaré map. Thus, we show that analysis via such Poincaré map allows to follow branches of both stable and unstable fixed points in bifurcation diagrams and hence to explore the complete dynamics of the controlled compass-gait biped model. We demonstrate the generation, other than the conventional local bifurcations in bipedal walking, i.e. the flip bifurcation and the saddle-node bifurcation, of a saddle-saddle bifurcation, a subcritical flip bifurcation and a new type of a local bifurcation, the saddle-flip bifurcation. In addition, to further understand the occurrence of the local bifurcations, we present an analysis with a two-parameter bifurcation diagram. Some new hidden walking dynamics are identified.

Local Bifurcation Analysis and Global Dynamics Estimation of a Novel 4-Dimensional Hyperchaotic System

In this paper, we present a novel 4-dimensional (4D) smooth quadratic autonomous hyperchaotic system with complex dynamics. In order to investigate the dynamics evolution of the system, the Lyapunov exponent spectrum, bifurcation diagram and various phase portraits are provided. The local dynamics of this hyperchaotic system, such as the stability, pitchfork bifurcation, and Hopf bifurcation of equilibrium point, are analyzed by using the center manifold theorem and bifurcation theory. About the global dynamics, the ultimate bound sets of the system are found by combining the Lyapunov function method and appropriate optimization method. Numerical simulations are given to demonstrate the emergence of the two bifurcations and show the ultimate boundary regions.

Analysis of local bifurcations in a channel subjected to non-uniform axial heating

The effect of non-uniform axial heating on density wave oscillations of a two-phase flow channel is studied. This study becomes relevant in view of designing and modelling of nuclear reactor heat removal systems as several reactor fuel rods (Boiling Water Reactors) exhibit non-uniform axial heat flux profiles. In this work a Nodalized Reduced Order Model (NROM) has been developed The unique feature of the model is that, using this model the stability behaviour of two-phase flow systems with a wide range of axial heat flux variations (single humped and double humped) can be studied. In addition to these, the model behaves as an auto-adaptive scheme, where the mesh sizes between consecutive nodes automatically vary during the system evolution according to the provided shape of axial heat flux. This helps to minimize the computational effort required to achieve significant accuracy to capture the state variable variations. The effects of single humped and various double humped axial heating profiles on local bifurcation have been studied. It has been observed in this study that in addition to the linear stability characteristic the nonlinear stability behaviour (depicted by the position of generalized Hopf points and occurrences of limit cycles) also changes significantly as the axial heat flux changes from uniform to non-uniform even though the total heat rate provided remains same. Moreover, a generalized description of the cause due to which these changes occur in stability behaviour of the system is also presented.

Non-linear dynamics and control of an automotive suspension system based on local and global bifurcation analysis

This paper details the non-linear dynamic behaviours and control of a non-linear semi-active suspension system using a quarter-car model under kinematic excitation by a road surface profile. The results of local and global bifurcation analysis indicate that the hysteretic non-linear characteristics of damping force cause the suspension system to exhibit codimension-two bifurcation, resulting in homoclinic orbits and a pitchfork bifurcation. The complex dynamic behaviour of automotive suspension systems was examined using a bifurcation diagram, phase portraits, a Poincare map, and frequency spectra. We also used Lyapunov exponent to identify the occurrence of chaotic motion and verify our analysis. Finally, a dither signal control was used to convert chaotic behaviours into periodic motion. Simulation results verify the effectiveness of the proposed control method.

Non-linear dynamics and control of an automotive suspension system based on local and global bifurcation analysis

This paper details the non-linear dynamic behaviours and control of a non-linear semi-active suspension system using a quarter-car model under kinematic excitation by a road surface profile. The results of local and global bifurcation analysis indicate that the hysteretic non-linear characteristics of damping force cause the suspension system to exhibit codimension-two bifurcation, resulting in homoclinic orbits and a pitchfork bifurcation. The complex dynamic behaviour of automotive suspension systems was examined using a bifurcation diagram, phase portraits, a Poincaré map, and frequency spectra. We also used Lyapunov exponent to identify the occurrence of chaotic motion and verify our analysis. Finally, a dither signal control was used to convert chaotic behaviours into periodic motion. Simulation results verify the effectiveness of the proposed control method.

Simultaneous Effects of Prey Defence and Predator Infection on a Predator-Prey System

Defensive behaviour and infection both are important factors in population dynamics. Defence may help to protect prey from its predator and infection can alter the quantitative dynamics of its hosts. We analyse here a predator-prey model in presence of prey defence, incorporated as the prey cover in the model and some parasitic infection in the predator population. The objective is to explore the consequences of a predator-prey interaction in presence of predator infection and prey cover. Proof of local stability and bifurcation analysis of all equilibrium points are given in terms of ecological and epidemiological reproduction numbers. Different simulation results are presented to illustrate the analytical findings. It is shown that infection acts like a stabilizing agent in an otherwise unstable predator-prey system. Prey defence may further enhance the stability, but an enzootic equilibrium may appear if defence crosses some critical value.