In this work, we consider a general conductance-based neuron model with the inclusion of the acetycholine sensitive, M-current. We study bifurcations in the parameter space consisting of the applied current, $I_{app}$ the maximal conductance of the M-current, $g_M$, and the conductance of the leak current, $g_L$. We give precise conditions for the model that ensure the existence of a Bogdanov-Takens (BT) point and show such a point can occur by varying $I_{app}$ and $g_{M}$. We discuss the case when the BT point becomes a Bogdanov-Takens-Cusp (BTC) point and show that such a point can occur in the three dimensional parameter space. The results of the bifurcation analysis are applied to different neuronal models and are verified and supplemented by numerical bifurcation diagrams generated using the package MATCONT. We conclude that there is a transition in the neuronal excitability type organized by the BT point and the neuron switches from Class-I to Class-II as conductance of the M-current increases.
In this paper, we will perform the parameter-dependent center manifold reduction near the generic and transcritical codimension two Bogdanov-Takens bifurcation in classical delay differential equations (DDEs). Using a generalization of the Lindstedt-Poincaré method to approximate the homoclinic solution allows us to initialize the continuation of the homoclinic bifurcation curves emanating from these points. The normal form transformation is derived in the functional analytic perturbation framework for dual semigroups (sun-star calculus) using a normalization technique based on the Fredholm al…
This paper provides for the first time correct third-order homoclinic predictors in $n$-dimensional ODEs near a generic Bogdanov-Takens bifurcation point. To achieve this, higher-order time approximations to the nonlinear time transformation in the Lindstedt-Poincaré method are essential. Moreover, a correct transform between approximations to solutions in the normal form and approximations to solutions on the parameter-dependent center manifold is needed. A detailed comparison is done between applying different normal forms (smooth and orbital), different phase conditions, and different pertu…
We derive the bifurcation set for a not previously considered three-parametric Bogdanov-Takens unfolding, showing that it is possible express its vector field as two different perturbed cubic Hamiltonians. By using several first-order Melnikov functions, we obtain for the first time analytical approximations for the bifurcation curves corresponding to homoclinic and heteroclinic connections, which along with the curves associated to local bifurcations organize the parametric regions with different structures of periodic orbits. As an application of these results, we study a family of 3D memris…
Nonlinear controlled plants with Bogdanov-Takens singularity may experience surprising changes in their number of equilibria, limit cycles and/or their stability types when the controllers slightly vary in the vicinity of critical parameter varieties. Each such a change is called a local bifurcation. We derive novel results with regards to truncated parametric normal form classification of the generalized cusp plants. Then, we suggest effective nonlinear bifurcation control law designs for precisely locating and accurately controlling many different types of bifurcations for two measurable pla…
We provide a complete symmetry-breaking bifurcation control for equivariant smooth differential systems with Bogdanov-Takens singularities. Controller coefficient space is partitioned by critical controller sets into different connected regions. The connected regions provide a classification for all qualitatively different dynamics of the controlled system. Hence, a state feedback controller design with four small controller coefficients is proposed for an efficient and full singular symmetry-breaking control. Our approach works well for nonlinear control systems with both controllable and unc…
In this paper, we consider the bifurcation of limit cycles for generic L-V system ($\dot{x}=y+x^2-y^2\pm\frac{4}{\sqrt{3}}xy,~\dot{y}=-x+2xy$) and B-T system ($\dot{x}=y,~\dot{y}=-x+x^2$) under perturbations of piecewise smooth polynomials with degree $n$. Here the switching line is $y=0$. By using Picard-Fuchs equations, we bound the number of zeros of first order Melnikov function which controls the number of limit cycles bifurcating from the center. It is proved that the upper bounds of the number of limit cycles for generic L-V system and B-T system are respectively $36n-65~(n\geq4),~37,57…
We study the Bazykin predator-prey model with predator intraspecific interactions and ratio-dependent functional response and show the existence and stability of two interior equilibrium points. We prove that the model displays a wide range of different bifurcations, such as saddle-node bifurcations, Hopf bifurcations, homoclinic bifurcations and Bogdanov-Takens bifurcations. We use numerical simulations to further illustrate the impact changing the predator per capita consumption rate has on the basin of attraction of the stable equilibrium points, as well as the impact of changing the effici…
This paper presents a modified Morris-Lecar model by incorporating the sodium inward current. The dynamical behaviour of the model in response to key parameters is investigated. The model exhibits various excitability properties as the values of parameters are varied. We have examined the effects of changes in maximum ion conductances and external current on the dynamics of the membrane potential. A detailed numerical bifurcation analysis is conducted. The bifurcation structures obtained in this study are not present in existing bifurcation studies of original Morris-Lecar model. The results i…
In this paper, we consider the dynamics of a predator-prey system of Gause type with cooperative hunting among predators and Holling III functional response. The known work numerically shows that the system exhibits saddle-node and Hopf bifurcations except homoclinic bifurcation for some special parameter values. Our results show that there are a weak focus of multiplicity three and a degenerate equilibrium with double zero eigenvalues (i.e., a cusp of codimension two) for general parameter conditions and the system can exhibit various bifurcations as perturbing the bifurcation parameters appr…
We examine the interaction of transcritical and saddle-node bifurcations in a predator-prey-nutrient system that is stressed by the presence of a toxicant affecting the prey. This model, formulated by Kooi et al. ({\sl Ecol. Model.} {\bf 212}(2008), 304--318), has a two-dimensional invariant sub system with zero predator density. In the sub system, a pair of prey-nutrient equilibria is created in a saddle-node bifurcation, while predator invasion in modelled by a transcritical bifurcation of one of this pair. Interactions of these bifurcations at codimension-two points give rise to bistable, p…
We discuss tipping phenomena (critical transitions) in nonautonomous systems using an example of a bistable ecosystem model with environmental changes represented by time-varying parameters [Scheffer et al., Ecosystems, 11 (2008), pp. 275--279]. We give simple testable criteria for the occurrence of nonautonomous tipping from the herbivore-dominating equilibrium to the plant-only equilibrium using global properties of the autonomous frozen system with fixed-in-time parameters. To begin with, we use classical autonomous bifurcation analysis to identify a codimension-three degenerate Bogdanov-Ta…
Spatiotemporal patterns are common in biological systems. For electrically-coupled cells previous studies of pattern formation have mainly used external forcing as the main bifurcation parameter. The purpose of this paper is to show that spatiotemporal patterns in electrically-coupled smooth muscle cells occur even in the absence of forcing. We study a reaction-diffusion system with the Morris-Lecar equations and observe a wide range of spatiotemporal patterns for different values of the model parameters. Some aspects of these patterns are explained via a bifurcation analysis of the system wit…
