Balanced Colorings and Bifurcations in Rivalry and Opinion Networks

Balanced colorings of networks classify robust synchrony patterns — those that are defined by subspaces that are flow-invariant for all admissible ODEs. In symmetric networks, the obvious balanced colorings are orbit colorings, where colors correspond to orbits of a subgroup of the symmetry group. All other balanced colorings are said to be exotic. We analyze balanced colorings for two closely related types of network encountered in applications: trained Wilson networks, which occur in models of binocular rivalry, and opinion networks, which occur in models of decision making. We give two examples of exotic colorings which apply to both types of network, and prove that Wilson networks with at most two learned patterns have no exotic colorings. We discuss in general terms how exotic colorings affect the existence and stability of branches for local bifurcations of the corresponding model ODEs, both to equilibria and to periodic states.

Bifurcations in 2D Spatiotemporal Maps

In this work, we give theoretical and numerical analyses for local bifurcations of 2D spatiotemporal discrete systems of the form [Formula: see text] where [Formula: see text] is a real nonlinear function, [Formula: see text] and [Formula: see text] are two independent integer variables, representing respectively a spatial coordinate and the time. On the basis of the spectral theory, we derive the conditions under which the local bifurcations such as flip and fold occur at the fixed points for some parameter values. As a case-study, a quite complex system, [Formula: see text]D spatiotemporal dynamic given by two coupled logistic maps, named [Formula: see text]D logistic coupled maps ([Formula: see text]D-LCM) is considered. The proposed map provides a reliable experimental and theoretical basis for identifying some cases of local bifurcations.

Bifurcations in a Predator–Prey Model of Leslie-Type with Simplified Holling Type IV Functional Response

In this paper, we complete the remaining investigation of local bifurcations in a predator–prey model of Leslie-type with simplified Holling type IV functional response. The system has at most three equilibria, and local bifurcations were completely investigated in the cases of one and three equilibria, but in the case of two equilibria the previous study was only on a fixed parameter. We extend the study in the case of two equilibria for all parameters, and find that the system exhibits Hopf bifurcations of codimensions 1 and 2, and Bogdanov–Takens bifurcations of codimensions 2 and 3. Previous results and our research show that the codimension of local bifurcations is at most 3, and both focus type and cusp type Bogdanov–Takens bifurcations of codimension 3 can occur.

Multi-Bifurcation Cascaded Bursting Oscillations and Mechanism in a Novel 3D Non-autonomous Circuit System with Parametric and External Excitation

In this paper, multi-timescale dynamics and the formation mechanism of a 3D non-autonomous system with two slowly varying periodic excitations are systematically investigated. Interestingly, the system shows novel multibifurcation cascaded bursting oscillations (MBCBOs) when the frequency of the two excitations is much lower than the mean frequency of the original system (MFOS). For instance, periodic, quasi-periodic and chaotic bursting oscillations induced by a variety of cascaded bifurcations are first observed, and the phenomenon of spiking transfer is also revealed. Besides, stability and local bifurcations of the system are comprehensively investigated to analyze the mechanism of the observed MBCBOs, in which bifurcation diagram, Lyapunov exponents, time series, phase portraits, and transformed phase diagrams are used. Finally, through a circuit simulation and hardware experiment, these complex dynamics phenomena are verified physically.

Modelling of a two prey and one predator system with switching effect

Prey switching strategy is adopted by a predator when they are provided with more than one prey and predator prefers to consume one prey over others. Though switching may occur due to various reasons such as scarcity of preferable prey or risk in hunting the abundant prey. In this work, we have proposed a prey-predator system with a particular type of switching functional response where a predator feeds on two types of prey but it switches from one prey to another when a particular prey population becomes lower. The ratio of consumption becomes significantly higher in the presence of prey switching for an increasing ratio of prey population which satisfies Murdoch’s condition [15]. The analysis reveals that two prey species can coexist as a stable state in absence of predator but a single prey-predator situation cannot be a steady state. Moreover, all the population can coexist only under certain restrictions. We get bistability for a certain range of predation rate for first prey population. Moreover, varying the mortality rate of the predator, an oscillating system can be obtained through Hopf bifurcation. Also, the predation rate for the first prey can turn a steady-state into an oscillating system. Except for Hopf bifurcation, some other local bifurcations also have been studied here. The figures in the numerical simulation have depicted that, if there is a lesser number of one prey present in a system, then with time, switching to the other prey, in fact, increases the predator population significantly.