International Journal of Bifurcation and Chaos最新文献

This work presents an efficient approach for computing the slow invariant manifold of the fourth-order canonical memristor-based Chua circuits using the flow curvature method. First, the magnetic-flux and charge characteristic curve is generated from the classical circuit with a piecewise-linear function. Then, the characteristic curve is generated from the circuit with the piecewise-linear function replaced by a cubic function. Further, the duality principle is applied to studying such memristor-based circuits in the three-dimensional flux-linkage and charge phase space and then in the four-dimensional current–voltage phase space. It is demonstrated that the slow invariant manifolds of these fourth-order memristor-based chaotic circuits can be more directly computed for the first case than the second.

In our earlier research (see [Katsanikas & Wiggins, 2021a, 2021b, 2023a, 2023b, 2023c]), we developed two methods for creating dividing surfaces, either based on periodic orbits or two-dimensional generating surfaces. These methods were specifically designed for Hamiltonian systems with three or more degrees of freedom. Our prior work extended these dividing surfaces to more complex structures such as tori or cylinders, all within the energy surface of the Hamiltonian system. In this paper, we introduce a new method for constructing dividing surfaces. This method differs from our previous work in that it is based on 3D surfaces or geometrical objects, rather than periodic orbits or 2D generating surfaces (see [Katsanikas & Wiggins, 2023a]). To explain and showcase the new method and to present the structure of these 3D surfaces, the paper provides examples involving Hamiltonian systems with three degrees of freedom. These examples cover both uncoupled and coupled cases of a quadratic normal form Hamiltonian system. Our current paper is the first in a series of two papers on this subject. This research is likely to be of interest to scholars and researchers in the field of Hamiltonian systems and dynamical systems, as it introduces innovative approaches to constructing dividing surfaces and exploring their applications.

In this study, we extend the two-dimensional host–parasitoid model to a one-host–two-parasitoid model, whose dynamic behaviors are more complex. As evidence, exploring the dynamic interaction between a host and its parasitoids provides significant insight into the biological control. Specifically, we demonstrate the existence of equilibrium points and explore their local stability properties, which are concerned with the effective biological control project. Furthermore, the transition between population fluctuations and the steady state is achieved via a bifurcation process, and we derive the occurrence conditions of the Neimark–Sacker bifurcation in the proposed system using an explicit criterion. To control population fluctuations and chaotic behaviors, two feedback control strategies are implemented in this system. Finally, the numerical simulations support our theoretical results and show the related biological phenomena.

We study directionally coupled phase-coherent chaotic oscillators in complex networks. We introduce an adjusted Lyapunov function that incorporates the frequencies of the oscillators and the interaction structure. Using the well-known Rössler system as an example, we address two optimization problems: frequency allocation and network design. Through numerical experiments, we demonstrate that the systematic synchrony can be effectively enhanced or inhibited by minimizing or maximizing the objective function, respectively. We then delve into the relationship between the structural and dynamical properties that lead to optimal synchronization. Interestingly, we observe a positive correlation between nodal in-degrees and frequency magnitudes, indicating that nodes with higher in-degrees tend to exhibit larger frequency magnitudes. On the other hand, we also find a negative correlation between nodal frequency and adjacent in-frequencies, suggesting that nodes with higher frequencies tend to be surrounded by nodes with lower frequency values. Finally, we explore the connections between degree correlations and optimal synchronization. We find that when minimizing the objective function, the presence of degree correlations always inhibits the systematic synchrony for frequency allocation, while the act of network design causes the correlations to become negative.

Long-term memory (LTM ) and short-term memory (STM ) and their evolution from one to the other are important mechanisms to understand brain memory. We use the Hodgkin–Huxley (HH ) model, a well-tested and closest model to biological neurons and synapses, to shine some light on LTM and STM memorization mechanisms. The role of ${\mathrm{\text{Na}}}^{+}$ and ${\mathrm{\text{K}}}^{+}$ion channels playing in LTM and STM process is carefully examined by using three different types of input signals, namely, a step DC voltage, a positive part of sinusoidal wave and periodic square signal with read voltage. Results are analyzed based on first-order${\mathrm{\text{K}}}^{+}$memristor and second-order${\mathrm{\text{Na}}}^{+}$memristor.

In real-world networks, due to complex topological structures and uncertainties such as time delays, uncontrolled systems may generate instability and complexity, thereby degrading network performance. This paper provides a detailed analysis of the stability, Hopf bifurcation, and complex dynamics of a networked system under delayed feedback control. Based on the linear stability method and Hopf bifurcation theorem, the stability of the equilibrium of the error system and the existence of Hopf bifurcation are studied. The stability of periodic solutions bifurcating from the trivial equilibrium is analyzed using normal form theory and central manifold theorem. Special focus is on the effects of the network topology and time delays on the stability and Hopf bifurcation. The theoretical results are also extended to the complex networks with asymmetric adjacent matrices. In addition, the controlled model exhibits complicated dynamical behavior via three types of codimension two bifurcations and period-doubling bifurcations that eventually lead to chaos. Numerical experiments have validated the theoretical results and indicated that delayed feedback control can effectively generate or annihilate the complicated behavior of complex networks.

Shortage of formal jobs, lack of skills in workforce and increasing human population proliferate the informal sector. This sector provides an opportunity to unskilled workers to gain skills along with earnings. In this paper, a deterministic nonlinear mathematical model is developed to study the effects of informal skill learning and job generation on unemployment. For the formulated system, feasibility of equilibria and their stability properties are discussed. A pertinent quantity (${{ℛ}}_0$), known as the reproduction number, is calculated and it is shown that the formulated system undergoes transcritical, saddle-node, Hopf and Bogdanov–Takens bifurcations on the variation of ${{ℛ}}_0$. The analytically obtained results are validated through numerical simulations. The results obtained from this study indicate that a substantial rate of job generation by self-employed individuals has a stabilizing effect on the system. Moreover, self-employment along with informal skill acquisition through engaging in informal work proves to be an effective measure in curbing the issue of unemployment in society.

Local activity could be the source for complexity. In this study, a multistable locally active memristor is proposed, whose nonvolatile memory, as well as locally active characteristics, is validated by the power-off plot and DC $V$–$I$ plot. Based on the two-dimensional Hindmarsh–Rose neuron and a one-dimensional Hopfield neuron, a simple neural network is constructed by connecting the two neurons with the locally active memristor. Coexisting multiple firing patterns under different initial conditions are investigated according to the controlled coupling factor. The results suggest that the system exhibits coexisting periodic and chaotic bursting with different firing patterns. Complex firing only occurs in the locally active area of the defined memristor, meanwhile the system shows a periodic oscillation in the passive area. Beyond this, the coupled neurons exhibit the specific phenomenon of attractor growing in the locally active region of the memristor. The circuit simulations by Power Simulation (PSIM) are included confirming the numerical simulations and theoretic analysis.

In this paper, we study the codimensions of Hopf bifurcation and Bogdanov–Takens bifurcation of a predator–prey model with alternative prey and prey refuges, which was proposed by Chen et al. [2023]. The results show that the predator–prey model can undergo a supercritical Hopf bifurcation or a Bogdanov–Takens bifurcation of codimension two under certain parameter conditions. It means that there are some predator–prey models with an alternative prey and prey refuges which have a limit cycle or a homoclinic loop. Moreover, it is also shown that the codimension of Hopf bifurcation is at most one and codimension of Bogdanov–Takens bifurcation is at most two.

By using model discretization of the piecewise constant argument method, a discrete amensalism model with nonselective harvesting and Allee effect is formulated. The dynamic analysis of the model is studied and the existence and stability of the equilibrium point are discussed. The fold bifurcation and flip bifurcation at the equilibrium point of the system are proved by using the bifurcation theory and the center manifold theorem. In order to control flip bifurcation and restore the system to a stable state, a hybrid control strategy of parameter perturbation and state feedback is adopted. Finally, the effectiveness of the theoretical results and the control strategy is verified by numerical simulations.