International Journal of Bifurcation and Chaos

Disorder in parameters appears to influence the collective behavior of complex adaptive networks in ways that might seem unconventional. For instance, heterogeneities may, unexpectedly, lead to enhanced regions of existence of stable synchronization states. This behavior is unexpected because synchronization appears, generically, in symmetric networks with homogeneous components. Related works have, however, misidentified cases where disorder seems to play a critical role in enhancing synchronization, where it is actually not the case. Thus, in order to clarify the role of disorder in adaptive networks, we use normal forms to study, mathematically, when and how the presence of disorder can facilitate the emergence of collective patterns. We employ parameter symmetry breaking to study the interplay between disorder and the underlying bifurcations that determine the conditions for the existence and stability of collective behavior. This work provides a rigorous justification for a certain barycentric condition to be imposed on the heterogeneity of the parameters while studying the synchronization state. Theoretical results are accompanied by numerical simulations, which help clarify incorrect claims of disorder purportedly enhancing synchronization states.

In prior work [Katsanikas & Wiggins, 2021a, 2021b, 2023c, 2023d], we introduced two methodologies for constructing Periodic Orbit Dividing Surfaces (PODS) tailored for Hamiltonian systems possessing three or more degrees of freedom. The initial approach, outlined in [Katsanikas & Wiggins, 2021a, 2023c], was applied to a quadratic Hamiltonian system in normal form having three degrees of freedom. Within this context, we provided a more intricate geometric characterization of this object within the family of 4D toratopes that describe the structure of the dividing surfaces discussed in these papers. Our analysis confirmed the nature of this construction as a dividing surface with the no-recrossing property. All these findings were derived from analytical results specific to the case of the Hamiltonian system discussed in these papers. In this paper, we extend our results for quartic Hamiltonian systems with three degrees of freedom. We prove for this class of Hamiltonian systems the no-recrossing property of the PODS and we investigate the structure of these surfaces. In addition, we compute and study the PODS in a coupled case of quartic Hamiltonian systems with three degrees of freedom.

In this paper, we study the dynamical complexity of points with emergence behavior but without weak face behavior, especially for points without physical-like behavior in certain dynamical systems such as transitive Anosov systems. We use the tools of saturated sets to prove that these points show strong dynamical complexity in the sense of entropy, density and distributional chaos. We obtain some observations of those results related to irregular sets and level sets. These results strengthen the previous results of [Catsigeras et al., 2019; Hou et al., 2023].

In this paper, we investigate the spatiotemporal dynamics in a diffusive two-species system with taxis term and general functional response, which means the directional movement of one species upward or downward the other one. The stability of positive equilibrium and the existences of Turing bifurcation, Turing–Hopf bifurcation and Turing–Turing bifurcation are investigated. An algorithm for calculating the normal form of the Turing–Hopf bifurcation induced by the taxis term and another parameter is derived. Furthermore, we apply our theoretical results to a cooperative Lotka–Volterra system and a predator–prey system with prey-taxis. For the cooperative system, stable equilibrium becomes unstable by taxis-driven Turing instability, which is impossible for the cooperative system without taxis. For a predator–prey system with prey-taxis, the dynamical classification near the Turing–Hopf bifurcation point is clearly described. Near the Turing–Hopf point, there are spatially inhomogeneous steady-state solution, spatially homogeneous/nonhomogeneous periodic solution and pattern transitions from one spatiotemporal state to another one.

The aim of this paper is to present a novel class of time-dependent controls to realize ultra-fast magnetization switching in nanomagnets driven by spin-torques produced by spin-polarized electric currents. Magnetization dynamics in such complex systems is governed by the Landau–Lifshitz–Slonczewski equation which describes the precessional motion of (dimensionless) magnetization vector on the unit-sphere. The relevant case of nanoparticles with uniaxial anisotropy having in-plane easy and intermediate axes as well as out-of-plane hard axis is considered. By exploiting the characteristic smallness of damping and spin-torque intensity, the complexity of the magnetic system’s dynamic is dealt with by employing tools borrowed from Hamiltonian Perturbation Theory. More precisely, the aforementioned controls are constructed via suitable perturbative tools in a way to realize approximate latitudinal solutions (i.e. motions on a sphere in which the out-of-plane magnetization component stays constant) with the effect to fast “switch” the system from one stationary state to another. The possibility to keep a (“small”) bounded value of the out-of-plane coordinate throughout this process of “transfer” turns out to be advantageous in the applications as it sensibly reduces the post-switching relaxation oscillations that may cause the failure of switching in real samples. Further relevant quantitative results on the behavior of the solutions during the pre- and post-switching stages (termed “expulsion” and “attraction”, respectively) are given as a by-product. A selection of validating numerical experiments is presented alongside the corresponding theoretical results.

As digital communication and storage continue to expand, the protection of image privacy information becomes increasingly critical. To safeguard sensitive visual information from unauthorized access, this paper proposes a novel image encryption scheme that integrates multiobjective Artificial Bee Colony (ABC) optimization algorithm and DNA coding. Multiple evaluation metrics including correlation relationship, Number of Pixel Change Rate (NPCR), Unified Average Changing Intensity (UACI), and information entropy are collaboratively optimized by the ABC algorithm. The proposed method begins with the application of the SHA-256 algorithm to generate keys and random sequences using chaotic systems. These sequences are then employed for shuffling, DNA coding, decoding, and diffusion, generating initial encrypted images. Subsequently, the encrypted images serve as individuals within the ABC algorithm to determine optimal parameters of the chaotic systems and the best ciphertext image. Simulation experiments demonstrate that the ciphertext images achieved excellent results in information entropy, pixel correlation coefficient, NPCR, and UACI. The integration of the multiobjective ABC optimization algorithm with DNA coding in our proposed image encryption scheme results in heightened security, as evidenced by superior performance in various metrics.

During the last decades, the study of discontinuous piecewise differential systems has become an interesting subject of research due to the important applications of this kind of systems to model natural phenomena. In the qualitative theory of differential equations, one of the interesting problems is the detection of the number of limit cycles and their configurations which remains open to date, except for very particular families of differential equations. Here, we are inspired to study the maximum number of limit cycles of the discontinuous piecewise differential systems separated by a nonregular line and formed by a linear center and one of the four classes of quadratic centers. The main tool used to prove our main results is based on the first integrals of such systems. All the computations of this paper are verified using the algebraic manipulator Mathematica.

This paper investigates a type of iterated implicit function systems composed of equations $F_n(x,y)=c$, where $F_n(x,y)$ is a continuous function, and $c$ is a constant. The existence of attractors of iterated implicit function systems is proved based on different equation conditions, including the equation $F_n(x,y)=c$ containing the implicit function or being ${\alpha }_n$-contractive about $y$. Meanwhile, we give definitions of implicit convergence of functions and monotone sequence of iterated implicit function systems. Finally, some properties of attractors of iterated implicit function systems are elucidated.

In this paper, we consider a spatially size–stage-structured population dynamics model with resting phase. The primary objective of this model is to study size structure, stage structure, resting phase and spatial location simultaneously in a single population system. At first, we reformulate the problem as an abstract nondensely defined Cauchy problem. Then, taking advantage of the integrated semigroup and bifurcation theories, we investigate the stability and Hopf bifurcation at the positive equilibrium of the model. Finally, numerical simulations are presented as evidence to support our analytical results. A discussion of related problems is also presented briefly.

In the ecological scenario, predators often risk their lives pursuing dangerous prey, potentially reducing their chances of survival due to injuries. Prey, on the other hand, try to strike a balance between reproduction rates and safety. In our study, we introduce a two-dimensional prey–predator model inspired by Tostowaryk’s work, specifically focusing on the domed-shaped functional response observed in interactions between pentatomid predators and neo-diprionid sawfly larvae. To account for the varying effectiveness of larval group defense, we incorporate a new component into the response equation. Our investigation delves into predator trade-off dynamics by adjusting the predator’s mortality rate to reflect losses incurred during encounters with dangerous prey and prey’s trade-off between safety and reproduction rate incorporating this domed-shaped functional response. Our model demonstrates bistability and undergoes various bifurcations, including transcritical, saddle-node, Hopf, Bogdanov–Takens, and Homoclinic bifurcations. Critical parameters impact both predator and prey populations, potentially leading to predator extinction if losses due to dangerous prey encounters become excessive, highlighting the risks predators face for their survival. Furthermore, the efficacy of group defense mechanisms can further endanger predators. Expanding our analysis to a spatially extended model under different perturbations, we explore Turing instability to explain the relationship between diffusion and encounter parameters through both stationary and dynamic pattern formation. Sensitivity to initial conditions uncovers spatiotemporal chaos. These findings provide valuable insights into comprehending the intricate dynamics of prey–predator interactions within ecological systems.

Competing populations within an ecosystem often release chemicals during the interactions and diffusion processes. These chemicals can have diverse effects on competitors, ranging from inhibition to stimulation of species’ growth. This work constructs a competition model that incorporates stimulatory substances, spatial effects, and multiple time lags to investigate the combined impact of these phenomena on competitors’ growth. When the stimulation rate from the produced chemicals falls within a suitable threshold interval, all species within the system can coexist. Under the species’ coexistence, their diffusive phenomenon leads to a spatially heterogeneous distribution, resulting in patchy structures (Turing patterns) within their habitat. As the parameter values exceed their thresholds, species begin to exhibit spatially periodic oscillations (spatial Hopf bifurcation). The presence of multiple delays and competitors’ diffusion contributes to spatially complex and heterogeneous behaviors (Turing–Hopf bifurcation). The results help us understand the underlying mechanisms behind these heterogeneous behaviors and enable us to mitigate their negative impact on species’ growth and harvest. Numerical simulations are used to measure the dynamics of competitors under different parameter conditions.

The computation of the normal form as well as its unfolding is a key step to understand the topological structure of a bifurcation. Though a lot of results have been obtained, it still remains unsolved for higher co-dimensional bifurcations. The main purpose of this paper is devoted to the computation of a codimension-3 zero-Hopf–Hopf bifurcation, at which a zero as well as two pairs of pure imaginary eigenvalues can be found from the matrix evaluated at the equilibrium point. Different distributions of eigenvalues are considered, which may behave in a non-semisimple form for 1:1 internal resonance. Based on the combination of center manifold and normal form theory, all the coefficients of normal forms and nonlinear transformations are derived explicitly in terms of parameters of the original vector field, which are obtained via a recursive procedure. Accordingly, a user friendly computer program using a symbolic computation language Maple is developed to compute the coefficients up to an arbitrary order for a specific vector field with zero-Hopf–Hopf bifurcation. Furthermore, universal unfolding parameters are derived in terms of the perturbation of physical parameters, which can be employed to investigate the local behaviors in the neighborhood of the bifurcation point. Here, we emphasize that though different norm forms based on different choices may exist, their topological structures are the same, corresponding to qualitatively equivalent dynamics.

For the optical soliton model in fifth-order weakly nonlocal nonlinear media, to find its exact explicit solutions, the corresponding traveling wave system is formulated as a planar dynamical system with a singular straight line. Then, by using techniques from dynamical systems and singular traveling wave theory developed by [Li & Chen, 2007] to analyze the planar system and find the corresponding phase portraits, the dynamical behavior of the amplitude component can be assessed. Under different parameter conditions, exact explicit solitary wave solutions, periodic wave solutions, kink, and anti-kink wave solutions, compacton solutions, as well as peakons and periodic peakons are found with precise formulations.

In this paper, a 5D Lorenz system is discussed. The discrete mappings are developed to solve the periodic motions in the 5D Lorenz system. Then the stability and bifurcations are determined by eigenvalue analysis. A bifurcation tree is presented to demonstrate that the discrete mapping method can provide not only stable orbits but also unstable motions. Finally, trajectory illustrations are given to show bifurcation influences on periodic orbits and homoclinic orbits in the 5D Lorenz system.

The Riccati polynomial differential systems are differential systems of the form $x^{\prime }=c_0(x)$, $y^{\prime }=b_0(x)+b_1(x)y+b_2(x)y^2$, where $c_0$ and $b_i$ for $i=0,1,2$ are polynomial functions. We characterize all the Riccati polynomial differential systems having an invariant algebraic curve. We show that the coefficients of the first four highest degree terms of the polynomial in the variable $y$ defining the invariant algebraic curve determine completely the Riccati differential system. A similar result is obtained for any Abel polynomial differential system.