International Journal of Bifurcation and Chaos

The quadratic map of the interval displays one attractor for each parameter value. Conservative maps of the plane display infinite coexistence of stability islands around periodic orbits. Between these two extremes, dissipative systems of the plane are known to have infinite coexistence of sinks as a generic property, yet very hard to detect. We investigate how more and more coexistence is gained as the area-contraction rate b → 1. In this paper, we show a sequence of simple sinks gaining coexistence, and investigate the convergence properties of its bifurcation values. The sinks are simple, or primary, due to their geometrical structure.

We address the question of finding the attractors of the extended complex Lorenz model (ℂLM), which is obtained by extending the space from ℝ³ to ℂ³, and defining the model by the same equations as the classical Lorenz model (LM). We have numerical evidence of two strong attractors unrelated to the Lorenz attractor. We calculate its Lyapunov exponents and show that two of them are 0, and the other four are double and negative. Hence the attractors are nonchaotic. We show that they have a quasi-periodic nature. To decipher the structure of these attractors, we introduce the imaginary Lorenz model (𝕀LM), which is defined in the same space ℂ³ by multiplying with the Lorenz equations. Both models locally commute, and with its help we account for the double Lyapunov exponent 0 and show evidence that the basin of attraction of each attractor is a big open set of ℂ³. The chaotic limit set L_ℂ ⊂ ℂ³ obtained from the classical Lorenz attractor L₀ of (LM) by moving it with the (𝕀LM) has two positive Lyapunov exponents, but only captures a set of 6D-volume 0 in its basin of attraction. Hence this attractor may be hyperchaotic in ℝ⁵.

We develop two algorithms for obtaining an index 1 saddle torus between two attractors of which basin boundary is smooth by using the bisection method. In the first algorithm, which we name "overall template method", we make a template approximating a whole attractor and use it for pattern matching. The attractor can be a periodic orbit, a torus and even a chaos. In the second algorithm, which we name "partial template method", we make a template approximating only a part of an attractor. In this algorithm, the attractor is restricted to a periodic orbit and a two-torus, but calculation time is much smaller than that of the overall template method. We demonstrate these two algorithms for two solutions, observed in a ring of six-coupled bistable oscillators. One is a saddle torus in the basin boundary of the two switching solutions, and the other is of the two quasi-periodic propagating wave solutions. We calculate the Lyapunov spectrum of the obtained saddle torus, and confirm one positive Lyapunov exponent.

In this paper we consider a family of dynamical systems that we call "arabesques", defined as closed chains of 2-element negative circuits. An n-dimensional arabesque system has n 2-element circuits, but in addition, it displays by construction, two n-element circuits which are both positive versus one positive and one negative, depending on the parity (even or odd) of the dimension n. In view of the absence of diagonal terms in their Jacobian matrices, all these dynamical systems are conservative and consequently, they cannot possess any attractor. First, we analyze a linear variant of them which we call "arabesque 0" or for short "A0". For increasing dimensions, the trajectories are increasingly complex open tori. Next, we inserted a single cubic nonlinearity that does not affect the signs of its circuits (that we call "arabesque 1" or for short "A1"). These systems have three steady states, whatever be the dimension, in agreement with the order of the nonlinearity. All three are unstable, as there cannot be any attractor in their state-space. The 3D variant (that we call for short "A1_3D") has been analyzed in some detail and found to display a complex mixed set of quasi-periodic and chaotic trajectories. Inserting n cubic nonlinearities (one per equation) in the same way as above, we generate systems "A2_nD". A2_3D behaves essentially as A1_3D, in agreement with the fact that the signs of the circuits remain identical. A2_4D, as well as other arabesque systems with even dimension, has two positive n-circuits and nine steady states. Finally, we investigate and compare the complex dynamics of this family of systems in terms of their symmetries.

Dense granular media exhibit rich phenomenology when subject to imposed stresses and strains. This is a result of the many degrees of freedom present in an assembly of grains and the nonlinear interactions between the grains. Their complex behavior include the self-organization of load-bearing columnar structures known as force chains across a wide range of spatial scales. Behavior akin to phase transitions from a strong solid-like to a weak liquid-like response can also be observed with shear bands, i.e. regions where force chains collectively buckle, being the signature microstructure in this transition from the stable to the failure regime. An inherent aspect of shear bands and dense granular failure is the phenomenon of dilatancy, i.e. expansion in volume, when the material is subjected to a combined compression and shear. To understand the origins of dilatancy, it is useful to consider the granular material as a mixture of two components: grains and the interstitial material filling the voids or pores between the grains. The grains within a dense granular material respond to applied loads by rearranging to create local zones which contract and dilate. Extant studies of this mechanical response are typically focused on the solid skeleton, in particular, the topology of the network representing the physical contacts between grains. Here, we propose an alternative perspective which is to consider network representations of the evolving anisotropic pore space. We demonstrate how to construct pore space networks that express the local size of voids about a grain through network edge weights. We investigate sectors of the loading history when a percolating giant component of the pore space network exists. By defining two weight functions for edge properties, we: (i) discover via a recurrence plot-based analysis a temporal time scale for jamming–unjamming (contractant-dilatant) dynamics in shear bands; and show that (ii) the formation of a persistent shear band in response to the deformation places grains in a configuration predisposed to the efficient transport of interstitial material as evidenced by the location of percolating shortest path routes through the most dilatant sites. A proper understanding of the micromechanics of pore evolution with respect to shear bands and dilatancy is key to a range of applications such as modeling ground water flow, dewatering systems, carbon capture and sequestration.

In this article, we deal with the problems of characterizing isochronous centers for real planar quasi-cubic homogeneous analytic system. The technique is based on reducing the quasi-cubic analytic system into an analytic system. With the help of common computer algebra software-MATHEMATICA, we compute the period constants of the origin and obtain the necessary isochronous center conditions for the transformed system. Finally, we give a proof of the sufficiency by various methods. Similar results are less so far. Our work is new in terms of research about quasi-cubic analytic system and consists of the existing results related to cubic polynomial system as a special case. What is worth pointing out is that we offer a kind of interesting phenomenon that the exponent parameter λ controls the nonanalyticity of the studied system (5).

The dynamical transitions resulting from time delay in single Hindmarsh–Rose system are investigated in the present paper. As the time delay varies, the change of the structure of slow manifold is first formulated by using the method of stability switch. Then, the delay-induced dynamical transitions are investigated through the analysis of geometric singular perturbation, and in several case studies, the mechanism of the dynamical transitions is illuminated. Numerical results demonstrate the validity of the theoretical results.

We model the innervation dynamics of interneurons in a cerebral cortex center A between the time of initial sensory input and acquisition of a sustained steady state. The model assumes that interneurons in A are heavily interconnected allowing synchronization. This invites modeling the dynamics by means of a discrete time map. The model takes into account the influence of excitatory and inhibitory cells and reflects the architecture of synapses along the axons. The acquisition of a sustained chaotic state is characterized by means of a natural invariant probability measure. The time to attain this probability measure can be estimated.

In this paper, two methods are proposed to construct multi-wing chaotic attractors based on a Lorenz-like autonomous chaotic system. The first is switching method which can directly multiply the wings of the system without segment linearization of the system. The second is coordinate transition method which is related to the initial value of the system. Moreover, theoretical analysis and simulation results show the effectiveness of these two methods.

A class of delayed ratio-dependent Gause-type predator–prey model is considered. We study the eigenvalue problem for the linearized system at the coexisting equilibrium. For a critical case when the characteristic equation has a single zero root and a simple pair of pure imaginary roots, a complete bifurcation analysis is presented by employing the center manifold reduction and the normal form method. We analyzed the influence of the time delay on the Hopf–Fold bifurcation and showed the occurrence of quasi-periodic motion and bursting behavior. This phenomenon is in line with the seasonal variation law of the population.

In this paper, the theory of regular chains and a triangular decomposition method relying on modular computations are presented in order to symbolically solve multivariate polynomial systems. Based on the focus values for dynamic systems obtained by using normal form theory, this method is applied to compute the limit cycles bifurcating from Hopf critical points. In particular, a quadratic planar polynomial system is used to demonstrate the solving process and to show how to obtain center conditions. The modular computations based on regular chains are applied to a cubic planar polynomial system to show the computation efficiency of this method, and to obtain all real solutions of nine limit cycles around a singular point. To the authors' best knowledge, this is the first article to simultaneously provide a complete, rigorous proof for the existence of nine limit cycles in a cubic system and all real solutions for these limit cycles.

Using a discrete-delay nonlinear dynamic system, we model the time evolution of a stock market price index and net stock of savings in mutual funds. The proposed deterministic model has a unique steady-state, so its time evolution is determined by nonlinear effects acting out of equilibrium. For this model, we find the local stability properties and the local bifurcations conditions, given the parameter space. Specifically, we find that in both versions of the model (with and without delay) a Neimark–Sacker bifurcation can occur. Moreover, we show that the system without delay has a chaotic behavior. Finally, we formulate the associated discrete stochastic model and establish the conditions for asymptotic stability. Several numerical simulations are finally performed for both the deterministic and the stochastic model to justify the theoretical results.

A modified CK direct method is generalized to find finite symmetry groups of nonlinear mathematical physics systems. For the (2 + 1)-dimensional Broer–Kaup–Kupershmidt (BKK) system, both the Lie point symmetry and the non-Lie symmetry groups are obtained by this method. While using the traditional Lie approach, one can only find the Lie symmetry groups. Furthermore, abundant localized structures of the BKK equation are also obtained from the non-Lie symmetry group.

In this paper, we propose in uncertain dynamical systems a novel method for identifying the region that includes all possible bifurcation boundaries. First, we formulate a robust bifurcation analysis problem for parameter-dependent differential equations with dynamic uncertainties. Next, to solve this problem, we give stability and instability conditions for the uncertain system. Finally, on the basis of these conditions, we propose a method for solving the robust bifurcation analysis problem.

In this paper we provide a geometric characterization of those locally compact Hausdorff topological groups which admit a faithful strongly chaotic continuous action on some Hausdorff space.

We study the asymptotic behavior of symbolic computing systems, notably one-dimensional cellular automata (CA), in order to ascertain whether and at what rate the number of complex versus simple rules dominate the rule space for increasing neighborhood range and number of symbols (or colors), and how different behavior is distributed in the spaces of different cellular automata formalisms. Using two different measures, Shannon's block entropy and Kolmogorov complexity, the latter approximated by two different methods (lossless compressibility and block decomposition), we arrive at the same trend of larger complex behavioral fractions. We also advance a notion of asymptotic and limit behavior for individual rules, both over initial conditions and runtimes, and we provide a formalization of Wolfram's classification as a limit function in terms of Kolmogorov complexity.

A general relation between knot theory and symbolic dynamics is studied for bimodal maps in this paper. When symbolic sequences of maps are expressed as knots, it is easy to see that the knot of a renormalizable sequence is composed of a bunch of periodic flows. In this setting, the generation of renormalizable knots can be simply operated in geometry and explicitly calculated in algebra. In this paper we provide a universal form of renormalizable knots, which can be decomposed into a sequence of elementary templates; especially, it is independent of the traditional *-products of symbolic dynamics. We present some examples and list the elementary templates of period not bigger than 6.

The results in this paper show that the cubic vector fields ẋ = -y + M(x, y) - y(x² + y²), ẏ = x + N(x, y) + x( x² + y²), where M, N are quadratic homogeneous polynomials, having simultaneously a center at the origin and at infinity, have at least 61 and at most 68 topologically different phase portraits. To this end, the reversible subfamily defined by M(x, y) = -γxy, N(x, y) = (γ - λ)x² + α²λy² with α, γ ∈ ℝ and λ ≠ 0, is studied in detail and it is shown to have at least 48 and at most 55 topologically different phase portraits. In particular, there are exactly five for γλ < 0 and at least 46 for γλ > 0. Furthermore, the global bifurcation diagram is analyzed.

Pattern formation is a ubiquitous phenomenon in the natural world. Previous studies showed that for an activator–inhibitor system without cross-diffusion, spatial patterns can be formed only when the diffusion of the inhibitor is significantly faster than that of the activator. However, cross-diffusion exists extensively in real systems, especially in biological systems. Here, we study a prototypic two-species negative feedback system with cross-diffusion. By performing stability analysis of equilibrium state, we find sufficient conditions for Turing instability. Both analytical and numerical results demonstrate that mutual diffusions of the two species can lead to the Turing pattern formation regardless of differences in self-diffusion coefficients. However, in the absence of the mutual diffusion or even if there is the cross-diffusion of only one species, the system cannot exhibit Turing patterns. Our results reveal the mechanism of Turing pattern formation in a class of reaction–diffusion systems, where mutual diffusion between species plays a key role.

Recently, the study on the dynamical behavior of complex dynamical systems has become a focal subject in the field of complexity. In particular, the system's adaptability and sensitivity have attracted increasing attention from various scientific communities. In this paper, we focus on some properties of complexity to gain a better understanding of it. Two descriptive mathematical definitions of attractors' adaptability and sensitivity are introduced from the viewpoint of dynamical systems. Then, these new descriptions are applied to analyze the adaptability and sensitivity of stable equilibrium points. In addition, a method is introduced for improving both the adaptability and sensitivity of a system with a stable equilibrium point.