International Journal of Bifurcation and Chaos

Motivated by the forced harmonic vibration of complex mechanical systems, we analyze the dynamics involving different waves in a double-well potential oscillator coupling amplitude modulation control of low frequency. The combination of amplitude modulation factor significantly enriches the dynamical behaviors on the formation of multiple-S-shaped manifold and multiple jumping phenomena that alternate between epochs of slow and fast motion. We can conduct bifurcation analysis to identify two harmonic vibrations. One is that the singular orbit makes multiple jumps to a fast trajectory segment from one attracting equilibrium to another as the expression of slow variable by using the DeMoivre formula. With the increase of tuning frequency, the system exhibits relaxation-type oscillations whose small amplitude oscillations are produced by nonlinear local cycles together with a distinct large amplitude cycle oscillation accounting for the Melnikov threshold values. The tuning frequency may not only affect the asymptotic expressions for the solution curves near fold singularities but also allow for the large amplitude orbit vibrations near fold-cycle singularities. Numerical analysis for computing critical manifolds and their intersections is used to detect the dynamical features in this paper.

The paper presents a discussion on an opinion about the stability margin towards an emergency in local climate dynamics from the bifurcation analysis viewpoint. With this purpose we propose to attract the practice-oriented bifurcation analysis, where the conflict-of-units between notions used to understand natural evolution processes and notions used to describe desirable artificial regimes is resolved by integrating analytics on the basis of modified bifurcation diagrams. The discussion focuses on the phenomenon of interannual temperature variability, where local annual maximums and minimums are analyzed with daily details in both time and temperature coordinates. This phenomenon is considered via the probable, periodical and regulator conceptions. Advantages of the regulator conception are verified by results of processing the data of temperature meteorological observations on daily means over the last 135 years. This conception is based on the HDS-hypothesis, in accordance to which local climate dynamics is determined by the natural competition between the amplitude quantization (restricted by the temperature Hysteresis) and time quantization (caused by the Double Synchronization). Thus an alternation between three elementary processes with the same period (year) and different patterns of annual warming–cooling cycles is supposed as a typical behavior for local climate systems, and the idea on high-dynamic local climate ensembles is developed instead of the conventional opinion on quasi-static local climate norms. Mechanisms of temperature changes due to abrupt shifts (so-called change-points) of the HDS-regulator parameters are distinguished from mechanisms of temperature changes due to bifurcations. The notion of a stability margin is used as a distance to an emergency and is visualized in the parametrical space. So, in spite of the mechanisms of temperature changes with/without bifurcations are different, their conflict-free sewing becomes conceptually possible in the context of the stability margin towards emergencies determined relatively bifurcation boundaries in the parametrical space. Since the discussed dynamics is not supposed to exist in terms of the traditional estimations concerning the observed local climate changes, then we believe that the paper would be interesting for scientists in the field of bifurcation analysis as well as for scientists and specialists, activity areas of which relate to the contemporary challenges connected with climate changes.

In this paper, we establish a cold chain dynamic game model including a milk manufacturer and two downstream oligopoly supermarkets under the wholesale price contract in the real world. The manufacturer is responsible for the production and cold transportation, and the two retailers sell the product. The Nash equilibrium points and the complexity of the system are discussed. The influence of the decision parameters and the stability of the system are studied by using complexity theory. We reveal the stable regions for the dynamic system. In addition, revenue sharing contract and profit sharing contract are two valuable contracts. In order to see how the two contracts would impact on the system’s equilibrium solution and the profits, we establish and analyze two new dynamic models for the cold chain. By the comparison of the analyses under three contracts, we find that the manufacturer’s effort of cold transportation will change under different contracts, and the profit distribution of the whole cold chain will be affected. Chaos control is also studied by the method of delay feedback control, in order to provide some management advice.

Data confidentiality is mandatory during transmission or when storing sensitive information, especially in financial, medical and military applications. In this context, several cipher solutions and techniques have been presented in the literature. However, existing solutions are mainly based on static structures, where the confusion and diffusion primitives are fixed and independent of the secret key. In this article, we propose a new block cipher scheme that is based on the Substitution Permutation Networks (SPN). The proposed cipher consists of three operations: round-key addition, substitution, and bits’ permutation. Moreover, the substitution operation is applied at the byte level and it is based on a dynamically generated S-box, while the diffusion primitives are applied at the bit level using a dynamically generated P-box. Such key-dependent design ensures better cryptographic strength and system performance when compared, for instance, to DES, 3DES, RC5, and PRESENT schemes, among others, due to its key expansion algorithm. Thorough analysis show that the proposed scheme exhibits a high degree of randomness, key and plain-text sensitivity, and it satisfies the avalanche effect. From a theoretical perspective, we have formulated the Output Feedback mode of operation as a discrete dynamical system on a topological space. We prove that the dynamics of this system (in terms of sensitivity to the initial vector, etc.) are directly related to the strong connectivity of a graph. By doing so, we are able to characterize the conditions under which this mode evolves chaotically, as defined in Devaney’s theory. In particular, such theoretical investigations allow us to link the avalanche effect and key sensitivity of the cipher with the sensitivity of the whole process, that is, with the mode of operation.

In the chaotic polynomial Lorenz-type systems (including Lorenz, Chen, Lü and Yang systems) and Rössler system, their equilibria are unstable and the number of the hyperbolic equilibria are no more than three. This paper shows how to construct a simple analytic (nonpolynomial) chaotic system that can have any preassigned number of equilibria. A special 3D chaotic system with no equilibrium is first presented and discussed. Using a methodology of adding a constant controller to the third equation of such a chaotic system, it is shown that a chaotic system with any preassigned number of equilibria can be generated. Two complete mathematical characterizations for the number and stability of their equilibria are further rigorously derived and studied. This system is very interesting in the sense that some complex dynamics are found, revealing many amazing properties: (i) a hidden chaotic attractor exists with no equilibria or only one stable equilibrium; (ii) the chaotic attractor coexists with unstable equilibria, including two/five unstable equilibria; (iii) the chaotic attractor coexists with stable equilibria and unstable equilibria, including one stable and two unstable equilibria/94 stable and 93 unstable equilibria; (iv) the chaotic attractor coexists with infinitely many nonhyperbolic isolated equilibria. These results reveal an intrinsic relationship of the global dynamical behaviors with the number and stability of the equilibria of some unusual chaotic systems.

Homoclinic orbits of a buckled beam subjected to transverse uniform harmonic excitation are investigated in the case of 1:1 internal resonance. The geometric singular perturbation method and Melnikov method are employed to show the existence of the one-bump and multi-bump homoclinic orbits that connect the equilibria in a resonance band of the slow manifold. Each bump is a fast excursion away from the resonance band, and the bumps are interspersed with slow segments near the resonance band. The results obtained imply the existence of the amplitude modulated chaos for the Smale horseshoe sense in the class of buckled beam systems.

In this paper, a 4D Lorenz-type multistable hyperchaotic system with a curve of equilibria is investigated by using differential geometry method, i.e. with KCC-theory. Due to the deviation curvature tensor and its eigenvalues, the curve of equilibria of this hyperchaotic system is proved analytically to be Jacobi unstable under a certain parameter condition, and a periodic orbit of this system is proved numerically to be also Jacobi unstable. Furthermore, the dynamics of contravariant vector field near the curve of equilibria and the periodic orbit are studied, respectively, and their results comply absolutely with the above analysis of Jacobi stability.

In this paper, the synchronization problem of a chaotic Aihara neuron network is considered. Based on the master stability function (MSF) analysis, some necessary conditions are proposed to investigate the complete synchronization of the electrical coupled chaotic Aihara neuron networks. Especially, the two-dimensional parameter-space plots are obtained numerically to visualize the possible synchronized region. For the chaotic Rulkov neuron network, the numerical simulations of the MSF show that complete synchronization of the electrical coupled Aihara neuron network can be achieved when the coupling strength lies in a suitable interval, while it is highly improbable to attain complete synchronization when all neurons are in the chaotic bursting. Moreover, the complete synchronization cannot be reached with further increase or decrease of coupling strength. These results are probably common features as far as discrete-time neuron networks are concerned. Finally, based on the drive-response active control method, an LMI-based criterion is obtained to ensure complete synchronization of the available chaotic Aihara neuron network. The spatiotemporal patterns measured by the state distribution of neurons are employed to validate the effectiveness of our control scheme.

The aim of this paper is to solve the density classification task (DCT), an extensively studied classical problem, using one-dimensional nonuniform Cellular Automata (CA) rules. A perfect solution of DCT requires searching for CA rules for binary strings of all possible lengths. But the generic problem is still open though the solution exists only for a specific fixed length CA. This paper provides two fundamental ideas to solve this problem in a better way. The first technique solves this problem using deterministic Turing machines which ultimately leads to generation of different CA rules under periodic boundary conditions. In the second technique, the existence of DCT solution by analyzing the state transition diagrams (STDs) of number conserving CA rules is investigated. The possibility of finding the exact solutions of DCT using Turing machine, STD and number conservation property of CA rules can be viewed as an improvement over the approximate solutions obtained by evolutionary techniques.

Quasiperiodicity and chaos in a ring of unidirectionally coupled sigmoidal neurons (a ring neural oscillator) caused by a single shortcut is examined. A codimension-two Hopf–Hopf bifurcation for two periodic solutions exists in a ring of six neurons without self-couplings and in a ring of four neurons with self-couplings in the presence of a shortcut at specific locations. The locus of the Neimark–Sacker bifurcation of the periodic solution emanates from the Hopf–Hopf bifurcation point and a stable quasiperiodic solution is generated. Arnold’s tongues emanate from the locus of the Neimark–Sacker bifurcation, and multiple chaotic oscillations are generated through period-doubling cascades of periodic solutions in the Arnold’s tongues. Further, such chaotic irregular oscillations due to a single shortcut are also observed in propagating oscillations in a ring of Bonhoeffer–van der Pol (BVP) neurons coupled unidirectionally by slow synapses.

We study nonlinearly coupled logistic map with feedback in the limit of large delay time ($\tau$). In the $90$’s, the mapping of delayed dynamical systems to the spatially extended system was pursued in the context of optical systems by Arecchi and coworkers. We map the delayed system to a coupled map lattice of size $\tau$ with a sequential update and strictly one-sided coupling. Apart from aiding visualization, this formal equivalence helps us identify “phases” in a delayed system. One of the easily identified phases is that of frozen antiferromagnetic phase which is abundant in phase space. As coupling parameter ($𝜖$) increases, the system goes from periodic cascades along the coarse-grain region to chaos. We use the number of domain walls as an order parameter for locating this transition. The domain walls undergo random walk with drift and annihilate each other. At critical parameter value, we observe the power-law decay of this parameter. The decay exponents fall within the directed Ising universality class in general. We also find the escape time distribution, i.e. time required for reaching frozen antiferromagnetic regime. It diverges as it approaches the critical point. The divergence of average laminar length shows features indicating crisis induced intermittency.

This paper deals with a new modified hyperchaotic van der Pol–Duffing (MVPD) snap oscillator. Various dynamical properties of the proposed system are investigated with the help of Lyapunov exponents, stability analysis of the equilibrium points and bifurcation plots. The existence of the Hopf bifurcation is established by analyzing the corresponding characteristic equation. It is also proved that the MVPD oscillator shows multistability with coexisting attractors. Various numerical simulations are conducted and presented to show the dynamical behavior of the MVPD system. To show that the system is hardware realizable, we derive the discrete model of the MVPD system using the Euler’s method and using the hardware–software cosimulation, the proposed MVPD system is implemented in Field Programmable Gate Arrays. It is shown that the output of the digital implementations of the MVPD systems matches the numerical analysis.

In this paper, systems of nonlinear integro-differential Volterra equations are examined that can be represented as feedback interconnections of linear time-invariant block and periodic nonlinearities. The interest in such systems is motivated by their numerous applications in mechanical, electrical and communication engineering; examples include, but are not limited to, models of phase-locked loops, pendulum-like mechanical systems, coupled vibrational units and electric machines. Systems with periodic nonlinearities are usually featured by multistability and have infinite sequences of (locally) stable and unstable equilibria; their trajectories may exhibit nontrivial (e.g. chaotic) behavior. We offer frequency-domain criteria, ensuring convergence of any solution to one of the equilibria points, and this property is referred to as the gradient-like behavior and corresponds to phase locking in synchronization systems. Although it is hard to find explicitly the equilibrium, attracting a given trajectory, we give a constructive estimate for the distance between this limit equilibrium and the initial condition. The relevant estimates are closely related to the analysis of cycle slipping in synchronization systems. In the case where the criterion of gradient-type behavior fails, a natural question arises — which nonconverging solutions may exist in the system and, in particular, how many periodic solutions it has. We show that a relaxation of the frequency-domain convergence criterion ensures the absence of high-frequency periodic orbits. The results obtained in the paper are based on the method of integral quadratic constraints that has arisen in absolute stability theory and stems from Popov’s techniques of “a priori integral indices”. We illustrate the analytic results by numerical simulations.

In this paper, we characterize all the Liouvillian first integrals of a cubic polynomial differential system that contains the van der Pol and the Duffing oscillators. It is also shown that the centers correspond to the Liouville integrable cases.

We answer the last question left open in [Kočan, 2012] which asks whether there is a relation between an infinite LY-scrambled set and $\omega$-chaos for dendrite maps. We construct a continuous self-map of a dendrite without any infinite LY-scrambled set but containing an uncountable $\omega$-scrambled set.

Complex-periodic spiral waves are investigated extensively in the oscillatory medium. In this paper, the linearly polarized electric field (LPEF) is employed to induce complex-periodic spiral waves in the excitable medium with abnormal dispersion. As the amplitude of LPEF is increased beyond a threshold, the simple-periodic spiral wave converts into an irregularly complex-periodic one, in which, the local dynamics exhibit several regular spikes followed by one missed spiking period. Furthermore, with the increase of the LPEF amplitude, the missed spiking period follows different numbers of regular spikes [so-called period-1 (P-1), period-2 (P-2), etc.], even a mix of different periods. Meanwhile, the wavelength of the spiral wave transits from a short to a longer one. The pure-periodic (from P-6 to P-2) spirals generally contain defect lines, across which the phase of local oscillation changes by $2\pi$. In contrast, there is no defect line in the mixed-periodic spiral waves. This finding indicates that the defect line is not a necessary feature for complex-periodic spiral waves. Moreover, three types of tip trajectories of pure-periodic spiral waves are identified depending on the periods. That is, the outward-petal meandering, the outward-petal meandering with slow modulation, and drifting tip motion, and the tip trajectories could be used to distinguish them from the complex-oscillatory spiral waves.

This paper deals with a kind of piecewise smooth equation which is linear in the dependent variable. We study the problem of lower bounds for the maximum number of limit cycles of such equations using Melnikov functions. First of all, using the first order Melnikov function, we prove that these differential equations have a sharp upper bound for the number of the limit cycles which bifurcate from the periodic orbits and cross the separation straight line. Furthermore, in some cases the maximum number of these limit cycles is three, up to any order analysis. In the end, we apply this result on a kind of piecewise smooth planar system which has a separation curve with ${𝕊}^1$ up to homomorphism.

In this paper, the membership function in fuzzy systems is used in the Diffusion Limited Aggregation (DLA) model to investigate the fractal diffusion of soot particles from diesel engine emissions. The transformation of the morphology of soot particle aggregates and the control of fractal diffusion of soot particles are investigated by analyzing the nonlinear relationship between the motion steps and angles of diffusing particles. The simulation results demonstrate that the morphology of the aggregates varies from loose to compact by changing the particles’ motion steps and angles in membership functions. Meanwhile, the Ballistic Aggregation (BA)-like aggregates are obtained. Furthermore, the control of the morphology of soot particle aggregates is realized, which makes the settlement of the aggregates become easier. This will provide a reference for further understanding the growth mechanism of soot particle diffusion and enhancing the purification technology of the soot particles.