International Journal of Bifurcation and Chaos

We propose a new three-dimensional map that demonstrates the two- and three-frequency quasi-periodicity. For this map, all basic quasi-periodic bifurcations are possible. The study was realized by using Lyapunov charts completed by plots of Lyapunov exponents, phase portraits and bifurcation trees illustrating the quasi-periodic bifurcations. The features of the three-parameter structure associated with quasi-periodic Hopf bifurcation are discussed. The comparison with nonautonomous model is carried out.

The nonlinear dynamical behavior of an atmospheric circulation in a beta-plane channel is examined on a five-spectral mode model, truncated from the Charney and DeVore quasi-geostrophic equation. Bifurcation and chaos are observed when subjected to a topographic driving disturbance and a thermally driving zonal source. An equilibrium state undergoes supercritical Hopf bifurcation and becomes a stable periodic state with respect to the magnitude of the thermally driving source, whereas the periodic state undergoes a subcritical Hopf bifurcation and transforms into a low-index equilibrium state with respect to the increasing topographic driving disturbance. The stable periodic state further develops into a pair of stable periodic states when increasing the thermally driving source. The first one with the period of 4.3 days exhibits an oscillation of strong and weak zonal flow patterns, whereas the second one with the period of 6.8 days demonstrates a fluctuation amongst weak zonal disturbance flow patterns. Moreover, the two periodic states transform respectively into chaos through separate period-doubling cascades with the further development of the thermally driving source.

An electronic oscillator circuit is designed by connecting an inductor in series with a locally-active PTC Memristor and a battery. The PTC Memristor is locally active on the negative resistance region of its DC $V_M$–$I_M$ curve. A DC operating point $Q$ is chosen on the locally-active region of the PTC Memristor and a small-signal equivalent circuit at $Q$ is derived via Taylor series. The small-signal admittance $Y_{CM}(s;V)$ of the composite one-port in Fig. 1 is derived using the small-signal equivalent circuit at $Q$, in series with an inductor whose value is chosen such that $Y_{CM}(i{\omega }_0;V)=0$ at some $s=i{\omega }_0$. The sinusoidal oscillation computed numerically from this circuit is shown to emerge from a supercritical Hopf bifurcation.

Memristive dynamics are generated utilizing active, weak-inversion, CMOS circuits. Class A and Class AB approaches to the generation of these dynamics are presented. The Class A circuit is composed of a grounded capacitor and two exponential transconductors (E-cells). It consumes a few nanowatts of power from a 1V power supply. The Class AB circuit also consists of a similar architecture with an added geometric mean splitter. Simulation results confirm the realization of pinched hysteresis loops exhibiting the zero crossing property of the memristor. The effects of varying input amplitude and frequency as well as the capacitor value are investigated. With the use of MOS capacitors in the place of monolithic capacitors, the area requirements of these emulators can be significantly diminished. Simulation results of circuits with both types of capacitors are presented. The simple emulators presented here offer the possibility of realizing nanoscale memristor dynamics when high yield memristor processes are not accessible.

This article introduces a novel four-dimensional autonomous multiscroll chaotic circuit which is derived from the actual simplest memristor-based chaotic circuit. A fourth circuit element — another inductor — is introduced to generate the complex behavior observed. A systematic study of the chaotic behavior is performed with the help of some nonlinear tools such as Lyapunov exponents, phase portraits, and bifurcation diagrams. Multiple scroll attractors are observed in Matlab, Pspice environments and also experimentally. We also observe the phenomenon of antimonotonicity, periodic and chaotic bubbles, multiple periodic-doubling bifurcations, Hopf bifurcations, crises and the phenomenon of intermittency. The chaotic dynamics of this circuit is realized by laboratory experiments, Pspice simulations, numerical and analytical investigations. It is observed that the results from the three environments agree to a great extent. This topology is likely convenient to be used to intentionally generate chaos in memristor-based chaotic circuit applications, given the fact that multiscroll chaotic systems have found important applications as broadband signal generators, pseudorandom number generators for communication engineering and also in biometric authentication.

The ion migration memristor is a nonlinear element with memory function and nanoscale size, it is considered as a potential candidate to reduce system power consumption and circuit size. When it works as the nonlinear part of the chaotic system, rich nonlinear curves will be produced, and at the same time, the complexity of chaotic systems and the randomness of signals will be enhanced. So in this paper, by Matlab numerical simulation, a new double-wing chaotic system based on an ion migration memristor is designed. In reality, there are many systems interfered inevitably by random noise, so in this paper the random bounded noises are also considered. The power spectrum, Lyapunov exponent spectrum, Poincaré map and bifurcation diagram are used to investigate its complex dynamic characteristics. Then, a SPICE-based analog circuit is presented to verify the feasibility of the system, for which the simulation results are consistent with the numerical simulation. Finally, the sliding mode variable structure control is applied to overcome the shortcomings of traditional control method, so that the chaotic orbits can be controlled to any fixed points or periodic orbits, and this provides an insight into chaos control in power electronics systems.

A smooth curve model of meminductor and its equivalent circuit have been designed, on the condition that the meminductor is commonly unavailable. The equivalent circuit can be used for breadboard experiments for various application circuit designs of meminductor. Based on the meminductor, a new chaotic oscillator is proposed. The dynamical behaviors of the oscillator are investigated, including equilibrium set, Lyapunov exponent spectrum, bifurcations and dynamical map of the system. Particularly, the amplitude controlling is analyzed and coexisting attractors are found for conditions of different parameters. Furthermore, the experimental results are given to confirm the correction of the proposed meminductor model and the meminductor-based oscillator.

This paper introduces a class of polynomial maps in Euclidean spaces, investigates the conditions under which there exist Smale horseshoes and uniformly hyperbolic invariant sets, studies the chaotic dynamical behavior and strange attractors, and shows that some maps are chaotic in the sense of Li–Yorke or Devaney. This type of maps includes both the Logistic map and the Hénon map. For some diffeomorphisms with the expansion dimension equal to one or two in three-dimensional spaces, the conditions under which there exist Smale horseshoes and uniformly hyperbolic invariant sets on which the systems are topologically conjugate to the two-sided fullshift on finite alphabet are obtained; for some expanding maps, the chaotic region is analyzed by using the coupled-expansion theory and the Brouwer degree theory. For three types of higher-dimensional polynomial maps with degree two, the conditions under which there are Smale horseshoes and uniformly hyperbolic invariant sets are given, and the topological conjugacy between the maps on the invariant sets and the two-sided fullshift on finite alphabet is obtained. Some interesting maps with chaotic attractors and positive Lyapunov exponents in three-dimensional spaces are found by using computer simulations. In the end, two examples are provided to illustrate the theoretical results.

As a commercial interface, three-phase voltage-source inverters (VSI) are commonly equipped for energy conversion to export DC power from most distributed generation (DG) to the AC utility. Not only do voltage-source converters take charge of converting the power to the loads but support the grid voltage at the point of common connection (PCC) as well, which is dependent on the condition of the grid-connected loads. This paper explores the border collision and its interacting mechanism among the VSI, resistive interacting loads and grids, which manifests as the alternating emergence of the inverting and rectifying operations, where the normal operation is terminated and a new one is assumed. Their mutual effect on the power quality under investigation will cause the circuital stability issue and further deteriorate the voltage regulation capability of VSI by dramatically raising the grid voltage harmonics. It is found in a design-oriented view that the border collision operation will be induced within the unsuitable parameter space with respect to transmission lines of AC grid, resistive loads and internal resistance of VSI. The physical phenomenon is also identified by the theoretical analysis. With numerical simulations for various circuit conditions, the corresponding bifurcation boundaries are collected, where the stability of the system is lost via border collision.

A macroeconomic model based on the economic variables (i) assets, (ii) leverage (defined as debt over asset) and (iii) trust (defined as the maximum sustainable leverage) is proposed to investigate the role of credit in the dynamics of economic growth, and how credit may be associated with both economic performance and confidence. Our first notable finding is the mechanism of reward/penalty associated with patience, as quantified by the return on assets. In regular economies where the EBITA/Assets ratio is larger than the cost of debt, starting with a trust higher than leverage results in the highest long-term return on assets (which can be seen as a proxy for economic growth). Therefore, patient economies that first build trust and then increase leverage are positively rewarded. Our second main finding concerns a recommendation for the reaction of a central bank to an external shock that affects negatively the economic growth. We find that late policy intervention in the model economy results in the highest long-term return on assets. However, this comes at the cost of suffering longer from the crisis until the intervention occurs. The phenomenon that late intervention is most effective to attain a high long-term return on assets can be ascribed to the fact that postponing intervention allows trust to increase first, and it is most effective to intervene when trust is high. These results are derived from two fundamental assumptions underlying our model: (a) trust tends to increase when it is above leverage; (b) economic agents learn optimally to adjust debt for a given level of trust and amount of assets. Using a Markov Switching Model for the EBITA/Assets ratio, we have successfully calibrated our model to the empirical data of the return on equity of the EURO STOXX 50 for the time period 2000–2013. We find that dynamics of leverage and trust can be highly nonmonotonous with curved trajectories, as a result of the nonlinear coupling between the variables. This has an important implication for policy makers, suggesting that simple linear forecasting can be deceiving in some regimes and may lead to inappropriate policy decisions.

Invariant algebraic surfaces are commonly observed in differential systems arising in mathematical modeling of natural phenomena. In this paper, we study the integrability and dynamics of quadratic polynomial differential systems defined in ${ℝ}^3$ having an elliptic paraboloid as an invariant algebraic surface. We obtain the normal form for these kind of systems and, by using the invariant paraboloid, we prove the existence of first integrals, exponential factors, Darboux invariants and inverse Jacobi multipliers, for suitable choices of parameter values. We characterize all the possible configurations of invariant parallels and invariant meridians on the invariant paraboloid and give necessary conditions for the invariant parallel to be a limit cycle and for the invariant meridian to have two orbits heteroclinic to a point at infinity. We also study the dynamics of a particular class of the quadratic polynomial differential systems having an invariant paraboloid, giving information about localization and local stability of finite singular points and, by using the Poincaré compactification, we study their dynamics on the Poincaré sphere (at infinity). Finally, we study the well-known Rabinovich system in the case of invariant paraboloids, performing a detailed study of its dynamics restricted to these invariant algebraic surfaces.

In this paper, we describe in detail the global and cocycle attractors related to nonautonomous scalar differential equations with diffusion. In particular, we investigate reaction–diffusion equations with almost-periodic coefficients. The associated semiflows are strongly monotone which allow us to give a full characterization of the cocycle attractor. We prove that, when the upper Lyapunov exponent associated to the linear part of the equations is positive, the flow is persistent in the positive cone, and we study the stability and the set of continuity points of the section of each minimal set in the global attractor for the skew product semiflow. We illustrate our result with some nontrivial examples showing the richness of the dynamics on this attractor, which in some situations shows internal chaotic dynamics in the Li–Yorke sense. We also include the sublinear and concave cases in order to go further in the characterization of the attractors, including, for instance, a nonautonomous version of the Chafee–Infante equation. In this last case we can show exponentially forward attraction to the cocycle (pullback) attractors in the positive cone of solutions.

Propagating modes in a class of nonic derivative nonlinear Schrödinger equations incorporating ninth order nonlinearity are investigated by the method of dynamical systems. Because the functions $\varphi (\xi )$ and $\psi (\xi )$ in the solutions $A(x,t)=[\varphi (\xi )+i\psi (\xi )]exp(i(px-\mathrm{\Omega }t))$, $(\xi =x-ct)$ satisfy a four-dimensional integral system having two first integrals (i.e. the invariants of motion), a planar dynamical system for the squared wave amplitude $\mathrm{\Phi }={\varphi }^2+{\psi }^2$ can be derived in the invariant manifold of the four-dimensional integrable system. By using the bifurcation theory of dynamical systems, under different parameter conditions, bifurcations of phase portraits and exact periodic solutions, homoclinic and heteroclinic solutions for this planar dynamical system can be given. Therefore, under some parameter conditions, solutions $A(x,t)$ and $\varphi (\xi ),\psi (\xi )$ can be exactly obtained. Thirty six exact explicit solutions of equation are derived.

Simple systems of third-order autonomous nonlinear differential equations can exhibit chaotic behavior. In this paper, we present a new class of chaotic flow with a square-shaped equilibrium. This unique property has apparently not yet been described. Such a system belongs to a newly introduced category of chaotic systems with hidden attractors that are interesting and important in engineering applications. The mathematical model is accompanied by an electrical circuit implementation, demonstrating structural stability of the strange attractor. The circuit is simulated with PSpice, constructed, and analyzed (measured).

Astrocytes have important functions in the central nervous system (CNS) and are significant in our understanding of the neuronal network. Astrocytes modulate neuronal firings at both single cell level of tripartite synapses and the neuron-glial network level. Astrocytes release adenosine triphosphate (ATP) and glutamate into the neuron-glial network. These gliotransmitters diffuse over the network to form long distance signals to regulate neuron firings. In this paper, we study a neuron-glial network model that includes a diffusion of astrocytic ATP and glutamate to investigate how long distance diffusion of the gliotransmitters affects the information processing in a neuronal network. We find that gliotransmitters diffusion can compensate for the failure of information processing of interneuron network firings induced by defectively coupled synapses. Moreover, we find that calcium waves in astrocyte network and firings in interneuron network are both sensitive to the glutamate diffusion rate and feedback intensities of astrocytes on interneurons.

The presence of hidden attractors in dynamical systems has received considerable attention recently both in theory and applications. A novel three-dimensional autonomous chaotic system with hidden attractors is introduced in this paper. It is exciting that this chaotic system can exhibit two different families of hidden attractors: hidden attractors with an infinite number of equilibrium points and hidden attractors without equilibrium. Dynamical behaviors of such system are discovered through mathematical analysis, numerical simulations and circuit implementation.

This paper focuses on the design of the novel chaotic masking scheme via message embedded synchronization. A general class of the systems allowing the message embedded synchronization is presented here, moreover, it is shown that the generalized Lorenz system belongs to this class. Furthermore, the secure encryption scheme based on the message embedded synchronization is proposed. This scheme injects the embedded message into the dynamics of the transmitter as well, ensuring thereby synchronization with theoretically zero synchronization error. To ensure the security, the embedded message is a sum of the message and arbitrary bounded function of the internal transmitter states that is independent of the scalar synchronization signal. The hexadecimal alphabet will be used to form a ciphertext making chaotic dynamics of the transmitter even more complicated in comparison with the transmitter influenced just by the binary step-like function. All mentioned results and their security are tested and demonstrated by numerical experiments.

In this paper, we study the dynamics of autonomous ODE systems with ${𝔻}_4$ symmetry. First, we consider eight weakly-coupled oscillators and establish the condition for the existence of stable heteroclinic cycles in most generic ${𝔻}_4$-equivariant systems. Then, we analyze the action of ${𝔻}_4$ on ${ℂ}^2$ and study the pattern of periodic solutions arising from Hopf bifurcation. We identify the type of periodic solutions associated with the pairs $(H,K)$ of spatiotemporal or spatial symmetries, and prove their existence by using the $H\mathrm{\text{mod}}K$ Theorem due to Hopf bifurcation and the ${𝔻}_4$ symmetry. In particular, we give a rigorous proof for the existence of a fourth branch of periodic solutions in ${𝔻}_4$-equivariant systems. Further, we apply our theory to study a concrete case: two coupled van der Pol oscillators with ${𝔻}_4$ symmetry. We use normal form theory to analyze the periodic solutions arising from Hopf bifurcation. Among the families of the periodic solutions, we pay particular attention to the phase-locked oscillations, each of them being embedded in one of the invariant manifolds, and identify the in-phase, completely synchronized motions. We derive their explicit expressions and analyze their stability in terms of the parameters.

This paper presents an unconventional theory and method for solving general nonlinear dynamical systems. Instead of the direct iterative methods, the discretized nonlinear system is first formulated as a global optimization problem via the least squares method. A newly developed canonical duality theory shows that this nonconvex minimization problem can be solved deterministically in polynomial time if a global optimality condition is satisfied. The so-called pseudo-chaos produced by linear iterative methods are mainly due to the intrinsic numerical error accumulations. Otherwise, the global optimization problem could be NP-hard and the nonlinear system can be really chaotic. A conjecture is proposed, which reveals the connection between chaos in nonlinear dynamics and NP-hardness in computer science. The methodology and the conjecture are verified by applications to the well-known logistic equation, a forced memristive circuit and the Lorenz system. Computational results show that the canonical duality theory can be used to identify chaotic systems and to obtain realistic global optimal solutions in nonlinear dynamical systems. The method and results presented in this paper should bring some new insights into nonlinear dynamical systems and NP-hardness in computational complexity theory.

Analytical predictions of the triangle and clover heteroclinic bifurcations in the problem of self-oscillations stability loss near 1:3 resonance are provided using the method of nonlinear time transformation. The analysis was carried out considering the slow flow of a self-excited nonlinear Mathieu oscillator corresponding to the normal form near this 1:3 strong resonance. Using the Hamiltonian system of the corresponding slow flow near this resonance, the unperturbed zero-order approximation of the heteroclinic connections is established. Conditions of persistence of homoclinic connections in the perturbed first-order approximation of the heteroclinic connections provide close analytical approximations of the triangle and clover heteroclinic bifurcation curves, simultaneously. The analytical predictions are compared to the results obtained by numerical simulations for validation.