International Journal of Bifurcation and Chaos

We provide a multiparameter analysis of dynamics in a nonautonomous system of two alternately exciting self-oscillatory elements that are able to demonstrate a uniformly chaotic attractor of Smale–Williams type in the stroboscopic Poincaré map. Parameter space charts of regular and chaotic regimes are presented. Possible scenarios of the onset of hyperbolic chaos are discussed. The numerical studies are supplemented by experimental results obtained for a laboratory electronic device.

Synchronization is a common phenomenon whereby a dynamical system follows the pacemaker provided by an external forcing. Often, such systems have multiple synchronization modes, which are equivalent solutions. We investigate the specific case of two to one synchronization produced by the periodic forcing of a van der Pol oscillator where two possible modes, shifted by one period of the modulation, exist. By studying the flow and the local Lyapunov exponents along the orbit we give an explanation of the noise induced jumps observed in a stochastic forced oscillator. While this investigation gives results that are specific to this system, the framework presented is more general and can be applied to any system showing similar jumping dynamics.

Two-dimensional parameter-space diagrams related to a driven Josephson junction are reported. Three cases are considered, namely those involving the external direct current as one of the parameters. Typical periodic structures embedded in a chaotic region are observed in all diagrams, organized in different ways: (i) As structures with a similar shape to the Arnold tongues of the circle map, in period-adding sequences, and (ii) as structures with other shapes, in arrangements including two mixed sets of period-adding sequences.

This paper describes two simple three-dimensional autonomous chaotic flows whose attractor dimensions can be adjusted continuously from $2.0$ to $3.0$ by a single control parameter. Such a parameter provides a means to explore the route through limit cycles, period-doubling, dissipative chaos, and eventually conservative chaos. With an absolute-value nonlinearity and certain choices of parameters, the systems have a vast and smooth continual transition path from dissipative chaos to conservative chaos. One system is analyzed in detail by means of the largest Lyapunov exponent, Kaplan–Yorke dimension, bifurcations, coexisting attractors and eigenvalues of the Jacobian matrix. An electronic version of the system has been constructed and shown to perform in accordance with expectations.

In previous papers [Isagi et al., 1997; Satake & Iwasa, 2000], a forest model was proposed. The authors demonstrated numerically that the mature forest could possibly exhibit annual reproduction (fixed point synchronization), periodic and chaotic synchronization as the energy depletion constant $d$ is gradually increased. To understand such rich synchronization phenomena, we are led to study global dynamics of a piecewise smooth map $f_{d,\beta }$ containing two parameters $d$ and $\beta$. Here $d$ is the energy depletion quantity and $\beta$ is the coupling strength. In particular, we obtain the following results. First, we prove that $f_{d,0}$ has a chaotic dynamic in the sense of Devaney on an invariant set whenever $d>1$, which improves a result of [Chang & Chen, 2011]. Second, we prove, via the Schwarzian derivative and a generalized result of [Singer, 1978], that $f_{d,\beta }$ exhibits the period adding bifurcation. Specifically, we show that for any $\beta >0$, $f_{d,\beta }$ has a unique global attracting fixed point whenever $d\le \frac{1}{(\beta +1){(\frac{\beta +1}{\beta +2})}^{\beta }}$ ($<1$) and that for any $\beta >0$, $f_{d,\beta }$ has a unique attracting period $k+1$ point whenever $d$ is less than and near any positive integer $k$. Furthermore, the corresponding period $k+1$ point instantly becomes unstable as $d$ moves pass the integer $k$. Finally, we demonstrate numerically that there are chaotic dynamics whenever $d$ is in between and away from two consecutive positive integers. We also observe the route to chaos as $d$ increases from one positive integer to the next through finite period doubling.

In this paper, we give several classical definitions of topological entropy (on a noncompact and noninvariant subset) for nonautonomous dynamical system. Furthermore, their relationships are established.

In this paper, a new method is proposed for the identification of the FitzHugh–Nagumo (FHN) model dynamics via deterministic learning. The FHN model is a classic and simple model for studying spiral waves in excitable media, such as the cardiac tissue, biological neural networks. Firstly, the FHN model described by partial differential equations (PDEs) is transformed into a set of ordinary differential equations (ODEs) by using finite difference method. Secondly, the dynamics of the ODEs is identified using the deterministic learning theory. It is shown that, for the spiral waves generated by the FHN model, the dynamics underlying the recurrent trajectory corresponding to any spatial point can be accurately identified by using the proposed approach. Numerical experiments are included to demonstrate the effectiveness of the proposed method.

A new chaotic system is proposed to generate multiscroll chaotic attractors. The major method used is for the step function to act as a nonlinear function. To prove that the proposed system can generate multiscroll chaotic attractors, the equilibrium point, the time domain waveform and the phase diagram of the proposed system are calculated. Finally, the design of the hardware circuit produces experimental results at a maximum of 8-scroll hardware. Theoretical analysis, simulation and hardware experimental results are fully matched, which further proves the existence of the proposed system and the physical realization. This provides the possibility for future applications in engineering.

This paper investigates the global lagged finite-time synchronization of the master-slave Lur’e systems subject to time delay of signal transmission. By designing a variable-substitution and feedback controller, a master-slave finite-time synchronization scheme for the Lur’e systems with time delay is built up. Two delay-independent global lagged finite-time synchronization criteria are proved in the forms of linear matrix inequalities (LMIs), and the corresponding settling time of synchronization is analytically estimated. The obtained LMI criteria are applied to Chua’s oscillators, obtaining some easily implemented algebraic criteria under various single-variable-substitution and feedback controller, which are then optimized to improve their conservative property. Finally, several numerical examples are illustrated to verify the effectiveness of the optimized criteria.

In this paper, a delay-coupled Duffing equation is studied. By the characteristic roots technique, sufficient conditions are obtained for Hopf bifurcation occurrence. And the spatio-temporal patterns of the bifurcating periodic solutions are also obtained, some examples are given to demonstrate the theoretical analysis. Especially, the obtained numerical simulation results show that there are hidden attractors in this delayed system, which can coexist with stable equilibrium or stable bifurcating orbits.

This work deals with the bifurcation phenomena that occur during the open-loop operation of a single-fed three-phase wound rotor induction motor. This paper demonstrates the occurrence of saddle-node bifurcation, hysteresis, supercritical saddle-node bifurcation, cusp and Hopf bifurcation during the individual operation of this electromechanical system. Some experimental results associated with the bifurcation phenomena are presented.

The existence of minimal pullback attractors in $L^2(\mathrm{\Omega })$ for a nonautonomous reaction–diffusion equation, in the frameworks of universes of fixed bounded sets and that given by a tempered growth condition, is proved in this paper, when the domain $\mathrm{\Omega }$ is a general nonempty open subset of ${ℝ}^N$, and $h\in L_{\mathrm{\text{loc}}}^2(ℝ;H^{-1}(\mathrm{\Omega }))$. The main concept used in the proof is the asymptotic compactness of the process generated by the problem. The relation among these families is also discussed.

In this paper, we study the dynamics in delayed nonlinear financial system, with particular attention focused on Hopf and double Hopf bifurcations. Firstly, we identify the critical values for stability switches, Hopf and double Hopf bifurcations. We show how the parameters affect the dynamical behavior of the system. Secondly, the normal forms near the Hopf and double Hopf bifurcations, as well as the classifications of local dynamics are analyzed. These bifurcations lead a chaotic system to be stable states, such as the coexistence of a pair of stable equilibria or a pair of stable periodic oscillations, and then chaos disappears. Numerical simulations are presented to verify the analytical predictions. Furthermore, detailed numerical analysis using MATLAB extends the local bifurcation analysis to a global picture, namely, a family of stable periodic solutions exist in a large region of delay and “chaos switchover” phenomenon appears. Therefore, in accordance with the above theoretical analysis, reasonable parameters can be designed in order to achieve various applications.

Unraveling parameters in real systems, such as physical systems and biological systems, is one of the central topics in model-driven research including computational systems biology. The existing adaptive synchronization technique, though theoretically proved to be valid for parameters identification in some representative models, is always showing unsatisfactory convergence rate when the number of parameters increases and real data are taken into account. In this paper, a piecewise adaptive rule with a fractional power based on finite time control is proposed to accelerate the rate for parameter identification. The global feasibility of the proposed rule is proved theoretically and illustrated by several examples of physical and biological significance.

This paper presents online methods for controlling local bifurcations of power grids with the goal of increasing bifurcation values (i.e. increasing load margins) via network topology optimization, a low-cost control. In other words, this paper presents online methods for increasing power transfer capability subject to static stability limit via switching transmission line out/in (i.e. disconnecting a transmission line or connecting a transmission line). To illustrate the impact of network topology on local bifurcations, two common local bifurcations, i.e. saddle-node bifurcation and structure-induced bifurcation on small power grids with different network topologies are shown. A three-stage online control methodology of local bifurcations via network topology optimization is presented to delay local bifurcations of power grids. Online methods must meet the challenging requirements of online applications such as the speed requirement (in the order of minutes), accuracy requirement and robustness requirement. The effectiveness of the three-stage methodology for online applications is demonstrated on the IEEE 118-bus and a 1648-bus practical power systems.

It is practically known that a recurrence plot, a two-dimensional visualization of time series data, can contain almost all information related to the underlying dynamics except for its spatial scale because we can recover a rough shape for the original time series from the recurrence plot even if the original time series is multivariate. We here provide a mathematical proof that the metric defined by a recurrence plot [Hirata et al., 2008] is equivalent to the Euclidean metric under mild conditions.

This work studies periodic orbits as action minimizers in the spatial isosceles three-body problem with mass $M=[1,m,1]$. In each period, the body with mass $m$ moves up and down on a vertical line, while the other two bodies have the same mass $1$, and rotate about this vertical line symmetrically. For given $m>0$, such periodic orbits form a one-parameter set with a rotation angle $\theta$ as the parameter.

Two new phenomena are found for this set. First, for each $m>0$, this set of periodic orbits bifurcate from a circular Euler (central configuration) orbit to a Broucke (collision) orbit as $\theta$ increases from $0$ to $\pi$. There exists a critical rotation angle ${\theta }_0(m)$, where the orbit is a circular Euler orbit if $0<\theta \le {\theta }_0(m)$; a spatial orbit if ${\theta }_0(m)<\theta <\pi$; and a Broucke (collision) orbit if $\theta =\pi$. The exact formula of ${\theta }_0(m)$ is numerically proved to be ${\theta }_0(m)=\frac{\pi }{4}\sqrt{\frac{4m+1}{m+2}}$. Second, oscillating behaviors occur at rotation angle $\theta =\pi /2$ for all $m\in [0.1,3]$. Actually, the orbit with $\theta =\pi /2$ runs on its initial periodic shape for only a few periods. It breaks the first periodic shape and becomes irregular in a moment. However, it runs close to a different periodic shape after a while. In a short time, it falls apart from the second periodic shape and runs irregularly again. Such oscillation continues as time $t$ increases. Up to $t=1.2\times 10^5$, the orbit is bounded and keeps oscillating between periodic shapes and irregular motions. Further study implies that, for each $m\in [0.1,3]$, the angle between any two consecutive periodic shapes is a constant. When $m>3$, similar oscillating behaviors are expected.

In the present paper, we perform a complete bifurcation analysis of a two-stage population model, in which the per capita birth rate and stage transition rate from juveniles to adults are density dependent and take the general Beverton–Holt functions. Our study reveals a rich bifurcation structure including codimension-one bifurcations such as saddle-node, Hopf, homoclinic bifurcations, and codimension-two bifurcations such as Bogdanov–Takens (BT), Bautin bifurcations, etc. Moreover, by employing the polynomial analysis and approximation techniques, the existences of equilibria, Hopf and BT bifurcations as well as the formulas for calculating their bifurcation sets have been provided. Finally, the complete bifurcation diagrams and associate phase portraits are obtained not only analytically but also confirmed and extended numerically.

In this paper, we study the vibrational resonance (VR) phenomenon as a useful mechanism for energy harvesting purposes. A system, driven by a low frequency and a high frequency forcing, can give birth to the vibrational resonance phenomenon, when the two forcing amplitudes resonate and a maximum in amplitude is reached. We apply this idea to a bistable oscillator that can convert environmental kinetic energy into electrical energy, that is, an energy harvester. Normally, the VR phenomenon is studied in terms of the forcing amplitudes or of the frequencies, that are not always easy to adjust and change. Here, we study the VR generated by tuning another parameter that is possible to manipulate when the forcing values depend on the environmental conditions. We have investigated the dependence of the maximum response due to the VR for small and large variations in the forcing amplitudes and frequencies. Besides, we have plotted color coded figures in the space of the two forcing amplitudes, in which it is possible to appreciate different patterns in the electrical power generated by the system. These patterns provide useful information on the forcing amplitudes in order to produce the optimal electrical power.

In this paper, we analyze chaotic dynamics for two-dimensional nonautonomous maps through the use of a nonautonomous version of the Conley–Moser conditions given previously. With this approach we are able to give a precise definition of what is meant by a chaotic invariant set for nonautonomous maps. We extend the nonautonomous Conley–Moser conditions by deriving a new sufficient condition for the nonautonomous chaotic invariant set to be hyperbolic. We consider the specific example of a nonautonomous Hénon map and give sufficient conditions, in terms of the parameters defining the map, for the nonautonomous Hénon map to have a hyperbolic chaotic invariant set.