International Journal of Bifurcation and Chaos

A detailed mathematical analysis of the two-dimensional hybrid model for the lateral dynamics of walking-like mechanical systems (the so-called hybrid inverted pendulum) is presented in this article. The chaotic behavior, when being externally harmonically perturbed, is demonstrated. Two rather exceptional features are analyzed. Firstly, the unperturbed undamped hybrid inverted pendulum behaves inside a certain stability region periodically and its respective frequencies range from zero (close to the boundary of that stability region) to infinity (close to its double support equilibrium). Secondly, the constant lateral forcing less than a certain threshold does not affect the periodic behavior of the hybrid inverted pendulum and preserves its equilibrium at the origin. The latter is due to the hybrid nature of the equilibrium at the origin, which exists only in the Filippov sense. It is actually a trivial example of the so-called pseudo-equilibrium [Kuznetsov et al., 2003]. Nevertheless, such an observation holds only for constant external forcing and even arbitrary small time-varying external forcing may destabilize the origin. As a matter of fact, one can observe many, possibly even infinitely many, distinct chaotic attractors for a single system when the forcing amplitude does not exceed the mentioned threshold. Moreover, some general properties of the hybrid inverted pendulum are characterized through its topological equivalence to the classical pendulum. Extensive numerical experiments demonstrate the chaotic behavior of the harmonically perturbed hybrid inverted pendulum.

A reaction–diffusion Gierer–Meinhardt system with homogeneous Neumann boundary condition on one-dimensional bounded spatial domain is considered in the present article. Local asymptotic stability, Turing instability and existence of Hopf bifurcation of the constant positive equilibrium are explored by analyzing in detail the associated eigenvalue problem. Moreover, properties of spatially homogeneous Hopf bifurcation are carried out by employing the normal form method and the center manifold technique for reaction–diffusion equations. Finally, numerical simulations are also provided in order to check the obtained theoretical conclusions.

A reaction–diffusion predator–prey system with homogeneous Dirichlet boundary conditions describes the lethal risk of predator and prey species on the boundary. The spatial pattern formations with the homogeneous Dirichlet boundary conditions are characterized by the Turing type linear instability of homogeneous state and bifurcation theory. Compared with homogeneous Neumann boundary conditions, we see that the homogeneous Dirichlet boundary conditions may depress the spatial patterns produced through the diffusion-induced instability. In addition, the existence of semi-trivial steady states and the global stability of the trivial steady state are characterized by the comparison technique.

It is mathematically challenging to analytically show that complex dynamical phenomena observed in simulations and experiments are truly chaotic. The Shil’nikov lemma provides a useful theoretical tool to prove the existence of chaos in three-dimensional smooth autonomous systems. It requires, however, the proof of existence of a homoclinic or heteroclinic orbit, which remains a very difficult technical problem if contigent on data. In this paper, for the Chen system with linear time-delay feedback, we demonstrate a homoclinic orbit by using a modified undetermined coefficient method and we propose a spiral involute projection method. In such a way, we identify experimentally the asymmetrical homoclinic orbit in order to apply the Shil’nikov-type lemma and to show that chaos is indeed generated in the Chen circuit with linear time-delay feedback. We also identify the presence of a single-scroll attractor in the Chen system with linear time-delay feedback in our experiments. We confirm that the Chen single-scroll attractor is hyperchaotic by numerically estimating the finite-time local Lyapunov exponent spectrum. By means of a linear scaling in the coordinates and the time, such a method can also be applied to the generalized Lorenz-like systems. The contribution of this work lies in: first, we treat the trajectories corresponding to the real eigenvalue and the image eigenvalues in different ways, which is compatible with the characteristics of the trajectory geometry; second, we propose a spiral involute projection method to exhibit the trajectory corresponding to the image eigenvalues; third, we verify the homoclinic orbit by experimental data.

In recent years, scholars studied and proposed some secure color image encryption algorithms. However, the majority of the published algorithms encrypted red, green and blue (called $R$, $G$, $B$ for short) components independently. In the paper, we propose a color image encryption scheme based on hyperchaotic system and permutation-diffusion architecture. The encryption algorithm utilizes a block permutation which is realized by mixing $R$, $G$, $B$ components to strengthen the dependence of each component. Besides, it can reduce time consumption. Then, the key streams generated by the hyperchaotic system are exploited to diffuse the pixels, the three components affect each other again. And in the diffusion process, we can get two totally different encrypted images even though we change the last pixel because the $G$ component is diffused in reverse order. The experimental results reveal that our algorithm possesses better abilities of resisting statistical attacks and differential attacks, larger key space, closer information entropy to 8, and faster encryption speed compared with other chaos-based color image encryption algorithms.

In this paper, a new structure of chaotic systems is proposed. There are many examples of differential equations with analytic solutions. Chaotic systems cannot be studied with the classical methods. However, in this paper we show that a system that has a simple analytical solution can also have a strange attractor. The main goal of this paper is to show examples of chaotic systems with a simple analytical solution that is unstable so that the chaotic orbit does not track it. We believe the structures presented here are new. Two categories of chaotic systems are described, and their dynamical properties are investigated. The proposed systems have analytic solutions that exist far from the equilibrium. Of course, all strange attractors are dense in unstable periodic orbits, but mostly the equations that describe these orbits are unknown and difficult to calculate. The analytical solutions provide examples where the orbits can be calculated despite their instability.

Based on the study on Jerk chaotic system, a multiscroll hyperchaotic system with hidden attractors is proposed in this paper, which has infinite number of equilibriums. The chaotic system can generate $N+M+2$ scroll hyperchaotic hidden attractors. The dynamic characteristics of the multiscroll hyperchaotic system with hidden attractors are analyzed by means of dynamic analysis methods such as Lyapunov exponents and bifurcation diagram. In addition, we have studied the synchronization of the system by applying an adaptive control method. The hardware experiment of the proposed multiscroll hyperchaotic system with hidden attractors is carried out using discrete components. The hardware experimental results are consistent with the numerical simulation results of MATLAB and the theoretical analysis results.

As a basic and crucial security requirement for Wireless Sensor Networks (WSNs), authentication is generally used to prevent various communication attacks such as Denial-of-Service (DoS) attack. A novel broadcast authentication framework is proposed in this paper, where an Identity-Based Signature scheme by using the Extended Chaotic Maps (ECM-IBS) is designed to authenticate all broadcast messages and specifically, a chaos-based hash function is used for message authentication in the WSNs. It is implemented using a WSN hardware device of CC2530 and its performance is analyzed under various methods. Performance analysis and experimental results show that the proposed ECM-IBS scheme has a quick signature generation speed, low energy consumption and short verification delay, and can be applied to WSN applications.

This paper investigates the control of multistability in a self-excited memristive hyperchaotic oscillator using linear augmentation method. Such a method is advantageous in the case of system parameters that are inaccessible. The effectiveness of the applied control scheme is revealed numerically through the nonlinear dynamical tools including bifurcation diagrams, Lyapunov exponent spectrum, phase portraits, basins of attraction and relative basin sizes. Results of such numerical methods reveal that the asymmetric pair of chaotic attractors which were coexisting with the symmetric periodic one in the system, are progressively annihilated as the coupling parameter is increasing. The main transitions observed in the control system are the coexistence of three distinct attractors for weak values of the coupling strength. Above a certain critical value of the coupling parameter, only two attractors are now coexisting within the system. Finally, for higher values of the control strength, the controlled system becomes regular and monostable.

In this paper, we incorporate immune systems into an HIV model, which considers both logistic target-cell proliferation and viral cell-to-cell transmission. We study the dynamics of this model including the existence and stability of equilibria. Based on the existence of equilibria, we focus on the backward bifurcation and forward bifurcation. Considering the stability of equilibria, Hopf bifurcation is discussed by identifying the basic reproduction number $R_0$ as bifurcation parameter. The direction and stability of Hopf bifurcation are investigated by computing the first Lyapunov exponent. Specially, the effects of immune response on the basic reproduction number $R_0$ and viral dynamics are addressed by deriving the sensitivity analysis. As a result, we find that the removal rate of infected cells by cytotoxic T lymphocytes (CTLs), ${\gamma }_1$, is the predominant factor of $R_0$. However, we conclude from numerical results that it is unfeasible to decrease $R_0$ by increasing the value of ${\gamma }_1$ constantly. Numerical simulation is also presented to demonstrate the applicability of the theoretical predictions. These dynamics are investigated by the proposed model to point out the importance and complexity of immune responses in fighting HIV replication.

As an emerging field for sampling paradigms, compressive sensing (CS) can sample and represent signals at a sub-Shannon–Nyquist rate. To realize CS from theory to practice, the random sensing matrices must be substituted by faster measurement operators that correspond to feasible hardware architectures. In recent years, binary matrices have attracted much research interest because of their multiplier-less and faster data acquisition. In this work, we aim to pinpoint the potential of chaotic binary sequences for constructing high-efficiency sensing implementations. In particular, the proposed chaotic binary sensing matrices are verified to meet near-optimal theoretical guarantees in terms of both the restricted isometry condition and coherence analysis. Simulation results illustrate that the proposed chaotic constructions have considerable sampling efficiency comparable to that of the random counterparts. Our framework encompasses many families of binary sensing architectures, including those formed from Logistic, Chebyshev, and Bernoulli binary chaotic sequences. With many chaotic binary sensing architectures, we can then more easily apply CS paradigm to various fields.

A Rao–Nam cryptosystem based on error correction code is proposed to provide both security and reliability. Since its security is drastically constrained by the limited error syndromes, in this paper, an improved Rao–Nam cryptosystem based on fractional order hyperchaotic system and Extended Difference Family–Quasi-Cyclic–Low-Density Parity-Check (EDF–QC–LDPC) codes is proposed to improve the security. A four-dimensional fractional order hyperchaotic system is constructed and is used to generate an excellent pseudorandom sequence. By replacing error syndromes with the pseudorandom sequence and permuting the coded message dynamically, the security of the Rao–Nam cryptosystem is enhanced greatly. The ability of the improved Rao–Nam cryptosystem against known attacks is analyzed and the error correction performance with different parameters is simulated. The results show that the proposed cryptosystem has a significant advantage of resisting the chosen-plaintext attack. Moreover, the proposed cryptosystem retains high capacity of error correction.

For a commutative nonautonomous dynamical system we show that topological transitivity of the nonautonomous system induced on probability measures (hyperspaces) is equivalent to the weak mixing of the induced systems. Several counter examples are given for the results which are true in autonomous but need not be true in nonautonomous systems. Wherever possible sufficient conditions are obtained for the results to hold true. For a commutative periodic nonautonomous system on intervals, it is proved that weak mixing implies Devaney chaos. Given a periodic nonautonomous system, it is shown that sensitivity is equivalent to some stronger forms of sensitivity on a closed unit interval.

The piecewise logistic map (PLM) is an improved version of logistic map, which is specifically designed for cryptographic application. However, the probability density distribution of PLM is not uniform. To overcome this shortage, a parameter-coupled piecewise logistic map (PCPLM) is presented. Using the PCPLM as a basic unit, we construct a four-dimensional chaotic model, which has even probability density distribution. The four-dimensional chaotic model based on PCPLMs also has a parallel structure, which is beneficial to enhancing the running efficiency. Finally, a novel pseudorandom number generator (PRNG) is presented based on this four-dimensional chaotic model. The security analysis and simulation tests on the proposed PRNG are performed. The results both confirm that the proposed PRNG is secure and efficient. Therefore, it can be used as a candidate for data security.

Let $h:F\to F$ be a continuous selfmap of a nontrivial compact metric space $(F,\rho )$. This paper derives the large deviations theorem in a sequence. It is shown that if $(h,\eta )$ satisfies the large deviations theorem in a sequence then it is an E-system which implies that $h$ is syndetically transitive, and that if the pair $(h,\eta )$ satisfies the large deviations theorem in a sequence and $F$ is an infinite space, then $h$ is syndetically sensitive, which implies that $h$ is ergodically sensitive. Moreover, it is proved that if the pair $(h,\eta )$ satisfies the large deviations theorem in a syndetic sequence, then $\eta$ is an ergodic measure with supp$(\eta )=F$. Our main results extend and improve the corresponding results in the literature.

The two-dimensional border collision normal form is considered. It is known that multiple attractors can exist in this piecewise smooth system. We show that in appropriate parameter regions there can be a robust transition from a stable fixed point to multiple coexisting attractors with toological dimensions equal to two.

This is a final part of the series of our papers devoted to a multivalued version of the (Sharkovsky type) Block cycle coexistence theorem. It improves our last general result in the sense that its part related to the usual ordering of positive integers becomes a full analogy of the standard single-valued case, while the alternative part related to the Sharkovsky ordering of positive integers is an analogy of the multivalued case for interval maps, provided there exists a fixed point. That is why we call the obtained theorem here as “sharp”. This theorem is still applied via the associated Poincaré translation operators to differential equations and inclusions on the circle. All the deterministic results are also randomized in an advantageous way.

This paper investigates two generalized two-component peakon type dual systems, which can be reduced to the same planar dynamical systems via the dynamical system approach and the theory of singular traveling wave systems, where one of them contains the two-component Camassa–Holm system. By bifurcation analysis on the corresponding traveling wave system, we obtain the phase portraits and derive possible exact traveling wave solutions that include solitary wave solution, peakon and anti-peakon, pseudo-peakon, periodic peakon, compacton and periodic wave solution. Our results are also applicable to the two-component Camassa–Holm equation.