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The fractal and chaos are bound tightly, and their relevant researches are well-established. Few of them, however, concentrate on the research of the possibility of combining fractal and chaotic systems for generating multi-scroll chaotic attractors. This paper presents a novel non-equilibrium point chaotic system, exhibiting extremely rich and complex hidden behaviors including chaos, hyper-chaos, multi-scroll attractors, extreme multi-stability, and initial offset-boosting behavior. The proposed system is combined with fractal transformation to observe a new class of multi-scroll attractors such as multi-ring attractors, separated-scroll attractors, and nested attractors. Particularly, the first swallow-like attractors are found. Moreover, another efficient method to generate a different class of chaotic attractors applies parabola transformation and triangle transformation. Additionally, the spectrum entropy (SE) complexity is also employed to discuss the complexity of the proposed system before and after fractal, resulting in chaotic sequences with the fractal transformation that has higher complexity. Finally, we develop a hardware platform to implement the presented attractors before and after fractal in a way to confirm the accuracy of the numerical simulations, providing a theoretical basis for the next application in image encryption.

We prove that in systems undergoing Hopf bifurcations, the effects of periodic forcing can be amplified by the shearing in the system to create sustained chaotic behavior. Specifically, strange attractors with SRB measures are shown to exist. The analysis is carried out for infinite dimensional systems, and the results are applicable to partial differential equations. Application of the general results to a concrete equation, namely the Brusselator, is given.

In Danca et al. (Int J Bifurc Chaos 26(02):1650038, 2016 ), it is shown that the Rabinovich–Fabrikant (RF) system admits self-excited and hidden chaotic attractors. In this paper, we further show that the RF system also admits a pair of symmetric transient hidden chaotic attractors. We reveal more extremely rich dynamics of this system, such as a new kind of “virtual saddles.”

In this work, a new fractional-order chaotic system with a single parameter and four nonlinearities is introduced. One striking feature is that by varying the system parameter, the fractional-order system generates several complex dynamics: self-excited attractors, hidden attractors, and the coexistence of hidden attractors. In the family of self-excited chaotic attractors, the system has four spiral-saddle-type equilibrium points, or two nonhyperbolic equilibria. Besides, for a certain value of the parameter, a fractional-order no-equilibrium system is obtained. This no-equilibrium system presents a hidden chaotic attractor with a `hurricane’-like shape in the phase space. Multistability is also observed, since a hidden chaotic attractor coexists with a periodic one. The chaos generation in the new fractional-order system is demonstrated by the Lyapunov exponents method and equilibrium stability. Moreover, the complexity of the self-excited and hidden chaotic attractors is analyzed by computing their spectral entropy and Brownian-like motions. Finally, a pseudo-random number generator is designed using the hidden dynamics.

This paper presents some numerical evidence of unusual dynamics and hidden chaotic attractors of the Rabinovich–Fabrikant system, with some insightful descriptions and discussions. From a generalized Hamiltonian energy perspective, the attractors could be analyzed in more details.

Two kinds of chaotic attractors with nontrivial topologies are found in a 4D autonomous continuous dynamical system. Since the equilibria of the system are located on both sides of the basic structures of these attractors, and the basic structures of these attractors are bond orbits, we call them bondorbital attractors. They have fractional dimensions and their Kaplan–Yorke dimensions are greater than 3. The generation mechanisms of the two types of attractors are explored and analyzed based on the Shilnikov’s theorems. The type I attractors generated by system with parameter P 1 possess coexistence features, and the type II attractors generated by system with parameter P 2 have the ability to realize consecutive bond orbits. Furthermore, the type I attractors have continuous attracting basins with diagonal distribution and can be caught by means of a method of shorting capacitors in hardware experiments, whereas the type II attractors possess discrete basins of attraction and are difficult to be captured in hardware experiments. The difference between the two types of bondorbital attractors and traditional self-excited attractors in generation method is analyzed, and we also verify that they are not hidden attractors. Based on the step function sequence f ( x ,  M ,  N ), the type II attractors with at most ( N + M + 1 )-fold structures can be generated in the system with parameter P 2 . Two sets of symmetric specific initial conditions are used to verify that the system with parameter P 2 can generate bondorbital attractors with fourfold and fivefold basic structures based on f ( x , 2, 2). Some characteristics of the two classes of bondorbital attractors are listed in tabular form.

We study a rough difference equation on a discrete time set, where the driving Hölder rough path is a realization of a stochastic process. Using a modification of Davie’s approach (Cong et al. in J. Dyn. Differ. Equ. 34:605–636, 2022) and the discrete sewing lemma, we derive norm estimates for the discrete solution. In particular, when the discrete time set is regular, the system generates a discrete random dynamical system. We also generalize a recent result in Duc and Kloeden (Numerical attractors for rough differential equations, 2021) on the existence and upper semi-continuity of a global random pullback attractor under the dissipativity and the linear growth condition for the drift.

We present a novel method for interpolating univariate time series data. The proposed method combines multi-point fractional Brownian bridges, a genetic algorithm, and Takens’ theorem for reconstructing a phase space from univariate time series data. The basic idea is to first generate a population of different stochastically-interpolated time series data, and secondly, to use a genetic algorithm to find the pieces in the population which generate the smoothest reconstructed phase space trajectory. A smooth trajectory curve is hereby found to have a low variance of second derivatives along the curve. For simplicity, we refer to the developed method as PhaSpaSto-interpolation, which is an abbreviation for phase-space-trajectory-smoothing stochastic interpolation. The proposed approach is tested and validated with a univariate time series of the Lorenz system, five non-model data sets and compared to a cubic spline interpolation and a linear interpolation. We find that the criterion for smoothness guarantees low errors on known model and non-model data. Finally, we interpolate the discussed non-model data sets, and show the corresponding improved phase space portraits. The proposed method is useful for interpolating low-sampled time series data sets for, e.g., machine learning, regression analysis, or time series prediction approaches. Further, the results suggest that the variance of second derivatives along a given phase space trajectory is a valuable tool for phase space analysis of non-model time series data, and we expect it to be useful for future research.

In this paper, it is found numerically that the previously found hidden chaotic attractors of the Rabinovich–Fabrikant system actually present the characteristics of strange nonchaotic attractors. For a range of the bifurcation parameter, the hidden attractor is manifestly fractal with aperiodic dynamics, and even the finite-time largest Lyapunov exponent, a measure of trajectory separation with nearby initial conditions, is negative. To verify these characteristics numerically, the finite-time Lyapunov exponents, ‘0-1’ test, power spectra density, and recurrence plot are used. Beside the considered hidden strange nonchaotic attractor, a self-excited chaotic attractor and a quasiperiodic attractor of the Rabinovich–Fabrikant system are comparatively analyzed.

It is believed that local activation is the origin of all complexities, and the locally active memristive synaptic neural network can generate complex chaotic dynamic behaviors, such as hyperchaotic, multi-scroll, multi-stability and hidden dynamical behaviors. However, there are few studies on the simultaneous occurrence of multiple complex dynamic behaviors in neural networks. No chaotic system of multi-scroll hyperchaotic hidden attractors based on neural network has been found yet. To solve the problem, in this paper, we propose a new locally active memristive Hopfield neural network (HNN) model based on a multi-segment function, which is affected by electromagnetic radiation and external current. The multi-scroll hyperchaotic hidden attractors are found in the memristive HNN for the first time. Theoretical analysis and numerical simulation results show that the memristive HNN model has no equilibrium point, and the number of multi-scroll attractors is controlled by the state equation parameters of the memristive synapse. In addition, the structures and number of scrolls are also affected by electromagnetic radiation and external current. At the same time, under the appropriate parameter conditions, by modifying the initial value of the system, the memristive HNN has a controllable number of coexisting attractors, showing extreme multi-stability. Finally, a memristive HNN analog circuit is designed. The hardware experiment results reproduce the multi-scroll dynamics phenomenon, which verifies the correctness of the theoretical analysis and numerical simulation.