Stability and stabilization of a class of fractional-order nonlinear systems for $$\varvec{0<}\,{\varvec{\alpha }} \,\varvec{< 2}$$

This paper investigates the stability and stabilization problem of fractional-order nonlinear systems for \(0<\alpha <2\). Based on the fractional-order Lyapunov stability theorem, S-procedure and Mittag–Leffler function, the stability conditions that ensure local stability and stabilization of a class of fractional-order nonlinear systems under the Caputo derivative with \(0<\alpha <2\) are proposed. Finally, typical instances, including the fractional-order nonlinear Chen system and the fractional-order nonlinear Lorenz system, are implemented to demonstrate the feasibility and validity of the proposed method.     

Aperiodically intermittent control for synchronization of switched complex networks with unstable modes via matrix $$\varvec{\omega }$$-measure approach

The study, by using aperiodically intermittent pinning control, is to synchronize switched delayed complex networks with unstable subsystems. Matrix \(\omega \)-measure and mode-dependent average dwell time method are used to achieve globally exponential synchronization for such system. By designing the useful switching rule and control scheme, we obtain the novel synchronization criteria, which improve the conventional results. Finally, simulation analysis demonstrates the advantages of proposed innovations.     

Sliding mode control for uncertain active vehicle suspension systems: an event-triggered $$\varvec{\mathcal {H}}_{\infty }$$ H ∞ control scheme

Nonlinear Dynamics - This paper considers the event-triggered sliding mode control problem of uncertain active vehicle suspension systems. A more comprehensive polytope approach is employed to...     

Observer-based quantized sliding mode $${\varvec{\mathcal {H}}}_{\varvec{\infty }}$$ control of Markov jump systems

An adjustable quantized approach is adopted to treat the \(\mathcal {H}_{\infty }\) sliding mode control of Markov jump systems with general transition probabilities. To solve this problem, an integral sliding mode surface is constructed by an observer with the quantized output measurement and a new bound is developed to bridge the relationship between system output and its quantization. Nonlinearities incurred by controller synthesis and general transition probabilities are handled by separation strategies. With the help of these measurements, linear matrix inequalities-based conditions are established to ensure the stochastic stability of the sliding motion and meet the required \(\mathcal {H}_{\infty }\) performance level. An example of single-link robot arm system is simulated at last to demonstrate the validity.     

Resonance $$\varvec{Y}$$ -type soliton,hybrid and quasi-periodic wave solutions of a generalized $$\varvec{(2+1)}$$ -dimensional nonlinear wave equation

Nonlinear Dynamics - We present a new kernel-based algorithm for modeling evenly distributed multidimensional datasets that does not rely on input space sparsification. The presented method...     

Finite-time stability analysis of fractional-order complex-valued memristor-based neural networks with time delays

In this paper, the problem of finite-time stability of fractional-order complex-valued memristor-based neural networks (NNs) with time delays is extensively investigated. We first initiate the fractional-order complex-valued memristor-based NNs with the Caputo fractional derivatives. Using the theory of fractional-order differential equations with discontinuous right-hand sides, Laplace transforms, Mittag-Leffler functions and generalized Gronwall inequality, some new sufficient conditions are derived to guarantee the finite-time stability of the considered fractional-order complex-valued memristor-based NNs. In addition, some sufficient conditions are also obtained for the asymptotical stability of fractional-order complex-valued memristor-based NNs. Finally, a numerical example is presented to demonstrate the effectiveness of our theoretical results.     

Nonlinear dynamics of a $$\varvec{\phi ^6}-$$ modified Duffing oscillator: resonant oscillations and transition to chaos

The nonlinear dynamics of a hybrid Rayleigh–Van der Pol–Duffing oscillator includes pure and impure quadratic damping are investigated. The multiple timescales method is used to study exhaustively various resonances states. It is noticed that the system presents nine resonances states. The frequency response curves of quintic, third and second superharmonic, and subharmonic resonances states are obtained. Bistability, hysteresis, and jump phenomenon are also obtained. It is pointed out that these resonance phenomena are strongly related to the nonlinear cubic and quadratic damping and to the external force. The numerical simulations are used to make bifurcation sequences displayed by the model for each type of oscillatory. It is noticed that the pure quadratic, impure cubic damping, and external excitation affect regular and chaotic states.     

Design of a steering control law for an autonomous underwater vehicle using nonlinear $$\mathscr {H}_{\infty }$$ state feedback technique

This paper proposes a new robust nonlinear \(\mathscr {H}_{\infty }\) state feedback (NHSF) controller for an autonomous underwater vehicle (AUV) in steering plane. A three-degree-of-freedom nonlinear model of an AUV has considered for developing a steering control law. In this, the energy dissipative theory is used which leads to form a Hamilton–Jacobi–Isaacs (HJI) inequality. The nonlinear \(\mathscr {H}_{\infty }\) control algorithm has been developed by solving HJI equation such that the AUV tracks the desired yaw angle accurately. Furthermore, a path following control has been implemented using the NHSF control algorithm for various paths in steering plane. Simulation studies have been carried out using MATLAB/Simulink environment to verify the efficacies of the proposed control algorithm for AUV. From the results obtained, it is concluded that the proposed robust control algorithm exhibits a good tracking performance ensuring internal stability and significant disturbance attenuation.     

Bifurcation of solitons,peakons, and periodic cusp waves for \varvec{\theta }-equation

In this paper, we study the bifurcation of traveling wave solutions for \(\theta \) -equation using the bifurcation method and qualitative theory of dynamical systems. Not only smooth solitons but also explicit peakons and periodic cusp waves are obtained. In addition, we show that a new kind of phase portrait may admit peakons and infinitely many periodic cusp waves, in contrast to the traditional phase portraits. To the best of our knowledge, until now, this phenomenon has not appeared in any other literature. Some results of the previous studies are extended for this study.     

Recurrence behavior for controllable excitation of rogue waves in a two-dimensional $$\varvec{\mathcal {PT}}$$ PT -symmetric coupler

Nonlinear Dynamics - A (2 + 1)-dimensional variable-coefficient coupled nonlinear Schrödinger equation with different diffractions in parity-time symmetric coupler is studied, and exact...     

Solutions for Linear Conservation Laws with Velocity Fields in $$L^{\rm \infty}$$

A Space-Time Integrated Least Squares (STILS) method is derived for solving the linear conservation law with a velocity field in . An existence and uniqueness result is given for the solution of this equation. A maximum principle is established and finally a comparison with a renormalized solution is presented.     

Broken and unbroken $$\varvec{\mathcal {PT}}$$ PT -symmetric solutions of semi-discrete nonlocal nonlinear Schrödinger equation

Nonlinear Dynamics - In this letter, we obtain multi-soliton solutions in terms of ratio of ordinary determinants for semi-discrete nonlocal nonlinear Schrödinger (sd-NNLS) equation by...     

Analytical and numerical solution of an $$\varvec{n}$$n-term fractional nonlinear dynamic oscillator

Nonlinear Dynamics - This work presents a novel method for the analytical and numerical solution of an n-term fractional nonlinear dynamical system. Two simple methods, commonly known to vibration...