Unlike linear vector fields, which exhibit simple dynamics, piecewise linear vector fields support significantly richer and more complex behavior. For instance, whereas planar linear vector fields are known to be devoid of limit cycles, their piecewise counterparts may exhibit limit cycles. In dimension three, although linear vector fields cannot support isolated invariant tori, it is natural to ask whether piecewise linear vector fields may admit such structures. In this context, the central objective of this paper is to establish the existence of both attracting and repelling invariant tori in 3D piecewise linear vector fields, which are referred to as limit tori. Motivated by this question, we develop a general analytical mechanism for detecting and characterizing isolated invariant tori in a broader class of piecewise smooth differential systems. This mechanism provides sufficient conditions for the existence of an invariant torus associated with a normally hyperbolic invariant closed curve of the first return map, arising through a Neimark–Sacker bifurcation. We use this mechanism to derive a family of 3D piecewise linear vector fields that exhibits isolated invariant tori, which are attracting or repelling depending on the parameter values and persist under small perturbations. To the best of our knowledge, this constitutes the first analytical detection of such attractors and repellers in piecewise linear vector fields.
The presence of time delays enriches the dynamics of chaotic systems and makes synchronization between multiple chaotic systems more challenging. To address the synchronization problem between multiple time-delayed chaotic systems, a generalized triple dislocated matrix projective synchronization (GTDMPS) framework is developed. Four different delayed 4-dimensional chaotic systems are considered, and their dynamical properties including equilibrium points, dissipation, Lyapunov exponents, Kaplan Yorke dimension and sensitivity to initial conditions are investigated. Nonlinear active control and Lyapunov stability theory are employed to achieve synchronization between multiple drive systems and a response system. The synchronized chaotic signals are subsequently used to generate key streams for affine cipher-based image encryption algorithm. The security performance of the proposed encryption scheme is evaluated through key space analysis, histogram analysis, correlation coefficient, information entropy, NPCR and UACI, resistance analysis against known-plaintext and chosen-plaintext attack and computational efficiency metrics. Numerical simulations and image encryption experiments demonstrate the effectiveness of the proposed synchronization framework and its applicability in secure image encryption.
This paper introduces dual field-theoretic Hamel’s integrator in the form of discrete mechanical systems that utilize dynamic variables measured relative to moving frames. As in the continuous setting, the use of moving frames results in a flexible and overall more efficient formalism. The integrator is derived in a systematic way using the formalism of composition maps. It is then utilized for numerical integration of geometrically exact beams.
We prove the global existence of the solution for one-dimensional cubic fractional ( m⩾ 2 ) nonlinear Schrödinger equations in modulation spaces M_p,p'^s_p for p close to 2 with no smallness condition on initial data, where s_p=(m-2)(1/2-1/p) . The proof adapts a splitting method inspired by the work of Vargas-Vega, Hyakuna-Tsutsumi, Chaichenets et al. to the modulation spaces and exploits polynomial growth of the fractional Schrödinger semigroup on modulation spaces M_p,p' with loss of regularity s_p .
Identifying the optimal stimulation threshold for regulating spatiotemporal bioelectrical activity remains challenging under discontinuous physiological factors. Accordingly, to explore the spatial complexity of neocortical neurons, a non-smooth feedback strategy of memristive autaptic current involving the threshold of membrane potential is proposed, and a delayed reaction–diffusion memristive Wilson neuron model is established under the homogeneous Neumann boundary condition. The existence and local dynamics of the equilibrium point of subsystems are analyzed. In detail, we derive the analytical criteria for saddle-node bifurcation, diffusion-driven Turing bifurcation, and delay-induced Hopf bifurcation. Furthermore, we obtain computable formulas for the stability and direction of spatially periodic solutions caused by Hopf bifurcation. Intriguingly, hidden extreme bistability and fascinating multiple stability switches are triggered under certain delayed self-feedback. Based on differential inclusion theory, we investigate the existence of sliding segments, stable pseudo-equilibrium points, boundary equilibrium points, and tangent points controlled by non-smooth thresholds. Besides, the evolutionary mechanisms of the spatiotemporal self-excited and hidden sliding attractors are revealed by the geometric dynamics approach, and the critical delay corresponding to global sliding-grazing bifurcation and buckling bifurcation is further determined. The obtained results provide substantial theoretical guidance for the diagnosis and rehabilitation of neurological diseases.
