We study upper bounds for the moment of permanent ε -saturation in finite metric graphs. The dynamics is generated by moving points travelling with unit speed along edges and branching into all outgoing directions whenever they reach a vertex. We first reformulate this branched dynamics in terms of birth times at vertices and prove a sufficient same-time criterion for permanent ε -saturation. The main rigorous estimate is obtained from a rotation, regarded as a two-interval exchange transformation. More precisely, if the graph contains two closed walks based at the initial vertex whose lengths have irrational ratio, then the covering properties of the corresponding rotation imply an explicit upper bound for the permanent saturation time. In particular, bounded-type rotations yield a bound of order ε ^-1 . We also construct a more general auxiliary interval exchange transformation on the set of oriented edges. This construction depends on cyclic orders at the vertices and organizes the ordered edge-state data of the graph. Since the branched graph dynamics is non-invertible, whereas an interval exchange transformation is invertible away from discontinuities, this auxiliary IET is not identified with the full graph dynamics. Instead, we formulate the additional birth-time transfer property required for recurrence estimates of the auxiliary IET to imply saturation bounds. We also discuss rotation-type and non-rotation examples of graph-induced self-similar IETs, together with numerical illustrations for star and complete graphs.
Our goal is to derive new general integral representations of the energies of variational extremals in terms of the data on the boundary. They can be interpreted as nonlinear analogs of the classical Clapeyron’s theorem in linear elasticity theory. We generalize the approach of E. Noether who linked variational symmetries with conservation laws, by introducing what we call “partial variational symmetries”—smooth 1-parameter families of transformations that exploit some special properties of the Lagrangian, for instance, homogeneity in one or more variables or scale invariance. We associate different known and new integral representations of the energies with particular partial variational symmetries. The effectiveness of the obtained results is illustrated through solutions of several problems of nonlinear elasticity related to metastability and quasiconvexification.
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.
