Linear response theory asserts that sufficiently small external biases produce currents proportional to the applied force and forms the theoretical foundation of nonequilibrium transport. Here we demonstrate that linear response can break down even in uniformly hyperbolic deterministic systems when hierarchical asymmetry is present. Using a minimal class of uniformly expanding chaotic maps with hierarchical multiscale structure, we show that progressively finer transport channels become dynamically active as the applied bias decreases. The resulting force current relation is monotone and exhibits a hierarchical, fractal-like organization of activation thresholds. As a consequence, the effective mobility diverges as F to 0, demonstrating breakdown of linear response despite strong chaos and uniform hyperbolicity. The effect arises from deterministic multiscale activation rather than intermittency, stochastic noise, or singular invariant measures. These results identify hierarchy as an independent deterministic mechanism for nonperturbative transport response and demonstrate that uniform hyperbolicity alone does not guarantee the validity of linear response.

We consider a generalization of the Kuramoto model in which phase oscillators are represented by unit vectors coupled by a matrix of constant coefficients. We show that, when the oscillators are driven by an external periodic force, several resonances appear, giving rise to Arnold tongues that can be observed as the intensity and frequency of the external force are varied. Applying the Ott-Antonsen ansatz we obtain equations for the module and phase of order parameter. As these equations are explicitly time-dependent, we resort to extensive numerical simulations to uncover the resonant modes and their associated Arnold tongues and devil's staircases. These results contrast with the original forced Kuramoto model, where only $1:1$ resonance is possible.

We introduce a two-phase quadratic integrate-and-fire (QIF) neuron whose membrane potential evolves according to two alternating Riccati equations within finite bounds. This simple extension removes the unphysical voltage divergence of the standard QIF model while producing realistic spike waveforms. Despite this modification, the system retains an exact low-dimensional description in the thermodynamic limit, governed by a single complex Riccati equation. Expressions for collective quantities such as the firing rate and mean voltage remain compact and analytically tractable. Because the formalism preserves the mathematical structure of the standard QIF ensemble, it inherits its many generalizations and can serve as a drop-in replacement in existing mean-field frameworks, providing a more biologically plausible yet still exactly solvable neuronal model.

We present a dynamical picture of kink-anti-kink scattering in a pair of special, Frankensteinian potentials made of piece-wise quadratic and linear pieces. Specifically, we focus on models that support kinks without skin and core regions. We propose an intuitive interpretation for these models as being essentially free massive theories with a built-in particle-pair like production mechanism that enters into the dynamics above certain field-value thresholds. We present results concerning the kink's characteristics depending on these thresholds and the distribution of bouncing windows. We show that the second model exhibits a phase-transition-like property in which the nature of collisions switches from disintegration into a massive wave to production of oscillons for large segments of initial velocities when the field threshold is low enough.

We consider a class of spontaneously broken $SU(2)$ gauge theories with adjoint scalar and look for exact magnetic monopole solutions in the Bogomol'nyi-Prasad-Sommerfield (BPS) limit. We find that some of the resulting solutions exhibit a new internal degree of freedom (a moduli space parameter) that controls the energy density profile of the monopole while keeping the total energy (mass) constant.