Related papers: Entrainment to Subharmonic Trajectories in Oscilla…
A time-varying nonlinear dynamical system is called a totally positive differential system (TPDS) if its Jacobian admits a special sign pattern: it is tri-diagonal with positive entries on the super- and sub-diagonals. If the vector field…
A matrix is called strictly sign-regular of order $k$ (denoted by $SSR_k$) if all its $k\times k$ minors are non-zero and have the same sign. For example, totally positive matrices, i.e., matrices with all minors positive, are $SSR_k$ for…
A matrix is called totally nonnegative (TN) if all its minors are nonnegative, and totally positive (TP) if all its minors are positive. Multiplying a vector by a TN matrix does not increase the number of sign variations in the vector. In a…
Positive systems play an important role in systems and control theory and have found many applications in multi-agent systems, neural networks, systems biology, and more. Positive systems map the nonnegative orthant to itself (and also the…
An oscillator is called isochronous if all motions have a common period. When the system is forced by a time-dependent perturbation with the same period the dynamics may change and the phenomenon of resonance can appear. In this context,…
Discrete time crystals (DTCs) are nonequilibrium phases of matter characterized by robust subharmonic order parameter dynamics. We report a new type of DTC in a periodically driven surface code, the subharmonic signature of which is only…
A linear dynamical system is called positive if its flow maps the non-negative orthant to itself. More precisely, it maps the set of vectors with zero sign variations to itself. A linear dynamical system is called $k$-positive if its flow…
A linear dynamical system is called $k$-positive if its dynamics maps the set of vectors with up to $k-1$ sign variations to itself. For $k=1$, this reduces to the important class of positive linear systems. Since stable positive linear…
The paper studies the relation between a nonlinear time-varying flat discrete-time system and the corresponding linear time-varying systems which are obtained by a linearization along trajectories. It is motivated by the continuous-time…
A discrete time crystal (DTC) is a remarkable non-equilibrium phase of matter characterized by the persistent sub-harmonic oscillations of physical observables in periodically driven many-body systems. Motivated by the question of whether…
Many-body quantum systems, under suitable conditions, exhibit time-translation symmetry breaking and settle in a discrete time crystalline (DTC) phase -- an out-of-equilibrium quantum phase of matter. The defining feature of DTC is a robust…
Is a spontaneous perpetual reversal of the arrow of time possible? The out-of-time-ordered correlator (OTOC) is a standard measure of irreversibility, quantum scrambling, and the arrow of time. The question may be thus formulated more…
A matrix $A$ is called totally positive (or totally non-negative) of order $k$, denoted by TP_k (or TN_k), if all minors of size at most $k$ are positive (or non-negative). These matrices have featured in diverse areas in mathematics,…
When the discrete time-translation symmetry of isolated, periodically driven systems is spontaneously broken, a new phase of matter can emerge. We review some recent developments on both the theoretical underpinnings and experimental…
A discrete time crystal (DTC) is a robust phase of driven systems that breaks the discrete time translation symmetry of the driving Hamiltonian. Recent experiments have observed DTC signatures in two distinct systems. Here we show nuclear…
An oscillating sequence of order $d$ is defined by the linearly disjointness from all $\{e^{2\pi i P(n)} \}_{n=1}^{\infty}$ for all real polynomials $P$ of degree smaller or equal to $d$. A fully oscillating sequence is defined to be an…
Consider a discrete dynamical system given by a square matrix $M \in \mathbb{Q}^{d \times d}$ and a starting point $s \in \mathbb{Q}^d$. The orbit of such a system is the infinite trajectory $\langle s, Ms, M^2s, \ldots\rangle$. Given a…
Discrete time crystals are periodically driven systems characterized by a response with periodicity $nT$, with $T$ the period of the drive and $n>1$. Typically, $n$ is an integer and bounded from above by the dimension of the local (or…
Continuous time crystals (CTCs) are characterized by sustained oscillations that break the time translation symmetry. Since the ruling out of equilibrium CTCs by no-go theorems, the emergence of such dynamical phases has been observed in…
Out-of-time-ordered-correlators (OTOCs) have been suggested as a means to diagnose chaotic behavior in quantum mechanical systems. Recently, it was found that OTOCs display exponential growth for the inverted quantum harmonic oscillator,…