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Register a new userPattern Selection and Super-patterns in the Bounded Confidence Model
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We study pattern formation in the bounded confidence model of opinion dynamics. In this random process, opinion is quantified by a single variable. Two agents may interact and reach a fair compromise, but only if their difference of opinion falls below a fixed threshold. Starting from a uniform distribution of opinions with compact support, a traveling wave forms and it propagates from the domain boundary into the unstable uniform state. Consequently, the system reaches a steady state with isolated clusters that are separated by distance larger than the interaction range. These clusters form a quasi-periodic pattern where the sizes of the clusters and the separations between them are nearly constant. We obtain analytically the average separation between clusters L. Interestingly, there are also very small quasi-periodic modulations in the size of the clusters. The spatial periods of these modulations are a series of integers that follow from the continued fraction representation of the irrational average separation L.
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The communication process in a situation of emergency is discussed within the Scheff theory of shame and pride. The communication involves messages from media and from other persons. Three strategies are considered: selfish (to contact friends), collective (to join other people) and passive (to do nothing). We show that the pure selfish strategy cannot be evolutionarily stable. The main result is that the community structure is statistically meaningful only if the interpersonal communication is weak.
Driven many-body systems have been shown to exhibit discrete time crystal phases characterized by broken discrete time-translational symmetry. This has been achieved generally through a subharmonic response, in which the system undergoes one oscillation every other driving period. Here, we demonstrate that classical time crystals do not need to resonate in a subharmonic fashion but instead can also exhibit a continuously tunable anharmonic response to driving, which we show can emerge through a coresonance between modes in different branches of the dispersion relation in a parametrically driven medium. This response, characterized by a typically incommensurate ratio between the resonant frequencies and the driving frequency, is demonstrated by introducing a time crystal model consisting of an array of coupled pendula with alternating lengths. Importantly, the coresonance mechanism is the result of a bifurcation involving a fixed point and an invariant torus, with no intermediate limit cycles. This bifurcation thus gives rise to a many-body symmetry-breaking phenomenon directly connecting the symmetry-unbroken phase with a previously uncharacterized phase of matter, which we call an anharmonic time crystal phase. The mechanism is shown to generalize to driven media with any number of coupled fields and is expected to give rise to anharmonic responses in a range of weakly damped pattern-forming systems, with potential applications to the study of nonequilibrium phases, frequency conversion, and acoustic cloaking.
Pattern formation from homogeneity is well-studied, but less is known concerning symmetry-breaking instabilities in heterogeneous media. It is nontrivial to separate observed spatial patterning due to inherent spatial heterogeneity from emergent patterning due to nonlinear instability. We employ WKBJ asymptotics to investigate Turing instabilities for a spatially heterogeneous reaction-diffusion system, and derive conditions for instability which are loc
A hybrid asymptotic-numerical theory is developed to analyze the effect of different types of localized heterogeneities on the existence, linear stability, and slow dynamics of localized spot patterns for the two-component Schnakenberg reaction-diffusion model in a 2-D domain. Two distinct types of localized heterogeneities are considered: a strong localized perturbation of a spatially uniform feed rate and the effect of removing a small hole in the domain, through which the chemical species can leak out. Our hybrid theory reveals a wide range of novel phenomena such as, saddle-node bifurcations for quasi-equilibrium spot patterns that otherwise would not occur for a homogeneous medium, a new type of spot solution pinned at the concentration point of the feed rate, spot self-replication behavior leading to the creation of more than two new spots, and the existence of a creation-annihilation attractor with at most three spots. Depending on the type of localized heterogeneity introduced, localized spots are either repelled or attracted towards the localized defect on asymptotically long time scales. Results for slow spot dynamics and detailed predictions of various instabilities of quasi-equilibrium spot patterns, all based on our hybrid asymptotic-numerical theory, are illustrated and confirmed through extensive full PDE numerical simulations.
The flow of information reaching us via the online media platforms is optimized not by the information content or relevance but by popularity and proximity to the target. This is typically performed in order to maximise platform usage. As a side effect, this introduces an algorithmic bias that is believed to enhance polarization of the societal debate. To study this phenomenon, we modify the well-known continuous opinion dynamics model of bounded confidence in order to account for the algorithmic bias and investigate its consequences. In the simplest version of the original model the pairs of discussion participants are chosen at random and their opinions get closer to each other if they are within a fixed tolerance level. We modify the selection rule of the discussion partners: there is an enhanced probability to choose individuals whose opinions are already close to each other, thus mimicking the behavior of online media which suggest interaction with similar peers. As a result we observe: a) an increased tendency towards polarization, which emerges also in conditions where the original model would predict convergence, and b) a dramatic slowing down of the speed at which the convergence at the asymptotic state is reached, which makes the system highly unstable. Polarization is augmented by a fragmented initial population.
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