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Spontaneous symmetry-breaking proves a mechanism for pattern generation in legged locomotion of animals. The basic timing patterns of animal gaits are produced by a network of spinal neurons known as a Central Pattern Generator (CPG). Animal gaits are primarily characterized by phase differences between leg movements in a periodic gait cycle. Many different gaits occur, often having spatial or spatiotemporal symmetries. A natural way to explain gait patterns is to assume that the CPG is symmetric, and to classify the possible symmetry-breaking periodic motions. Pinto and Golubitsky have discussed a four-node model CPG network for biped gaits with ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ symmetry, classifying the possible periodic states that can arise. A more specific rate model with this structure has been analyzed in detail by Stewart. Here we extend these methods to quadruped gaits, using an eight-node network with ${\mathbb{Z}}_4\times {\mathbb{Z}}_2$ symmetry proposed by Golubitsky and coworkers. We formulate a rate model and calculate how the first steady or Hopf bifurcation depends on its parameters, which represent four connection strengths. The calculations involve a distinction between “real” gaits with one or two phase shifts (pronk, bound, pace, trot) and “complex” gaits with four phase shifts (forward and reverse walk, forward and reverse buck). The former correspond to real eigenvalues of the connection matrix, the latter to complex conjugate pairs. The partition of parameter space according to the first bifurcation, ignoring complex gaits, is described explicitly. The complex gaits introduce further complications, not yet fully understood. All eight gaits can occur as the first bifurcation from a fully synchronous equilibrium, for suitable parameters, and numerical simulations indicate that they can be asymptotically stable.

In this paper, the effects of a bias term modeling a constant excitation force on the dynamics of an infinite-equilibrium chaotic system without linear terms are investigated. As a result, it is found that the bias term reduces the number of equilibrium points (transition from infinite-equilibria to only two equilibria) and breaks the symmetry of the model. The nonlinear behavior of the system is highlighted in terms of bifurcation diagrams, maximal Lyapunov exponent plots, phase portraits, and basins of attraction. Some interesting phenomena are found including, for instance, hysteretic dynamics, multistability, and coexisting bifurcation branches when monitoring the system parameters and the bias term. Also, we demonstrate that it is possible to control the offset and amplitude of the chaotic signals generated. Compared to some few cases previously reported on systems without linear terms, the plethora of behaviors found in this work represents a unique contribution in comparison with such type of systems. A suitable analog circuit is designed and used to support the theoretical analysis via a series of Pspice simulations.

There are few explicit examples in the literature of vector fields exhibiting observable chaos that may be proved analytically. This paper reports numerical experiments performed for an explicit two-parameter family of $𝕊𝕆(2)\oplus {\mathbb{Z}}_2$-symmetric vector fields whose organizing center exhibits an attracting heteroclinic network linking two saddle-foci. Each vector field in the family is the restriction to ${𝕊}^3$ of a polynomial vector field in ${ℝ}^4$. We investigate global bifurcations due to symmetry-breaking and we detect strange attractors via a mechanism called Torus-Breakdown. We explain how an attracting torus gets destroyed by following the changes in the unstable manifold of a saddle-focus.

Although a complete understanding of the corresponding bifurcation diagram and the mechanisms underlying the dynamical changes is out of reach, we uncover complex patterns for the symmetric family under analysis, using a combination of theoretical tools and computer simulations. This article suggests a route to obtain rotational horseshoes and strange attractors; additionally, we make an attempt to elucidate some of the bifurcations involved in an Arnold tongue.

The Self-Organized Hydrodynamics model of collective behavior is studied on an annular domain. A modal analysis of the linearized model around a perfectly polarized steady-state is conducted. It shows that the model has only pure imaginary modes in countable number and is hence stable. Numerical computations of the low-order modes are provided. The fully nonlinear model is numerically solved and nonlinear mode-coupling is then analyzed. Finally, the efficiency of the modal decomposition to analyze the complex features of the nonlinear model is demonstrated.

A historical dispute in the conceptual underpinnings of evolution is the validity of the gene centered view of evolution (Dawkins, 1989; Sober and Lewontin, 1982). We transcend this debate by formalizing the gene centered view as a dynamic version of the mean field approximation. This establishes the conditions under which it is applicable and when it is not. In particular, it breaks down for trait divergence which corresponds to symmetry breaking in evolving populations.

I propose using the group $\text{SL}(4,ℝ)$ as a generalization of the Dirac group $\text{SL}(2,ℂ)$ used in quantum mechanics, in an attempt to match the symmetries of the ‘internal’ spacetime of elementary particles to the symmetries of the ‘external’ spacetime of general relativity. While my conclusions do not amount to a new model or theory, they do highlight some specific points where there are apparent conflicts between the mathematics and the physics of the standard model, that might repay closer investigation. As evidence that this mathematical approach may have interesting things to say about physics, I propose two mass equations that seem not to have been observed previously.

Consciousness in higher animals, by virtue of its 100 ms time constant, is a necessarily greatly simplified and stripped-down version of more complex multiple tunable workspace cognition/ regulation dyads like wound healing, immune function, gene expression, institutional function and the like. These more complex dynamic entities emerged through evolutionary exaptation of the inevitable information crosstalk between coresident cognitive modules. In consequence of the debrided nature of consciousness, it should not be difficult to construct a fast, single workspace “conscious machine” that mimics the human tunable neuronal global workspace system. Tied to a “backbrain” AI that has learned hyperrapid stereotypic pattern responses to some particular set of likely challenges, the result is an elementary “emotional” conscious machine. A clever designer, however, may want to use available high-speed electronics to mimic the more capable multiple-workspace/ workforce systems inherently less susceptible to inattentional blindness and related failings of overfocus and thrashing. Contrary to current social constructions, however, the ultimate utility of such machines remains obscure. Here, we explore these matters in formal detail, restricting argument to the asymptotic limit theorems of information and control theories.

Real-world cognitive structures — embodied biological, machine or composite entities — are inherently unstable by virtue of the “topological information” imposed upon them by external circumstance, adversarial intent, and other persistent “selection pressures”. Consequently, under the Data Rate Theorem (DRT), they must be constantly controlled by embedding regulators. For example, blood pressure and the stream of consciousness require persistent delicate regulation in higher organisms. Here, using the Rate Distortion Theorem of information theory, we derive a form of the DRT of control theory that characterizes such instability for adiabatically stationary nonergodic systems and uncover novel forms of cognitive dynamics under stochastic challenge. These range from aperiodic stochastic amplification to Yerkes–Dodson signal transduction and outright system collapse. The analysis, deliberately closely adapted from recent purely biological studies, leads toward new statistical tools for data analysis, uncovering groupoid symmetry-breaking phase transition analogs to Fisher Zeros in physical systems that may be important for studies of machine intelligence under real-world, hence embodied, interaction. The challenges facing construction, operation, and stabilization of high-order “workspace” or “multiple-workspace” machine cognition, perhaps backed by rapid pattern-matching “emotional” AI, whether explicitly recognized as conscious or not, will require parallel construction of new analytic machinery. This work provides one example, solidly based on the asymptotic limit theorems of information and control theories.

In this article we explore analytically the zero-temperature physics of a collection of structureless and spinless interacting bosons, focusing on the transformation properties of the ground state wave function under translations. The properties of this wave function may give an insight into the physical mechanism which gives rise to the liquid-solid transition taking place in samples of ⁴He atoms at T = 0K. In the first part we show that, whenever the number of particles is finite, the ground state is translationally invariant at any density. In the thermodynamic limit, when the density is high enough, the extensivity of the Bragg peaks in the static structure factor reveals a degeneracy of the ground state: there exist states with periodic local density modulations with the same energy as the translationally invariant ground state, and this is a scenario of translational symmetry breaking.

The fermionic properties of quarks and leptons are determined by their weak interactions, the ultimate sources of which are their nonzero weak charges, and they are distinguished from each other by the strong interaction, which applies to quarks but not to leptons. There is no obvious reason why electric charge should be relevant to the distinction, but quarks are all required, in the conventional picture, to have electric charges, while leptons are not. It is as though the strong interaction requires electric charge to be present even though it is defined to be electric charge independent. It is proposed that the solution of this mystery lies in subtle aspects of the SU(3) × SU(2) × U(1) symmetry breaking of the three interactions and is a consequence of the way in which this symmetry breaking arises.