Orbital carriers and inheritance in discrete-time quadratic dynamics

Gallas, Jason A. C.

doi:10.1142/s0129183120501004

Cited by 3 publications

(12 citation statements)

References 26 publications

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“…where r is the cyclic [nd×nd] version of the time translation operator (77), and f (Φ) acts within d-dimensional blocks (58) along the diagonal. We recognize the argument (58) of this (nd)-dimensional Dirac delta function as the defining equation (5) of the system,…”

Section: Hill's Formula For the Trace Of An Evolution Operatormentioning

confidence: 99%

A chaotic lattice field theory in one dimension

Liang,

Cvitanović

2022

Preprint

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Motivated by Gutzwiller's semiclassical quantization, in which unstable periodic orbits of low-dimensional deterministic dynamics serve as a WKB 'skeleton' for chaotic quantum mechanics, we construct the corresponding deterministic skeleton for infinite-dimensional lattice-discretized scalar field theories. In the field-theoretical formulation, there is no evolution in time, and there is no 'Lyapunov horizon'; there is only an enumeration of lattice states that contribute to the theory's partition sum, each a global spatiotemporal solution of system's deterministic Euler-Lagrange equations.The reformulation aligns 'chaos theory' with the standard solid state, field theory, and statistical mechanics. In a spatiotemporal, crystallographer formulation, the timeperiodic orbits of dynamical systems theory are replaced by periodic d-dimensional Bravais cell tilings of spacetime, each weighted by the inverse of its instability, its Hill determinant. Hyperbolic shadowing of large cells by smaller ones ensures that the predictions of the theory are dominated by the smallest Bravais cells.The form of the partition function of a given field theory is determined by the group of its spatiotemporal symmetries, that is, by the space group of its lattice discretization, best studied on its reciprocal lattice. Already 1-dimensional lattice discretization is of sufficient interest to be the focus of this paper. In particular, from a spatiotemporal field theory perspective, 'time'-reversal is a purely crystallographic notion, a reflection point group, leading to a novel, symmetry quotienting perspective of time-reversible theories and associated topological zeta functions.

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Section: Hill's Formula For the Trace Of An Evolution Operatormentioning

confidence: 99%

A chaotic lattice field theory in one dimension

Liang,

Cvitanović

2022

Preprint

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“…Even using modest numerical approximations of the orbital points one can obtain exact cluster equations as may be validated by comparing its roots with numerical values generated by iteration of the equations of motion. 1 We have encountered no case where the above procedure failed to produce exact expressions for orbital clusters.…”

Section: Preperiodic Generation Of Period Five Orbitsmentioning

confidence: 99%

“…Altogether, there are nine orbits of period six, listed in Table 1 of Ref. 1 The orbits form four groups, corresponding to the four factors composing S 6 (σ), which defines the sum of their orbital points:…”

Section: Preperiodic Generation Of Periods Six Seven and Eightmentioning

confidence: 99%

“…As discussed in Refs. 1,5 , apart from the intrinsic interest in detecting interdependent orbits, stratification is potentially significant to, e.g., rearrange trajectories sums in trace formulas underlying a semiclassical interpretation of atomic physics spectra. The existence of stratification was originally detected by extracting orbits from polynomials of very high degrees obtained by iterating Eq.…”

Section: Introductionmentioning

confidence: 99%

“…(12), we get o 4,2 (x) and o 4,3 (x), the pair of orbits conjugated over Q( √ 17). For details, see Refs 1,5. .…”

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confidence: 99%

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Preperiodicity and systematic extraction of periodic orbits of the quadratic map

Gallas

2020

Int. J. Mod. Phys. C

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Iteration of the quadratic map produces sequences of polynomials whose degrees explode as the orbital period grows more and more. The polynomial mixing all 335 period-12 orbits has degree [Formula: see text], while for the [Formula: see text] period-20 orbits the degree rises already to [Formula: see text]. Here, we show how to use preperiodic points to systematically extract exact equations of motion, one by one, without any need for iteration. Exact orbital equations provide valuable insight about the arithmetic structure and nesting properties of towers of algebraic numbers which define orbital points and bifurcation cascades of the map.

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Orbits of families of discrete dynamical systems evolving in the natural numbers

Campos Cantón

2025

Chaos: An Interdisciplinary Journal of Nonlinear Science

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In this paper, we give a class of one-dimensional discrete dynamical systems with state space N+. This class of systems is defined by two parameters: one of them sets the number of nearest neighbors that determine the rule of evolution, and the other parameter determines a segment of natural numbers Λ={1,2,…,b}. In particular, we investigate the behavior of a class of one-dimensional maps where an integer moves to an other integer given by the sum of the nearest neighbors minus a multiple of b∈N+. We find the coexistence of fixed points and periodic cycles. Two single parameter families of maps are introduced and their dynamics in the segment of natural sequence Λ. Furthermore, an order of the numbers of the set Λ−b is given by these families. Last, we present a connection of the N+ generated by the orbits of a particular case.

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Orbital carriers and inheritance in discrete-time quadratic dynamics

Cited by 3 publications

References 26 publications

A chaotic lattice field theory in one dimension

A chaotic lattice field theory in one dimension

Preperiodicity and systematic extraction of periodic orbits of the quadratic map

Orbits of families of discrete dynamical systems evolving in the natural numbers

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