Unimodal Maps Research Articles

Why are some hysteresis loops shaped like a butterfly?

The contribution of this paper is a framework for relating butterfly-shaped hysteresis maps to simple (single-loop) hysteresis maps, which are typically easier to model and more amenable to control design. In particular, a unimodal mapping is used to transform simple loops to butterfly loops. For the practically important class of piecewise monotone hysteresis maps, we provide conditions for producing butterfly-shaped maps and examine the properties of the resulting butterflies. Conversely, we present conditions under which butterfly-shaped maps can be converted to simple piecewise monotone hysteresis maps to facilitate hysteresis compensation and control design. Examples of a preloaded two-bar linkage mechanism and a magnetostrictive actuator illustrate the theory and its utility for understanding, modeling, and controlling systems with butterfly-shaped hysteresis.

Universal theory of dynamical chaos in nonlinear dissipative systems of differential equations

The article presents a new universal theory of dynamical chaos in nonlinear dissipative systems of differential equations, including autonomous and nonautonomous ordinary differential equations (ODE), partial differential equations, and delay differential equations. The theory relies on four remarkable results: Feigenbaum's period doubling theory for cycles of one-dimensional unimodal maps, Sharkovskii's theory of birth of cycles of arbitrary period up to cycle of period three in one-dimensional unimodal maps, Magnitskii's theory of rotor singular point in two-dimensional nonautonomous ODE systems, acting as a bridge between one-dimensional maps and differential equations, and Magnitskii's theory of homoclinic bifurcation cascade that follows the Sharkovskii cascade. All the theoretical propositions are rigorously proved and illustrated with numerous analytical examples and numerical computations, which are presented for all classical chaotic nonlinear dissipative systems of differential equations.

Feigenbaum Graphs: A Complex Network Perspective of Chaos

The recently formulated theory of horizontal visibility graphs transforms time series into graphs and allows the possibility of studying dynamical systems through the characterization of their associated networks. This method leads to a natural graph-theoretical description of nonlinear systems with qualities in the spirit of symbolic dynamics. We support our claim via the case study of the period-doubling and band-splitting attractor cascades that characterize unimodal maps. We provide a universal analytical description of this classic scenario in terms of the horizontal visibility graphs associated with the dynamics within the attractors, that we call Feigenbaum graphs, independent of map nonlinearity or other particulars. We derive exact results for their degree distribution and related quantities, recast them in the context of the renormalization group and find that its fixed points coincide with those of network entropy optimization. Furthermore, we show that the network entropy mimics the Lyapunov exponent of the map independently of its sign, hinting at a Pesin-like relation equally valid out of chaos.

The full renormalization horseshoe for unimodal maps of higher degree: exponential contraction along hybrid classes

We prove exponential contraction of renormalization along hybrid classes of infinitely renormalizable unimodal maps (with arbitrary combinatorics), in any even degree d. We then conclude that orbits of renormalization are asymptotic to the full renormalization horseshoe, which we construct. Our argument for exponential contraction is based on a precompactness property of the renormalization operator (“beau bounds”), which is leveraged in the abstract analysis of holomorphic iteration. Besides greater generality, it yields a unified approach to all combinatorics and degrees: there is no need to account for the varied geometric details of the dynamics, which were the typical source of contraction in previous restricted proofs.

A discrete dynamical system for the Marx model

The Marx model for the profit rate r depending on the exploitation rate e and on the organic composition of the capital k is studied using a dynamical approach. Introducing a discrete dynamical system in the model and supposing that both k and e depend on the profit rate of the previous cycle we get a discrete dynamical system for r, , which is a family of unimodal maps depending on the parameter a, where a is the exploitation rate at profit zero. It is interesting to note that the model can model the phenomenon known as Kondratiev waves in economic systems.

Chaos and Unpredictability in Evolutionary Dynamics in Discrete Time

A discrete-time version of the replicator equation for two-strategy games is studied. The stationary properties differ from those of continuous time for sufficiently large values of the parameters, where periodic and chaotic behavior replace the usual fixed-point population solutions. We observe the familiar period-doubling and chaotic-band-splitting attractor cascades of unimodal maps but in some cases more elaborate variations appear due to bimodality. Also unphysical stationary solutions can have unusual physical implications, such as the uncertainty of the final population caused by sensitivity to initial conditions and fractality of attractor preimage manifolds.

Combination Laws for Scaling Exponents and Relation to the Geometry of Renormalization Operators

Renormalization group has become a standard tool for describing universal properties of different routes to chaos—period-doubling in unimodal maps, quasiperiodic transitions in circle maps, dynamics on the boundaries of Siegel disks, destruction of invariant circles of area-preserving twist maps, and others. The universal scaling exponents for each route are related to the properties of the corresponding renormalization operators. We propose a Principle of Approximate Combination of Scaling Exponents (PACSE) that organizes the scaling exponents for different transitions to chaos. Roughly speaking, if the combinatorics of a transition is a composition of two simpler combinatorics, then the scaling exponents of the combined combinatorics is approximately equal to the product of the scaling exponents, both in the parameter space and in the configuration space, corresponding to each of these two combinatorics. We state PACSE quantitatively as precise asymptotics of the scaling exponents for combined combinatorics, and give convincing numerical evidence for it for each of the four dynamical systems mentioned above. We propose an explanation of PACSE in terms of the dynamical properties of the renormalization operators—in particular, as a consequence of certain transversal intersections of the stable and unstable manifolds of the operators corresponding to different transition to chaos.

A Cr unimodal map with an arbitrary fast growth of the number of periodic points

AbstractIn this paper we present a surprising example of a Cr unimodal map of an interval f:I→I whose number of periodic points Pn(f)=∣{x∈I:fnx=x}∣ grows faster than any ahead given sequence along a subsequence nk=3k. This example also shows that ‘non-flatness’ of critical points is necessary for the Martens–de Melo–van Strien theorem [M. Martens, W. de Melo and S. van Strien. Julia–Fatou–Sullivan theory for real one-dimensional dynamics. Acta Math.168(3–4) (1992), 273–318] to hold.

Scaling law in saddle-node bifurcations for one-dimensional maps: a complex variable approach

The study of transient dynamical phenomena near bifurcation thresholds has attracted the interest of many researchers due to the relevance of bifurcations in different physical or biological systems. In the context of saddle-node bifurcations, where two or more fixed points collide annihilating each other, it is known that the dynamics can suffer the so-called delayed transition. This phenomenon emerges when the system spends a lot of time before reaching the remaining stable equilibrium, found after the bifurcation, because of the presence of a saddle-remnant in phase space. Some works have analytically tackled this phenomenon, especially in time-continuous dynamical systems, showing that the time delay, τ, scales according to an inverse square-root power law, τ∼(μ−μ c )−1/2, as the bifurcation parameter μ, is driven further away from its critical value, μ c . In this work, we first characterize analytically this scaling law using complex variable techniques for a family of one-dimensional maps, called the normal form for the saddle-node bifurcation. We then apply our general analytic results to a single-species ecological model with harvesting given by a unimodal map, characterizing the delayed transition and the scaling law arising due to the constant of harvesting. For both analyzed systems, we show that the numerical results are in perfect agreement with the analytical solutions we are providing. The procedure presented in this work can be used to characterize the scaling laws of one-dimensional discrete dynamical systems with saddle-node bifurcations.

On C*-Algebras from Interval Maps

Given a unimodal interval map f, we construct partial isometries acting on Hilbert spaces associated to the orbit of each point. Then we prove that such partial isometries give rise to representations of a C*-algebra associated to the subshift encoding the kneading sequence of the critical point. This construction has the advantage of incorporating maps with a non necessarily Markov partition (e.g. Fibonacci unimodal map). If we are indeed in the presence of a finite Markov partition, then we prove that these new representations coincide with the (previously considered by the authors) representations arising from the Cuntz–Krieger algebra of the underlying (finite) transition matrix.

Typical points for one-parameter families of piecewise expanding maps of the interval

For one-parameter families of piecewise expanding maps of the intervalwe establish sufficient conditions such that a given point in the interval is typicalfor the absolutely continuous invariant measure for a full Lebesgue measure set of parameters.In particular, we consider $C^{1,1}(L)$-versions of $\beta$-transformations,piecewise expanding unimodal maps, and Markov structure preserving one-parameter families.For families of piecewise expanding unimodal maps we show that the turning point is almost surely typicalwhenever the family is transversal.

Concave unimodal maps have no majorisation relations between their ergodic measures

If T is a concave unimodal map on the unit interval [0,1] and {x: T n (x) = 1 for some n} is dense in [0,1], we prove that all T-invariant ergodic Borel probability measures are mutually incomparable with respect to the partial order of majorisation. This contrasts sharply with the situation for other interval maps previously studied.

Adding machines, kneading maps, and endpoints

Given a unimodal map f, let I = [ c 2 , c 1 ] denote the core and set E = { ( x 0 , x 1 , … ) ∈ ( I , f ) | x i ∈ ω ( c , f ) for all i ∈ N } . It is known that there exist strange adding machines embedded in symmetric tent maps f such that the collection of endpoints of ( I , f ) is a proper subset of E and such that lim k → ∞ Q ( k ) ≠ ∞ , where Q ( k ) is the kneading map. We use the partition structure of an adding machine to provide a sufficient condition for x to be an endpoint of ( I , f ) in the case of an embedded adding machine. We then show there exist strange adding machines embedded in symmetric tent maps for which the collection of endpoints of ( I , f ) is precisely E . Examples of this behavior are provided where lim k → ∞ Q ( k ) does and does not equal infinity, and in the case where lim k → ∞ Q ( k ) = ∞ , the collection of endpoints of ( I , f ) is always E .

Parapuzzle of the multibrot set and typical dynamics of unimodal maps

We study the parameter space of unicritical polynomials f_c : z ↦ z^d + c . For complex parameters, we prove that for Lebesgue almost every c , the map f_c is either hyperbolic or infinitely renormalizable. For real parameters, we prove that for Lebesgue almost every c , the map f_c is either hyperbolic, or Collet–Eckmann, or infinitely renormalizable. These results are based on controlling the spacing between consecutive elements in the “principal nest” of parapuzzle pieces.

Complex dynamics near a Hopf bifurcation with symmetry: A parameter study

The eight-dimensional normal form for a Hopf bifurcation with 𝕆(2) × 𝕆(2)-symmetry is investigated for a range of parameters in which all basic periodic solutions residing in two-dimensional fixed point subspaces are unstable, and the dynamics is bounded. By using symmetry-adapted variables, the dimension of the phase space is reduced to four. In the reduced phase space, periodic solutions are revealed as fixed points and quasiperiodic solutions as periodic orbits. Two parameter regimes are identified: A region in which trajectories in the reduced phase space are attracted by fixed points and periodic orbits and bistability occurs, and a region with complex dynamics. The characteristics of the complex dynamics are chaos, intermittency and periodic windows terminating in period doubling cascades, and both periodic and chaotic attractors occur in symmetric and asymmetric forms. The phase space geometry is suggestive of a kind of in–out intermittency involving two invariant subspaces. By using an appropriate cross-section, a reduced description of the dynamics is given in terms of a three-dimensional Poincaré map. Plots of the iterates of this map indicate that they reside in a one-dimensional manifold. In the initial range of the region with complex dynamics, the bifurcation diagrams (Poincaré map iterates versus a parameter) show the typical characteristics of unimodal map families. The symmetry of attractors is diagnosed using normalized asymmetry measures calculated from the iterates of the Poincaré map.

Non-autonomous periodic systems with Allee effects

A new class of maps called unimodal Allee maps are introduced. Such maps arise in the study of population dynamics in which the population goes extinct if its size falls below a threshold value. A unimodal Allee map is thus a unimodal map with three fixed points, a zero fixed point, a small positive fixed point, called threshold point, and a bigger positive fixed point, called the carrying capacity. In this paper, the properties and stability of the three fixed points are studied in the setting of non-autonomous periodic dynamical systems or difference equations. Finally, we investigate the bifurcation of periodic systems/difference equations when the system consists of two unimodal Allee maps.

Countable limit sets of unimodal maps

We show that for any positive integer n, there exists a unimodal map whose turning point has a countable ?-limit set of depth n. We also show that the quadratic map F(x) = 1 ? 2x 2 admits countable ?-limit sets of depth n for all positive integer n.

Can cerebellar input calibrate collicular topographic maps?

There is substantial evidence that repeating cerebellar microcircuit implements an adaptive filter algorithm suitable for fine tuning a wide range of sensorimotor skills [1]. Although it is known that multimodal map layer of superior colliculus receives a massive cerebellar input which directly influences collicular output cells [2], there has been little speculation as to function of this influence. We suggest here that these cerebellar inputs play a role in calibrating accuracy of collicular topographic maps. The cerebellum is a plausible candidate for this role for a number of reasons. It is known to be crucial for accuracy of saccades generated by colliculus. It is also known that cerebellar lesions impair prism adaptation of eye movements. A recent imaging study [3] found direct evidence of cerebellar activation during prism adaptation and suggests that the cerebellum is particularly involved in [establishing] a correct spatial mapping among visuomotor and sensorimotor coordinates systems. A modeling study [3] of collicular visuomotor mapping concludes that transformation occurs in brainstem with cerebellum adjusting it for accuracy, stating that ... importance of cerebellum has been neglected in previous modeling studies. Here we investigate whether adaptive filter model of cerebellar microcircuit which has been so successful in conventional sensorimotor contexts can be applied without change to very different computational problem of calibrating a topographic map driving an orienting response. We propose a model in which (i) unimodal sensory topographic maps constitute a probabilistic representation of target position and that these maps are optimally combined to produce a multimodal map [4] whose peak activity drives orienting response, (ii) cerebellar input can bias position of peak map activity (e.g. by a process such as attentional gain modulation [5]), (iii) cerebellum receives sensory information that generates topographic maps on its mossy fibre inputs, and (iv) information about orienting errors caused by miscalibration is made available to cerebellum on its climbing fibre inputs and drives cerebellar learning. We demonstrate in simulation that this mechanism can successfully calibrate topographic maps and go on to investigate its computational properties. For example we investigate a fundamental calibration ambiguity in which sensory maps can be miscalibrated in such a way that their effects on combined estimate cancel. We show that this ambiguity is resolved in a plausible way if error signals are gated whenever a sensor fails to observe target; a similar gating process has been observed in some motor behaviors [6].We also demonstrate that optimality of cerebellar learning rule [7] ensures that Purkinje cell synapses carrying cross-talk between sensors are driven to silence, greatly reducing need to hard-wire connectivity of parallel fiber inputs to cerebellar microzones.

Cryptanalysis of a family of self-synchronizing chaotic stream ciphers

Unimodal maps have been broadly used as a base of new encryption strategies. Recently, a stream cipher has been proposed in the literature, whose keystream is basically a symbolic sequence of the (one-parameter) logistic map or of the tent map. In the present work a thorough analysis of the keystream is made which reveals the existence of some serious security problems.

Stationary distributions of sums of marginally chaotic variables as renormalization group fixed points

We determine the limit distributions of sums of deterministic chaotic variables in unimodal maps assisted by a novel renormalization group (RG) framework associated to the operation of increment of summands and rescaling. In this framework the difference in control parameter from its value at the transition to chaos is the only relevant variable, the trivial fixed point is the Gaussian distribution and a nontrivial fixed point is a multifractal distribution with features similar to those of the Feigenbaum attractor. The crossover between the two fixed points is discussed and the flow toward the trivial fixed point is seen to consist of a sequence of chaotic band mergers.