]www.polyphys.mat.ethz.ch

# Two-Worlds Interpretation of Quantum Mechanics

Hans Christian Öttinger<EMAIL_ADDRESS>[ ETH Zürich, Quantum Center and
Department of Materials, HCP F 43.1, CH-8093 Zürich, Switzerland

###### Abstract

The stochastic nature of quantum mechanics is more naturally reflected in a
bilinear two-process representation of density matrices rather than in squared
wave functions. This proposition comes with a remarkable change of the
entanglement mechanism: entanglement does not originate from superpositions of
wave functions, but results from the bilinear structure of density matrices.
Quantum interference is not an additive superposition mechanism, but rather a
multiplicative phenomenon. The proposed bilinear representation of density
matrices is given in terms of two stochastic jump processes. These ideas are
illustrated for the Einstein-Podolsky-Rosen and double-slit experiments. The
expression of the stochastic nature of quantum mechanics in terms of random
variables rather than their probability distributions facilitates an
ontological viewpoint and leads us to a two-worlds interpretation of quantum
mechanics.

##  I Introduction

The stochastic nature of quantum mechanics calls for an appropriate setting
for describing randomness. The most natural description of quantum states is
given by a density matrix $\rho$ on a Hilbert space for a system of interest,
by which the average of any observable $A$, which is a self-adjoint operator
on the Hilbert space, can be obtained as a trace, $\langle A\rangle={\rm
tr}(\rho A)$. In the Schrödinger picture, the evolution of the density matrix
$\rho_{t}$ describing the time-dependent state of a quantum system is given by
the von Neumann equation

$i\hbar\frac{d\rho_{t}}{dt}=[H,\rho_{t}],$ (1)

where $\hbar$ is the reduced Planck constant, $H$ is the Hamiltonian of the
system, and the square brackets denote the commutator. Except for its greater
flexibility in choosing initial conditions, which are not restricted to pure
states, the von Neumann equation is equivalent to the Schrödinger equation.

As an alternative probabilistic tool, Schrödinger’s complex wave function
$\psi_{t}$ is less directly associated with the stochastic nature of quantum
mechanics than the density matrix $\rho_{t}$. One needs the additional rule
that the squared modulus of the wave function, $|\psi_{t}|^{2}$, should be
interpreted as a probability density, whereas (1) is a linear evolution
equation for the primary probabilistic quantity of interest, $\rho_{t}$.

Quantum master equations for density matrices have the further advantage that
they are perfectly suited not only for describing reversible dynamics, but
also for dissipative quantum systems [1, 2]. A class of robust quantum master
equations for dissipative systems has been obtained by quantizing the
geometric structures behind classical nonequilibrium thermodynamics [3, 4].
The same type of thermodynamic master equations has also been found by means
of projection-operator methods for coarse-graining reversible systems [5].

The most fundamental equations of nature are generally believed to be
reversible, whereas irreversibility is considered to be an emergent
phenomenon. However, as we presently do not know the most fundamental
equations of nature and, even if we happened to know them at some point, we
could never be sure because new observations could always be made, it might be
more appropriate to build all physical theories by default on irreversible
equations. One should actually consider reversible equations as an
idealization that needs to be justified whenever it might be appropriate.
Therefore, density matrices governed by quantum master equations appear to
provide the most natural setting for describing the evolution of quantum
systems.

Meaningful deterministic equations for the evolution of probabilistic tools
should characterize stochastic processes. We here propose to consider
stochastic processes, which provide the most common indeterministic models
[6], as the more fundamental level of description. This is particularly
appealing because quantum physics involves ontic randomness, contrary to the
epistemic randomness associated with coarse graining, say in classical
statistical mechanics.

When one solves the linear Schrödinger equation for the wave function of an
interacting system, superposition of states occurs inevitably. In the
subsequent section, we introduce a bilinear representation of quantum master
equations in terms of two independently evolving stochastic jump processes,
for which superposition no longer is an essential feature (Section II). This
representation is further motivated by discussing serious limitations of
quantum mechanics, which can be overcome in the quantum field theoretic
setting of particle physics (Section III). We then discuss two experiments,
the standard interpretation of which relies heavily on superposition states,
in the proposed two-process representation of quantum mechanics: the Einstein-
Podolsky-Rosen Gedankenexperiment (Section IV) and the double-slit experiment
(Section V). The final conclusions (Section VI) focus on the interpretation of
quantum mechanics, ontological considerations, and the elimination of
paradoxes by natural limitations for the applicability of quantum mechanics.

##  II Two-process unravelings

The general strategy of so-called unravelings is to introduce stochastic
processes in Hilbert space such that a density matrix evolving according to
the von Neumann equation (1) can be extracted in terms of suitable averages.
This idea of passing from probabilistic tools like density matrices to
equivalent stochastic processes was originally motivated by computer
simulations for dissipative quantum systems [1], but it is useful also for
reversible systems and conceptual clarifications. The passage from
probabilities to stochastic objects is particularly relevant to ontology [7].

For dissipative systems, unravelings typically employ continuous evolution to
represent reversible dynamics and a combination of stochastic jump processes
with continuous correction terms to reproduce dissipative dynamics [1]. In the
context of dissipative quantum field theory [8], it has been proposed to treat
also reversible interactions by jumps. The usefulness of this idea has been
elaborated in detail for simulating purely reversible quantum dynamics in [9].

The treatment of interactions as jumps requires a splitting of the full
Hamiltonian into free and interacting contributions, $H=H^{\rm free}+H^{\rm
int}$. We further assume that there exists a distinguished basis $\cal B$ of
orthonormal eigenstates of the free Hamiltonian $H^{\rm free}$. The basis
vectors are denoted by $b_{n}$ and the corresponding eigenvalues of $H^{\rm
free}$ are given by $E_{n}$. For unravelings in terms of two stochastic
processes $\left|\phi\right\rangle_{t}$ and $\left|\psi\right\rangle_{t}$ in
Hilbert space, one wishes to reproduce the density matrix $\rho_{t}$ by the
following expectation,

$\rho_{t}=E\Big{(}\left|\phi\right\rangle_{t}\left\langle\psi\right|_{t}\Big{)},$
(2)

where we have used Dirac’s bra-ket notation. In this setting we can impose the
strict superselection rule that the states $\left|\phi\right\rangle_{t}$ and
$\left|\psi\right\rangle_{t}$ can only be multiples of basis vectors from
$\cal B$. Reasons for believing in the existence of such a distinguished basis
and a strict superselection rule emerge from the origins of quantum mechanics
in elementary particle physics, or quantum field theory, and will be discussed
in Section III.

The strict superselection rule has the useful side-effect of reducing the
enormous number of possibilities for constructing unravelings. It naturally
guides us to the following construction of piecewise continuous trajectories
with interspersed jumps.

The stochastic processes for the bra-vector $\left|\psi\right\rangle_{t}$ and
the ket-vector $\left|\phi\right\rangle_{t}$ in (2) evolve independently, but
according to the same rules. Therefore, it is sufficient to describe the
construction of $\left|\phi\right\rangle_{t}$. At any time $t$, the stochastic
state vector $\left|\phi\right\rangle_{t}$ is a multiple of some basis vector
$b_{n}\in{\cal B}$. Between jumps, the process is governed by the free
Schrödinger equation,

$i\hbar\frac{d}{dt}\left|\phi\right\rangle_{t}=H^{\rm
free}\left|\phi\right\rangle_{t},$ (3)

which simply introduces a phase factor $e^{-iE_{n}\Delta t/\hbar}$ if
$\left|\phi\right\rangle_{t}$ is a multiple of $b_{n}$ and $\Delta t$ is the
time between jumps. Quantum jumps reproduce the effect of the interaction
Hamiltonian $H^{\rm int}$. They are characterized by a rate parameter $r$ for
the occurrence of jumps and by the transition probabilities $p_{ln}$ for
jumping from a given state $b_{n}$ to any particular state $b_{l}$ (with
$\sum_{l}p_{ln}=1$). If the process before a jump at time $t$ is in the state
$\left|\phi\right\rangle_{t}=c_{t}b_{n}$, then the state
$\left|\phi^{\prime}\right\rangle_{t}$ after the jump is given by the
stochastic jump rule

$\displaystyle\left|\phi\right\rangle_{t}=c_{t}b_{n}\rightarrow\left|\phi^{\prime}\right\rangle_{t}$
$\displaystyle=$ $\displaystyle\frac{1}{p_{ln}}\left(\frac{1}{i\hbar
r}\left\langle b_{l}\right|H^{\rm
int}\left|b_{n}\right\rangle+\delta_{ln}\right)c_{t}b_{l}$ (4)
$\displaystyle\mbox{with probability }p_{ln}.$

The rate parameter $r$ should be chosen such that $\hbar r$ is a
characteristic interaction energy of the system and the transition
probabilities $p_{ln}$ are given by the proportionality

$p_{ln}\propto\big{|}\left\langle b_{l}\right|H^{\rm
int}\left|b_{n}\right\rangle+i\hbar r\,\delta_{ln}\big{|}^{J}.$ (5)

For $J=1$ the changes of the prefactors in (4) are as uniform as possible,
whereas the choice $J=2$ can be motivated by Fermi’s golden rule.

We still need to specify the initial conditions for the stochastic processes
$\left|\phi\right\rangle_{t}$ and $\left|\psi\right\rangle_{t}$. The proper
initial conditions for the Einstein-Podolsky-Rosen and double-slit experiments
are discussed in Sections IV and V. In both cases, also the initial conditions
for the two processes of the unraveling are independent random variables.
Methods for constructing stochastic unravelings of equilibrium initial states
at given temperatures, with special emphasis on the ground state at zero
temperature, can be found in [10].

The continuous free evolution shows the importance of allowing for scalar
multiplication. The jumps eliminate the need for any addition of states.
Superpositions of base vectors do not occur in the two-process unraveling. The
discussion of concrete experiments clarifies how entanglement nevertheless
arises in the two-process unraveling.

In summary, we have constructed two stochastic processes
$\left|\phi\right\rangle_{t}$ and $\left|\psi\right\rangle_{t}$, each
satisfying a strict superselection rule. The density matrix obtained as a
correlation function of these processes according to (2) satisfies the von
Neumann equation (1), that is, the very same equation as obtained from the
Schrödinger equation for density matrices associated with pure states. The
stochastic representation of the density matrix is not restricted to pure
states and, as it is based on two stochastic processes instead of one
deterministic wave function, it offers an alternative interpretation of
quantum mechanics based on a new implementation of entanglement.

##  III Existence of distinguished bases

In conventional quantum mechanics one is free to do basis changes, which
typically involve superpositions. Therefore, we need to justify our assumption
of the existence of a distinguished basis $\cal B$ in the superposition-free
formulation of quantum mechanics based on two-process unravelings. To this end
one needs to recognize the limitations and the approximate nature of quantum
mechanics for systems consisting of a fixed number of particles and to look at
its deeper origin in quantum field theory.111We here use the term _quantum
mechanics_ for systems with a fixed, finite number of degrees of freedom,
whereas _quantum field theory_ deals with a variable number of degrees of
freedom, including the limit of infinitely many degrees of freedom. _Quantum
theory_ includes both mechanics and field theory.

Dirac’s marvelous quantization recipe formally suggests that we can quantize
any classical Hamiltonian system, preferably in canonical coordinates, by
replacing classical Poisson brackets by quantum commutators. However, the
quantization of say planetary motion can hardly be particularly meaningful. A
rigid restriction of quantum theory to elementary particles, which would be
the other extreme, is unpractical because we can never be sure that we know
the truly elementary particles. For example, nobody hesitates to apply quantum
mechanics to protons although they are not elementary particles.

A revealing textbook example for the application of quantum mechanics is the
hydrogen atom, which leads to the famous prediction for the line spectrum of
atomic hydrogen. Note that the Hamiltonian for this problem contains the
Coulomb potential of classical electrostatics. In that sense, quantum
mechanics possesses a semi-classical character, which is usually not pointed
out. Particularly questionable is the occurrence of electron-proton
interactions at a distance.

In a more rigorous treatment of the hydrogen atom, the Coulomb interaction
should result from an exchange of soft photons with momenta of the order of
the electron momentum and wave lengths of the order of the atomic size between
the electron and the proton, or between the electron and the quarks in a
proton. We then are in the domain of quantum field theory, allowing for the
creation and annihilation of photons. Quantum mechanics for systems with a
fixed number of particles can only be an approximate theory that arises in the
low-energy limit of quantum field theory, when particle creation and
annihilation are suppressed (which is hardly possible for massless particles).
Quantum mechanics should emerge from the quantum field theory of elementary
particles and their interactions after making suitable approximations. This
remark implies that special relativity can be properly taken into account at
the starting point for justifying quantum mechanics. Of course, it may turn
out that there are more fundamental theories of elementary particles from
which one could obtain both quantum mechanics and quantum field theory by
suitable approximations.

In quantum field theory, Fock space comes with a natural basis $\cal F$ [11,
12, 8]. The Fock basis vectors describe states with well-defined free particle
content, where free particles are in momentum eigenstates (we assume
regularization by a momentum lattice to avoid the problems associated with a
continuous basis; physically this means that space is finite). In quantum
field theory, the strict superselection rule implies that a system can never
be in a superposition state of different particle contents, where quantum
particles can be distinguished by a number of intrinsic properties. In making
approximations for passing from quantum field theory to quantum mechanics one
should make sure that a distinguished basis $\cal B$ emerges from the natural
Fock basis $\cal F$ of quantum field theory. For example, the treatment of a
proton as an elementary particle on the atomic length scale (or low energies)
in quantum mechanics relies on color confinement. Having an eye on what makes
sense from the perspective of particle physics restricts the situations in
which quantum mechanics can be applied in a deep and meaningful way.

Different options for choosing a distinguished basis may arise even in quantum
field theory. For example, in the Schwinger model for electrodynamics in $1+1$
dimensions [13, 14, 15], one can switch from the standard Fock space to an
alternative Fock space based on photons and bound electron-positron pairs (see
Section 3.3.7 of [8]), which is similar to regarding protons as elementary
particles and actually illustrates the emergence of a new distinguished basis
via confinement.

When quantum field theory is developed on the Fock space of momentum
eigenstates, one cannot know the position of a free quantum particle at a
given time. However, whenever collisions between typically three or four
particles occur, we know that all the colliding particles are at the same
position. This remark explains the characteristic particle tracks observed in
collider experiments when a high-energy collision is followed by many low-
energy collisions in a detector ruled by a certain correlation structure of
collision events [7].

In the next two sections, we discuss the Einstein-Podolsky-Rosen and double-
slit experiments in terms of two-process unravelings in order to illustrate
the basic ideas. For these discussions we use basis vectors inspired by the
Fock space of particle physics with the corresponding superselection rule. In
the Einstein-Podolsky-Rosen experiment, the allowed states consist of pairs of
photons with well-defined wave lengths and helicities; in the double-slit
experiment, the allowed states consist of electrons with well-defined momenta.

##  IV Einstein-Podolsky-Rosen Gedankenexperiment

The Einstein-Podolsky-Rosen (EPR) Gedankenexperiment [16] was designed to
reveal the occurrence of actions-at-a-distance as a sign of the incompleteness
of quantum mechanics. We consider the EPR experiment in the version for
photons, which is appealing for both theoretical arguments and experimental
realization [17, 18, 19, 20, 21, 22]. We here prefer photons over electrons
because their circular polarization states are characterized with respect to
the unambiguous direction of their motion, whereas the electron spin states
are usually characterized with respect to a preferred direction in space
(because their direction of motion is not Lorentz invariant and is actually
undefined in the rest frame). The arbitrariness of the chosen direction of
space would have to be addressed with undesirable changes of bases.

Pairs of photons with different wave lengths ($\lambda_{1}=551.3\,{\rm nm}$
and $\lambda_{2}=422.7\,{\rm nm}$) moving with the same circular polarization
in opposite directions can be created in the decay of properly excited calcium
atoms [17, 18]. Whereas the actual decay occurs through an intermediate state
within some $5\,{\rm ns}$, we here use the idealization of a single collision
event in which a calcium excitation is annihilated and two photons are
created.

In the discussion of the EPR experiment, one usually focuses entirely on
polarizations states and neglects spatial information. Spatial information is
associated with the annihilation of a calcium excitation into two photons and
with the detection of the photons by collisions in a photomultiplier. In
between, we do not have any spatial information. However, one may assume a
certain correlation structure for collision events, for example, implying that
the two photons from a single excitation hit two equally distant detectors at
equal times (in the laboratory). Filters can be introduced such that photons
with wave length $\lambda_{1}$ arrive only in one detector, say the left one,
whereas photons with wave length $\lambda_{2}$ arrive only in the other
detector, say the right one.

The following orthonormal basis vectors for the two-photon states are natural:
$b_{1}=\left|5513,1\right\rangle\,\left|4227,1\right\rangle$ and
$b_{2}=\left|5513,-1\right\rangle\,\left|4227,-1\right\rangle$, where each
photon is characterized by its wave length in Ångström and by its helicity.
The eigenvalues for these eigenstates of the free Hamiltonian are independent
of helicity, $E_{1}=E_{2}$. When a calcium excitation decays, the two
processes of the stochastic unraveling are independently initialized with
probability $1/2$ in the states $\sqrt{2}\,b_{1}$ or $-\sqrt{2}\,b_{2}$.
According to the basic idea of unravelings in (2), this implies the initial
density matrix

$\rho_{\rm EPR}=\frac{1}{2}\,\left(\begin{matrix}1&-1\\\
-1&1\end{matrix}\right),$ (6)

resulting from the independence of the bra- and ket-states. Note that this
density matrix is of rank $1$, that is, it corresponds to a pure state in
standard quantum mechanics. The pure state associated with the density matrix
(6) is the maximally entangled Bell singlet state $(b_{1}-b_{2})/\sqrt{2}$,
which is the usual starting point for the discussion of the EPR experiment. In
the unraveling, this initial condition is produced without any superposition
of the natural basis states. In the language of Schrödinger’s famous
dramatization, both the bra-cat and the ket-cat of the two-worlds
interpretation are either dead or alive, but they may be in different states.

Figure 1: Schematic illustration of the EPR experiment in the two-world
interpretation (see text for details).

The representation of the density matrix (6) in the two-process unraveling
appears at the center of the schematic illustration of the EPR experiment in
Figure 1. The photons moving in opposite directions with wave lengths
$\lambda_{1}$ and $\lambda_{2}$ are indicated by left and right arrows,
respectively. The free motion leads to complex phase factors that do not
depend on the helicity states, so that the bra-ket average for the density
matrix of the helicity states remains unchanged during the free evolution.
When the photons hit the detectors at equal distances from the site of their
creation, they still have their original, equal helicities: $+1$ and $-1$ with
equal probabilities, independently in the two worlds.

Each of the detectors consists of a beam-splitting linear polarizer and two
photomultipliers counting the photons in the two linear polarization states.
These polarizers are cubes made of two glass prisms with suitable dielectric
thin films on the sides stuck together. The polarizers can be rotated by the
angles $\theta_{1}$, $\theta_{2}$ around the optical axis with respect to a
reference direction. As the photons are moving in opposite directions, also
the rotations are performed in opposite directions, so that
$\theta_{1}+\theta_{2}$ actually is the angle between the two polarizers. We
are interested in the probabilities for finding the photons in the two optical
units in parallel ($\|$) and perpendicular ($\perp$) polarization states. In
the natural, intrinsic basis of circular polarization states, these
probabilities are the averages of the following observables, which are
obtained by transformation from linear to circular polarization states,

$A_{\|\,\|}(\theta_{1},\theta_{2})=A_{\perp\,\perp}(\theta_{1},\theta_{2})=\frac{1}{4}\,\left(\begin{matrix}1&e^{2i(\theta_{1}+\theta_{2})}\\\
e^{-2i(\theta_{1}+\theta_{2})}&1\end{matrix}\right)$ (7)

and

$\displaystyle A_{\|\,\perp}(\theta_{1},\theta_{2})$ $\displaystyle=$
$\displaystyle A_{\perp\,\|}(\theta_{1},\theta_{2})$ (8) $\displaystyle=$
$\displaystyle\frac{1}{4}\,\left(\begin{matrix}1&-e^{2i(\theta_{1}+\theta_{2})}\\\
-e^{-2i(\theta_{1}+\theta_{2})}&1\end{matrix}\right).\qquad$

These averages performed with the density matrix (6) are used to detect a
violation of Bell’s inequalities [23, 24], which are nowadays no longer
interpreted as actions-at-a-distance but rather as nonlocal correlations in
quantum systems [18, 19]. Such correlations arise from the multiplicative
interplay of the two independent stochastic processes of the two-world
interpretation. As the density matrices obtained from the maximally entangled
Bell singlet state of standard quantum mechanics and from the two-process
unraveling coincide, so do the respective predictions for the correlations.

It is important to note that the definition of the observables (7) and (8)
implies the following strong _postulate_ : When photons pass through
macroscopic polarizers, all polarization and phase shift effects coincide with
those for classical electromagnetic waves. Also filters must act on individual
photons in the same way as on electromagnetic waves.

##  V Double-slit experiment

The double-slit experiment provides most convincing evidence in favor of wave-
like properties of quantum particles. As for waves of water or electromagnetic
waves, one likes to add amplitudes and to square the result to obtain
intensities exhibiting interference effects. We discuss some details in the
spirit of Sections 1-3 to 1-5 in Volume III of the Feynman Lectures on Physics
[25]. A similar discussion, but from a more philosophical perspective, can be
found in Section 7.1 of [22]. An idealized experimental setup for the double-
slit experiment with electrons is sketched in Figure 2.

Figure 2: Double-slit experiment with electrons explained in terms of wave
functions (symbolically illustrated by the interference of waves).

The electron gun in Figure 2 could simply be a tungsten wire emitting
electrons when heated by an electric current, surrounded by a metal box with a
hole in it. A voltage between the wire and the box accelerates the electrons,
where all electrons leaving the box through the hole possess nearly the same
kinetic energy $E$. The wall with two parallel slits of equal width typically
is a thin metal plate. Attached to the absorbing wall, there is a movable
electron detector, say a Geiger counter, to measure the distribution of
electron hits as a function of position on the absorbing wall.

The electrons are found to hit the absorbing wall as individual events
occurring at random positions $x$. If only Slit I (or II) is open, one finds
the probability density $P_{\rm I}$ (or $P_{\rm II}$) for the spatial
distribution of events on the absorber, which may also be regarded as
intensities (see Figure 2). If both slits are open, one observes the startling
interference pattern $P_{\rm I,\,II}$ for the distribution of discrete
electron hits on the screen, which is taken as evidence for a “particle-wave
duality.” Electrons do not simply go through either one or the other slit so
that a wave-like pattern with constructive and destructive interference
arises. Note that also the single-slit probability densities $P_{\rm I}$ and
$P_{\rm II}$ are affected by wave-like diffraction.

###  V.1 Standard interpretation

Following [25], the spin of the electron is neglected in our discussion of the
double-slit experiment. The probability density $P_{\rm I,\,II}$ obtained with
both slits open is clearly not the sum of the probability densities $P_{\rm
I}$ and $P_{\rm II}$ for each slit alone. As for waves of water or
electromagnetic waves, interference effects can be described by adding
amplitudes rather than adding probability densities or intensities,

$P_{\rm I,\,II}=|\psi_{\rm I}+\psi_{\rm II}|^{2}\quad\neq\quad|\psi_{\rm
I}|^{2}+|\psi_{\rm II}|^{2}=P_{\rm I}+P_{\rm II},$ (9)

where the complex amplitudes $\psi_{\rm I}$ and $\psi_{\rm II}$ are the wave
functions for the single-slit experiments, obtained by solving the
corresponding Schrödinger equations. The probability densities sketched in
Figure 2 actually correspond to the intensity profiles obtained from the
Fraunhofer diffraction equation of classical wave optics for large distances
from the slits. Such a calculation is based on Huygens’ idea that every point
on a wavefront acts as a source of a spherical wave and that the sum of all
these spherical waves determines the further propagation of the wavefront.

Experimental data for cold neutrons can be found in [26]. Less detailed
experimental data for electrons are available in a much older paper [27] (in
German), or in its partial translation into English [28].

Explanations of the type sketched above are generally presented and readily
accepted as a sound and convincing story of quantum interference by both
physicists and philosophers. However, such explanations may be considered as
rather symbolic, raising a number of questions. For example, note that
$\psi_{\rm I}$ and $\psi_{\rm II}$ in (9) are time-independent wave functions
evaluated on the absorbing wall, that is, on a boundary of the domain. On what
domain should one actually solve the time-independent Schrödinger equation?
Between the walls or also around the electron gun? What kind of boundary
conditions are required to solve the second-order time-independent Schrödinger
equation? Is the wave function on the absorbing wall needed as a boundary
condition or is it a predictive result of the full solution? Is the sum
$\psi_{\rm I}+\psi_{\rm II}$ consistent with the boundary conditions for the
double-slit experiment? Furthermore note that the phase shifts, which are at
the heart of interference, are in direct correspondence to the flight times
for the electrons. How can then a time-independent solution provide the whole
story? Why shouldn’t small differences in arrival times reveal through which
slit an electron has passed? Does a discrete electron hit on the absorbing
wall occur at one instant in time or is it smeared in time due to
contributions from the two slits? A detailed theoretical discussion that goes
far beyond the usual textbook arguments can be found in [29].

Maybe the above questions could be addressed most convincingly within Bohmian
mechanics [30, 31, 32, 33, 34]. Whereas all these questions are nonchalantly
ignored for the standard interpretation based on wave functions, this is no
longer possible for the two-process unraveling.

###  V.2 Two-process interpretation

Each of the independent jump processes of the two-process unraveling consists
of the following five steps: creation of an electron at the exit hole of the
electron gun, free evolution of the electron to one of the two slits,
effective collision of the electron in the corresponding narrow slit, free
evolution of the electron to the absorbing wall, and absorption of the
electron by the wall. As in the standard interpretation, we neglect the
electron spin, and we assume that all electrons leave the gun with the same
(nonrelativistic) energy $E$. The magnitude of the electron momentum $\bm{k}$
is then restricted by $\bm{k}^{2}=2mE$, where $m$ is the electron mass. We
further assume that there is a large but finite number of possible
orientations of $\bm{k}$ and that both the set of possible momentum states and
their frequencies of occurrence respect the up-down symmetry of the experiment
illustrated in Figure 3. Finally, all relevant momenta possess a positive
component to the right.

To elaborate the stochastic jump processes in more detail, we introduce the
sets ${\cal K}_{\rm I},{\cal K}_{\rm II}$ which consist of the momenta
pointing from the exit of the electron gun to slit I or II, respectively, and
the set ${\cal K}$ containing all momenta that can occur in an effective
elastic collision in a slit. Simultaneously but independently for the two
processes of the unraveling, an electron is initiated with a momentum
$\bm{k}\in{\cal K}_{\rm I}\cup{\cal K}_{\rm II}$ so that it can pass through
one of the slits. After the time determined by the distance of the electron
gun from the wall with slits and the component of the momentum $\bm{k}$ normal
to the wall, the electron passes a slit. As a result of a high interaction
rate with macroscopic matter at the narrow slit, the electron jumps into a
state $\bm{k}^{\prime}\in{\cal K}$, where we make the simplifying assumption
of a single effective elastic collision, that is $|\bm{k}^{\prime}|=|\bm{k}|$.
The flight time to the absorbing wall is determined by the distance between
the parallel walls and by the normal component of $\bm{k}^{\prime}$. When the
electron hits the absorbing wall, it is assumed to be stopped by an inelastic
collision and its momentum jumps to $\bm{0}$.

Figure 3: Double-slit experiment with electrons explained by two-process
interpretation.

Equipped with the construction of the independent jump processes of the
stochastic unraveling, how can we find the probability for an electron to hit
the detector? As the jump process involves only momentum states, we need to
consider also the geometry of the experimental setup. For a given position of
the detector, we can construct the set ${\cal D}$ of all pairs of momenta
$(\bm{k},\bm{k}^{\prime})$ for which the electron ends up in the detector. The
number of possible momentum states should be so large that, even for a
detector with diameter small compared to the width of the slits, the set
${\cal D}$ contains many pairs. Figure 3 shows four such
$(\bm{k},\bm{k}^{\prime})$ pairs and the corresponding piece-wise linear
“trajectories” that end in the detector. We put the term “trajectories” in
quotation marks because the position of an electron is known only when
collisions occur, that is, in the electron gun, at a slit, and on the
absorbing wall, thus defining a $(\bm{k},\bm{k}^{\prime})$ pair; in between,
the free evolution of the electrons comes with a certain correlation structure
between collision events implying flight times and phase factors expressed by
piece-wise linearity, just as in the EPR experiment. Two “trajectories” per
slit is the minimum number required to account for single-slit diffraction.
Any finite number of “trajectories” per slit can be treated in the same way
and even the limit to infinitely many “trajectories” may be considered. If,
and only if, the “trajectories” of both jump processes are contained in ${\cal
D}$, there can be a nonzero contribution to the probability for the electron
to end up in the detector. Note that the absorption of the electron for the
two processes does not occur at exactly the same time; a non-vanishing
contribution to the density matrix arises only after both processes have
reached the absorbing wall.

The relevance of $(\bm{k},\bm{k}^{\prime})$ pairs shows that the calculation
of probability densities depends on the sequences of events occurring in the
jump processes. Such an analysis goes beyond the evolution of the density
matrix according to the von Neumann equation (1) and can only be performed in
the context of the more detailed stochastic unravelings of the density matrix.
An appealing framework for analyzing such histories of events is provided in
[35].

We are now in a position to evaluate the probability for an electron to hit
the detector. For the pair $j=(\bm{k},\bm{k}^{\prime})\in{\cal D}$, the
probability $p_{j}$ is given by the product of the probabilities for
generating and scattering the electron in the corresponding momentum states
$\bm{k}$ and $\bm{k}^{\prime}$, respectively. The complex phase factor
$\phi_{j}$ is determined by the length of the corresponding piece-wise linear
“trajectory” or, equivalently, by the flight time. The probability for a pair
of independent “trajectories” $j,l$ is given by the product $p_{j}\,p_{l}$,
and the overall probability for electron detection suggested by the density
matrix (2) is given by

$\sum_{j,l\in{\cal D}}p_{j}p_{l}\,\phi_{j}\phi_{l}^{*}=\bigg{|}\sum_{j\in{\cal
D}}p_{j}\phi_{j}\,\bigg{|}^{2}.$ (10)

What looks like the square of a sum actually results from the product of all
independent combinations of possibilities. The independent processes of the
unraveling imply the same result as the corresponding superposition of
properly weighted states [in (9), equal weights are assumed]. Without loss of
generality, the phase factors at the hole of the electron gun have been taken
as unity [random phase factors would average out in (10)].

It is remarkable that the numerical calculation performed in [26] for
comparison to experimental diffraction data for neutrons is performed in a
fully analogous way. It is also based on the phase factors resulting from the
sum of the lengths of the linear paths from the electron source to a point in
one of the slits and from there to the detector, and their final result
corresponds to the right-hand side of (10), thus confirming the assumed
correlation structure between collision events. The left-hand side reveals
that we here do not obtain this result by superposition, but from the bilinear
representation of the density matrix in terms of stochastically independent
bra- and ket-vectors.

##  VI Conclusions

The proposed two-worlds interpretation of quantum mechanics leads to the same
equation for density matrices as standard quantum mechanics, which is the von
Neumann equation (1). In standard quantum mechanics, this equation for density
matrices is inferred from the Schrödinger equation for wave functions. In the
two-worlds interpretation, the von Neumann equation is obtained from two
stochastic jump processes in Hilbert space. Although the same equation arises
in both approaches, they nevertheless are significantly different. The two-
worlds interpretation introduces a new entanglement mechanism and heavily
restricts the quantum systems to which the von Neumann equation can be
applied, thus reducing the risk of paradoxes.

We have proposed to use unravelings of density matrices in terms of stochastic
jump processes to account for the stochastic nature of quantum mechanics in
the most direct way. Continuous free Schrödinger evolution is interrupted by
stochastic quantum jumps that reproduce the proper interaction effects. The
bilinear representation (2) of density matrices in terms of two independently
evolving stochastic processes is the most general, natural and versatile
option. The enormous freedom in constructing two-process unravelings is
restricted by a strong superselection rule, which essentially eliminates
superposition states from quantum mechanics. This strict superselection rule,
which can be justified by obtaining quantum mechanics as a limit of quantum
field theory, requires that the content of fundamental particles in a quantum
system is well-defined; superpositions of particle states with different
characteristic properties are forbidden. In the proposed formulation of
quantum mechanics, entanglements no longer arise from superposition, but
rather from the dyadic pairing and averaging of the two stochastic state
vectors of our unravelings in the fundamental representation (2) of the
density matrix as a correlation function.

Superposition states are at the origin of many paradoxes in quantum mechanics.
Therefore, a superposition-free implementation of quantum entanglement offers
the possibility to eliminate paradoxes. This is a consequence of restricting
the applicability of quantum mechanics by the strong superselection rule that
results from the origin of quantum mechanics in the quantum field theory of
elementary particle physics.

For example, in Schrödinger’s cat version of the EPR experiment, the
superposition state involves a complex macroscopic system. In the tradition of
“Wigner’s friend” [36], Frauchiger and Renner [37] assume the applicability of
quantum mechanics to even more complex superposition states that include
agents who are themselves using quantum theory, and they then reveal
inconsistencies in conventional quantum mechanics. However, for these
situations the applicability of quantum mechanics cannot be justified by
making a connection to quantum field theory and verifying the strong
superselection rule.

The stochastic jumps of our unravelings reproduce the interaction of standard
quantum mechanics. Being equivalent to the interaction part of the Schrödinger
equation, they reflect the intrinsic stochastic nature of quantum mechanics,
not an additional feature. This situation is fundamentally different from the
GRW approach [38], in which spontaneous collapses of wave functions are an
additional stochastic jump feature on top of the full Schrödinger dynamics
[39]. In the GRW approach, superpositions are suppressed only in the passage
from microscopic to macroscopic systems [40].

Despite the linguistic similarity between the two-worlds and many-worlds [41,
42] interpretations of quantum mechanics, they are very different views.
Unlike in the many-worlds interpretation with a branching structure into many
separate worlds, a combination or overlay of two worlds, or maybe better of
two semi-worlds, determines the behavior of a single full world in the present
interpretation. In the many-worlds view, “world” is often replaced by
“universe” whereas, in the two-worlds interpretation, we deal with two
stochastic processes in the Hilbert space of a quantum “system.” These two
processes are rigorously ruled by classical probability theory, where
probability is an ontic feature of quantum theory. The Markov property of its
stochastic processes suggests to consider the two-worlds interpretation as a
hidden-variable theory. Note that the nature of this hidden-variable theory is
very different from deterministic Bohmian mechanics [30, 31, 32, 33, 34].

Although it is beyond the scope of this paper to develop a general theory of
measurement, it should be pointed out that a theory of measurable multitime
correlation functions has been developed in [1]. For the semilinear quantum
master equations of thermodynamic origin, which are linear for scalar
multiplication but, in general, nonlinear for addition, this theory of
measurable correlations has been generalized in [8]. One can further ask the
question whether density matrices can be measured. This question can, for
example, be addressed by quantum-state tomography [43, 44]. Concerning the
measurement problem formulated in a concise way by Maudlin [45] as the mutual
inconsistency of three claims associated with the standard formulation of
quantum mechanics in terms of wave functions, none of these three claims is
made in the two-worlds interpretation.

“Density-matrix realism” as an appealing alternative to “wave-function
realism” has been discussed in great detail in [46]. We here go a step further
and consider the stochastic unraveling of a density matrix as the fundamental
representation of a quantum system. Unravelings provide an opportunity of a
new interpretation of quantum mechanics, and this new form of realism may be
called “unravelism.” Complete knowledge of a quantum state requires two
stochastic state vectors. These two “semi-worlds” are what there ultimately
exists, and they play together to characterize the quantum state of the ”full
world,” including entanglements. In the context of dissipative quantum field
theory, this opportunity has already been explored in [8, 7]. The magic number
“two” in the two-world interpretation corresponds to the number “2” in the
expression $|\psi|^{2}$, which relates the wave function to probability. The
role of Born’s rule is taken over by the bilinear expression (2) for the
probabilistic tool density matrix in terms of two more fundamental stochastic
objects.

It might be desirable to couple the semi-worlds of the two-worlds
interpretation (bringing two views of the world together is a challenging
task, not only for drunkards). Such a coupling indeed arises as soon as we
pass from reversible to irreversible quantum systems [1, 8]. As we argued in
the introduction, irreversible dynamics should be considered as the most
natural choice, whereas reversible dynamics can be justified only in
exceptional situations. For thermodynamic equilibrium states, it has been
observed that the two processes of the unraveling may not be independent [10,
9]. This should not be surprising because thermodynamic equilibrium states are
reached by some kind of irreversible process, where the precise form or
strength of the process is irrelevant.

A quantum field theory that incorporates dissipative smearing at very short
length scales has been elaborated in hcoqft. It has been argued that the
length scale at which dissipative smearing sets a limit to physical
resolvability could be identified with the Planck length. As the Planck length
involves Newton’s gravitational constant it may be concluded that
irreversibility is associated with gravity and, therefore, that also the
coupling between the two worlds of an unraveling can be a gravitational
effect. The foundations for including gravity into dissipative quantum field
theory have been laid in [47, 48].

###### Acknowledgements.

I would like to thank Michael Esfeld, Carlo Rovelli, Andrea Oldofredi, Simon
Friederich, Alexei Bazavov, Romain Chessex and Alexander Weyman for helpful
comments on a first draft of this paper. I gratefully acknowledge inspiring
discussions with Jürg Fröhlich and Amine Rusi El Hassani on the operator-
algebraic ETH approach (Events–Trees–Histories) to quantum theory.

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$g^{\prime\prime\prime}_{3}(x)=6s\neq 0,$ (4.27)

for all cases. Therefore

  1. 1.

If $x_{2}=x_{0}=0,\,x_{1}\neq 0$, the zero fixed point has multiplicity of two
and we have that $g^{\prime}_{3}(0)=1$, $g^{\prime\prime}_{3}(0)=-2sx_{1}\neq
0$ and $g^{\prime\prime\prime}_{3}(0)=6s\neq 0$; therefore, by theorem 1.7,
the zero fixed point is an _unstable_ fixed point. Applying theorem 1.9 we get
the particular cases of semistability.

  2. 2.

If $x_{1}=x_{2}\neq 0$, this is fixed point has multiplicity of two and we
have that $g^{\prime}_{3}(x_{1})=1$, $g^{\prime\prime}_{3}(x_{1})=2sx_{1}\neq
0$ and $g^{\prime\prime\prime}_{3}(0)=6s\neq 0$; therefore, by theorem 1.7,
this fixed point is _unstable_ ; moreover, the semistability cases are
inferred from theorem 1.9 again.

  3. 3.

If $x_{0}=x_{1}=x_{2}=0$, the zero fixed point has multiplicity of three and
we have that $g^{\prime}_{3}(0)=1$, $g^{\prime\prime}_{3}(0)=0$ and
$g^{\prime\prime\prime}_{3}(0)=6s\neq 0$; therefore, by theorem 1.7, if $M>0$
the zero fixed point is _unstable_ and if $M<0$ it is _asymptotically stable_.

∎

###  4.7 Cubic examples

Here we will deal with specific parametrizations for the fixed points $x_{1}$
and $x_{2}$ in order to clarify the above findings and to demonstrate how we
can construct bifurcation diagrams with specific predetermined properties with
cubic maps. We will consider $M>0$ unless otherwise stated explicitly.

######  Example 4.1.

First, consider linear parametrizations for both $x_{1}$ and $x_{2}$ as

$x_{1}(\lambda)=-\lambda,\quad x_{2}(\lambda)=\lambda.$ (4.28)

The result is plotted in the lower panel of figure 4.1, where we see that the
middle fixed point is $x_{0}=0$ always, so we expect this to be the only
stable fixed point, until the separation between this and the other points
breaks the stability condition and the period doubling bifurcation cascade
sets on. The corresponding stability conditions are shown in the middle panel
of figure 4.1, where we confirm that the curve for $x_{0}$ is the only one
within the stability band $(-2,\,0)$ until $\lambda\approx 1.45$ where the
curve crosses the barrier of -2 and drops from then, causing $x_{0}$ to
bifurcate. The corresponding bifurcation diagram is shown in the upper panel
of figure 4.1, where we confirm the stated above.

Figure 4.1: Bifurcation diagram (upper), stability conditions plot (middle)
and fixed points plot (lower) for the linear parametrization of the fixed
points example of the CCM.

######  Example 4.2.

Next, based on the fact that the stability conditions depend upon the product
of the distances between the stable fixed point and the other two fixed
points, we take square root parametrizations of the fixed points and so that
the product is linear. We take the parametrizations as

$x_{1}(\lambda)=\sqrt{\lambda},\quad x_{2}(\lambda)=-\sqrt{\lambda},$ (4.29)

The plot of the fixed points is shown in lower panel of figure 4.2, where we
see again that the middle fixed point is $x_{0}$, so this will be the only
attracting fixed point as long as its corresponding stability curve is within
the stability region. In the middle panel of figure 4.2 we see that such
stability curve goes out of said region at $\lambda=2$ and $x_{0}$ continues
to bifurcate (period doubling) thereof. The corresponding bifurcation diagram
is shown in the upper panel of figure 4.2, which is very much like the one of
example 4.1 above, but looks “stretched” since the progression towards
bifurcation is now linear, instead of quadratic, and given by the stability
conditions curves.

Figure 4.2: Bifurcation diagram (upper), stability conditions plot (middle)
and fixed points plot (lower) for the square root parametrizations of the
fixed points example of the CCM.

######  Example 4.3.

Now we will explore the full range of stability regions by making a linearly
varying fixed point pass through the regions defined by the constant $x_{0}$
and a constant $x_{1}$. We define then

$x_{1}(\lambda)=2,\quad x_{2}(\lambda)=6\lambda+1,$ (4.30)

We then obtain the plot of the lower panel of figure 4.3, for the selected
range of interest of the parameter $\lambda$. In the middle panel of the same
figure we can see the stability curves for the fixed points, where we see that
initially, from left to right, all fixed points are unstable then,
progressively, $x_{0}$, $x_{2}$ and $x_{1}$ become stable, the latter one
losing stability for still greater values of $\lambda$. The corresponding
bifurcation diagram is shown in the upper panel, where we can see how first,
the stable fixed point is $x_{0}=0$, since it is the middle one, but begins in
the chaotic region and goes “reversal” towards being stable; then, as $x_{2}$
crosses through zero, it becomes the middle stable fixed point and, when it in
turn crosses the constant $x_{2}$, this latter one becomes the stable fixed
point, again loosing stability when $x_{2}$ crosses the stability band for
$x_{1}$.

Figure 4.3: Bifurcation diagram (upper), stability conditions plot (middle)
and fixed points plot (lower) for the constant and linear mixed
parametrizations of the fixed points example of the CCM.

######  Example 4.4.

This example is analogous to example 3.5.5 of chapter 3, where we had the
$x_{1}$ fixed point of the CQM tend asymptotically to a value of approximately
$-b_{\infty}$. To achieve the same qualitative phenomenon in the CCM, we can
define the roots as

$x_{1}(\lambda)=1.817-e^{-\lambda},\quad x_{2}(\lambda)=-1.817+e^{-\lambda}$
(4.31)

so that $x_{2}=-x_{1}$ and the fixed points are symmetric about the zero fixed
point; their product, which gives the stability condition for the zero fixed
point, will then be $-3.301489+3.634e^{-\lambda}+e^{-2\lambda}$ which will
asymptotically tend to $-3.301489\approx-b_{\infty}$, as
$\lambda\rightarrow\infty$; this allowed us to calculate the bifurcation
values of table 4.1. The lower panel of figure 4.4 shows the graphs of the
fixed points, the middle panel shows the stability conditions for each fixed
point and the upper panel shows the corresponding bifurcation diagram. This
example allows us to determine the bifurcation values $c_{k}$ of the CCM with
more precision; the values obtained are shown in table 4.1.

Figure 4.4: Bifurcation diagram (upper), stability conditions plot (middle)
and fixed points plot (lower) for the exponential asymptotic parametrizations
of the fixed points example 4.4 for the CCM.

##  Chapter 5 n-th degree polynomial maps

Consider again a one-dimensional discrete dynamical system defined by

$y_{n+1}=f(y_{n};\lambda)$ (5.1)

Where $f$ is a polynomial in one real variable $y$ with real fixed points and
whose coefficients depend smoothly on the real parameter $\lambda$. Depending
on the form of $f$ we have arrived previously to the definitions of the
General, Linear Factors and Canonical Forms of the polynomial quadratic and
cubic maps. Next, we will define precisely what we understand by the General
and Canonical Maps of $n$-th degree.

######  Definition 5.1 (General Polynomial Map).

The General Polynomial Map of $n$-th degree (GPM-$n$) is defined by

$f_{n}(y):=y+(-1)^{n-1}P_{f_{n}}(y)$ (5.2)

where

$P_{f_{n}}(y):=(-1)^{n-1}\sum_{i=0}^{n}a_{i}\,y^{i}.$ (5.3)

$P_{f_{n}}$ is called the _Fixed Points Polynomial_ associated to $f_{n}$.

It is obvious that any $n$-th degree real polynomial in one variable can be
put into the General Form by means of adjusting the value of the $a_{1}$
coefficient properly in the fixed points polynomial. This is the broadest
class of real polynomials of finite degree.

It is clear that the roots of $P_{f_{n}}$ are the fixed points of $f_{n}$. In
the case of $n$ odd, the fundamental theorem of algebra guarantees the
existence of at least one real fixed point. Let
$y_{i}\in\mathbb{C},\,i\in\\{0,1,\,...,\,n-1\\}$, be the $n$ roots of
$P_{f_{n}}$, then $(y-y_{i})$ is a factor of $P_{f_{n}}$ by the factor
theorem, therefore we can rewrite $P_{f_{n}}$ as

$P_{f_{n}}(y)=M(y-y_{0})\cdots(y-y_{n-1})=M\prod_{j=0}^{n-1}(y-y_{j}),\quad
M\in\mathbb{R},$ (5.4)

and then define the

######  Definition 5.2 (Linear Factors Form).

Let $f_{n}$ be the GPM-$n$ and $y_{j}$, $j\in\\{0,\,...,\,n-1\\}$, its $n$
fixed points. Then we can write

$\displaystyle f_{n}(y)$
$\displaystyle=y+(-1)^{n-1}\,M\,\prod_{i=0}^{n-1}(y-y_{i}),$ (5.5)
$\displaystyle=y+(-1)^{n-1}\mathrm{sgn}(M)|M|\prod_{i=0}^{n-1}(y-y_{i}),$
$\displaystyle=y+(-1)^{n-1}s\tilde{M}\prod_{i=0}^{n-1}(y-y_{i}),$

with the definitions of $s$ and $\tilde{M}$ used in chapters 3 and 4. We call
this the _Linear Factors Form_ of $f_{n}$.

We can directly verify that for $n=2,\,3$ we obtain the corresponding Linear
Factors Form of the Quadratic and Cubic Maps, described on pages 3.2 and 4.2,
respectively. Once we know the $n$ fixed points of a map $f_{n}$, it is
straightforward to write its linear factors form. The motivation behind the
$(-1)^{n-1}$ factor is that we want that, for purely aesthetic reasons, if
$M>0$, the fixed points are real and $0<y_{0}<y_{1}<\cdots<y_{n-1}$, we have
that $f^{\prime}_{n}(0)\geq 0$, which is thus accomplished.

We will now restrict this set of polynomials to those whose fixed points
polynomials have only real roots, i.e. maps with real fixed points only,
though not necessarily distinct. Let us make this precise by the following

######  Definition 5.3 (Canonical Polynomials Set).

The Canonical Polynomials Set, denoted $P_{C}[y]$, is

$P_{C}[y]:=\\{f\in\mathbb{R}[y]\,|\,\text{$P_{f}$ has only real roots.}\\}$
(5.6)

where $\mathbb{R}[y]$ is the set of polynomials with real coefficients on the
variable $y$. Likewise, $P_{C}^{n}[y]$ denotes
$P_{C}[y]\bigcap\mathbb{R}_{n}[y]$ where $\mathbb{R}_{n}[y]$ is the set of
polynomials of degree $n$ with real coefficients on the variable $y$.

The set $P_{C}$ has been our main work ground for the analysis in this work
and, as it turns out, its elements can be put in a much nicer form, easier to
understand.

We can further reduce the complexity of this set of maps by means of the
transformation

$y=T_{n}(x):=s\tilde{M}^{-\frac{1}{n-1}}x+y_{0}$ (5.7)

where $y_{0}$ is a real fixed point of the map in its linear product form.
Notice that $T_{n}$ is linear, therefore it has an inverse

$x=T_{n}^{-1}(y)=s\tilde{M}^{\frac{1}{n-1}}(y-y_{0}).$ (5.8)

$T_{n}$ is in fact a homeomorphism. We will drop the subscript $n$ when
referring to the transformation for no specific degree. Applying this
transformation to $y$ we reach the

######  Definition 5.4 (Canonical Polynomial Map).

The Canonical Polynomial Map of $n$-th degree (CPM-$n$) is

$g_{n}(x):=x+(-1)^{n-1}\,s^{n}\,x\,\prod_{i=1}^{n-1}(x-x_{i}),\quad n\geq 2,$
(5.9)

where

$x_{i}=s\tilde{M}^{\frac{1}{n-1}}(y_{i}-y_{0}),$

and $y_{j}$ are the $n$ fixed points of the corresponding linear factors form
map of $n$-th degree, (at least) $y_{0}$ being real.

It is clear from the definition that $x_{0}=0$ always. Notice also that the
$x_{i}$ result from evaluating $T_{n}^{-1}$ in the corresponding $y_{i}$. We
can easily prove that not only does the canonical map result from applying
$T$, but that the canonical map is in fact $T$-conjugate to the linear factors
form.

######  Proposition 5.1.

Let $f_{n}$ and $T_{n}$ and $g_{n}$ as defined above, having $f_{n}$ at least
one real fixed point; let $y_{0}$ be this real fixed point, without loss of
generality. Then $f_{n}$ is $T_{n}$-conjugate to $g_{n}$.

###### Proof.

It is clear that $T_{n}$ is a homeomorphism since it is linear. Then, we must
only prove that $T_{n}\circ f_{n}=g_{n}\circ T_{n}$, i.e.
$f_{n}(T_{n}(x))=T_{n}(g_{n}(x))$. We then have

$\displaystyle f_{n}(T_{n}(x))$
$\displaystyle=f_{n}(s\tilde{M}^{-\frac{1}{n-1}}x+y_{0})$ (5.10)
$\displaystyle=s\tilde{M}^{-\frac{1}{n-1}}x+y_{0}+(-1)^{n-1}\,s\tilde{M}\,\prod_{i=0}^{n-1}(s\tilde{M}^{-\frac{1}{n-1}}x+y_{0}-y_{i})$
$\displaystyle=s\tilde{M}^{-\frac{1}{n-1}}x+(-1)^{n-1}\,s^{2}\tilde{M}\,\tilde{M}^{-\frac{1}{n-1}}\,x\,\prod_{i=1}^{n-1}(s\tilde{M}^{-\frac{1}{n-1}}x+y_{0}-y_{i})+y_{0}$
$\displaystyle=s\tilde{M}^{-\frac{1}{n-1}}x+(-1)^{n-1}\,s^{n-1}\tilde{M}^{-\frac{1}{n-1}}\,x\,\prod_{i=1}^{n-1}\left[x-s\tilde{M}^{\frac{1}{n-1}}(y_{i}-y_{0})\right]+y_{0}$
$\displaystyle=s\tilde{M}^{-\frac{1}{n-1}}x+(-1)^{n-1}\,s^{n-1}M^{-\frac{1}{n-1}}\,x\,\prod_{i=1}^{n-1}\left(x-x_{i}\right)+y_{0}$
$\displaystyle=s\tilde{M}^{-\frac{1}{n-1}}\left[x+(-1)^{n-1}\,s^{n}x\,\prod_{i=1}^{n-1}\left(x-x_{i}\right)\right]+y_{0}$
$\displaystyle=T_{n}(g_{n}(x)).$

Where we have used $s^{2}=1$ and $s=s^{-1}$. ∎

This turns out to be very useful, since we know that topological conjugacy is
an equivalence relation that _preserves the property of chaos_ (see theorem
1.13 in page 1.13). This means that the analysis of stability and chaos (i.e.
the “dynamics”) of real polynomial maps with real fixed points is reduced to
the study of the canonical polynomial maps defined above, since we can always
take any polynomial in $P_{C}[x]$ to its canonical form by means of $T$,
determine the stability properties and then go back to the original
polynomial. A commutative diagram of the conjugacy is

$\begin{CD}y@=y@>{f_{n}}>{}>f_{n}(y)@=f_{n}(y)\\\
@A{T}A{}A@V{}V{T^{-1}}V@A{T}A{}A@V{}V{T^{-1}}V\\\
x@=x@>{}>{g_{n}}>g_{n}(x)@=g_{n}(x)\end{CD}$ (5.11)

###  5.1 Stability and chaos in the canonical map of degree n

The derivative of $g_{n}$, recalling $x_{0}=0$ to simplify notation, is

$g_{n}^{\prime}(x)=1+(-1)^{n-1}\,s^{n}\,\sum_{j=0}^{n-1}\prod_{i=0,\,i\neq
j}^{n-1}(x-x_{i}).$ (5.12)

Evaluating (5.12) in the fixed point $x_{k}$ we get the eigenvalue function
for each $x_{k}$

$\displaystyle\phi_{k}(\lambda)=g_{n}^{\prime}(x_{k}(\lambda))$
$\displaystyle=1+(-1)^{n-1}\,s^{n}\,\prod_{i=0,\,i\neq
k}^{n-1}(x_{k}(\lambda)-x_{i}(\lambda))$ (5.13)
$\displaystyle=1+s^{n}\,\prod_{i=0,\,i\neq
k}^{n-1}(x_{i}(\lambda)-x_{k}(\lambda)).$

Then, the asymptotic stability condition $|g_{n}^{\prime}(x_{k})|<1$ implies
that

$-2<s^{n}\,\prod_{i=0,\,i\neq k}^{n-1}(x_{i}-x_{k})<0.$ (5.14)

From (5.14) we can recover all the stability conditions for the fixed points
of the Canonical Quadratic and Cubic Maps.

which leads us to the following

######  Definition 5.5 (Product Distance Function).

Let $g_{n}$ be the Canonical Polynomial Map of $n$-th degree and
$x_{0}=0,\,x_{1},\,...,\,x_{n-1}$ its $n$ fixed points, all of which depend
upon the parameter $\lambda$. Let $x_{k}$ be a real fixed point among the
latter. Then

$D_{n,k}(\lambda):=s^{n}\,\prod_{i=0,\,i\neq
k}^{n-1}(x_{i}(\lambda)-x_{k}(\lambda)),\quad k\in\\{0,\,...,\,n-1\\},\,n\geq
2,$ (5.15)

is called the _Product Distance Function (PDF)_ of $x_{k}$.

The definition is motivated by the fact that $D_{n,k}$ is a product of the
distances to $x_{k}$ of each of the other $n-1$ fixed points and that this
quantity is fundamental in determining the stability of the fixed points.
These distances are positive when $x_{i}>x_{k}$ and is negative when
$x_{i}<x_{k}$. We have stressed the dependence on the parameter $\lambda$ in
the definition of $D_{n,k}$ so that its character as a function is clear,
stemming from the corresponding dependence on $\lambda$ of the fixed points.
In this way, the stability condition for the $k$-th fixed point is reduced to

$-2<D_{n,k}(\lambda)<0.$ (5.16)

Since $D_{n,k}$ must be negative in order for $x_{k}$ to be stable as a
sufficient condition, and an _odd_ number of factors $(x_{i}-x_{k})$ must be
negative for the product in $D_{n,k}$ to be negative, it follows that, if $n$
is even or $M>0$, an odd number of negative factors $(x_{i}-x_{k})$ is a
sufficient condition for the hyperbolic fixed point $x_{k}$ to be stable; i.e.
if $M>0$, an _odd_ number of fixed points must lie below $x_{k}$ and,
consequently, an even or zero (respectively, odd) number of fixed points must
lie above $x_{k}$ if $n$ is even (respectively, odd). By similar arguments, we
can prove

######  Proposition 5.2 (Necessary conditions for the stability of $x_{k}$).

Let $g_{n}$, $D_{n,k}$ be defined as above and $x_{k}$ be a hyperbolic real
fixed point of $g_{n}$. The following are necessary conditions for $x_{k}$ to
be an asymptotically stable fixed point:

  * •

if $M>0$ or $n$ is even, an odd number of fixed points must have values lower
than $x_{k}$; or

  * •

if $n$ is odd and $M<0$, zero or an even number of fixed points must have
values lower than $x_{k}$.

We must remark that the above conditions are _not_ sufficient for a fixed
point to be an attractor. The sufficient condition, however, is stated as

######  Theorem 5.3 (Sufficient condition for the stability of $x_{k}$).

Let $g_{n}$, $D_{n,k}$ be defined as above and $x_{k}$ be a hyperbolic real
fixed point of $g_{n}$. Then, a necessary and sufficient condition for $x_{k}$
to be an attractor is that

$-2<D_{n,k}(\lambda)<0.$ (5.17)

Below the value of -2 there are other “stability bands” that lead to further
period doubling bifurcations of the fixed points as they are crossed, but they
must be calculated numerically and, as we have seen, depend on the degree $n$
of the polynomial.

###  5.2 Example

We will now give an example in which the above theory is applied to different
degree polynomial maps.

######  Example 5.1 (Quartic Maps).

Using definition 5.2 for $n=4$, we have that

$f_{4}(y)=y-M\,(y-y_{0})(y-y_{1})(y-y_{2})(y-y_{3}).$ (5.18)

Suppose $f_{4}$ has at least one real fixed point. Without loss of generality,
suppose this fixed point is $y_{0}$. Then

$T_{4}(x)=s\tilde{M}^{-1/3}\,x+y_{0}.$

Making the substitution $y=T_{4}(x)$ we can verify that we get

$g_{4}(x)=x-x(x-x_{1})(x-x_{2})(x-x_{3}),$

where

$x_{i}=s\tilde{M}^{1/3}(y_{i}-y_{0}),\quad i\in\\{0,\,1,\,2,\,3\\},$

The stability a real fixed point $x_{k}$ is given by the product distance
function

$D_{4,k}(\lambda)=\prod_{i=0,\,i\neq k}^{3}(x_{i}(\lambda)-x_{k}(\lambda)),$

whose value must remain between minus two and zero in order for $x_{k}$ to be
asymptotically stable; that is, if all fixed points are real,

$\displaystyle-2$ $\displaystyle<D_{4,0}(\lambda)=x_{1}\,x_{2}\,x_{3}$
$\displaystyle<0,$ (5.19) $\displaystyle-2$
$\displaystyle<D_{4,1}(\lambda)=-x_{1}(x_{2}-x_{1})(x_{3}-x_{1})$
$\displaystyle<0,$ $\displaystyle-2$
$\displaystyle<D_{4,2}(\lambda)=-x_{2}(x_{1}-x_{2})(x_{3}-x_{2})$
$\displaystyle<0,$ $\displaystyle-2$
$\displaystyle<D_{4,3}(\lambda)=-x_{3}(x_{1}-x_{3})(x_{2}-x_{3})$
$\displaystyle<0,$

for 0, $x_{1},\,x_{2}$ and $x_{3}$ to be asymptotically stable fixed points,
respectively.

For example, let

$\displaystyle x_{1}(\lambda)$ $\displaystyle=\lambda,$ (5.20) $\displaystyle
x_{2}(\lambda)$ $\displaystyle=-\lambda,$ $\displaystyle x_{3}(\lambda)$
$\displaystyle=2\,\lambda.$

The plots of these fixed points with their corresponding parametric dependence
on $\lambda$ are shown in the lower panel of figure 5.1.

In light of proposition 5.2 we expect only $x_{0}$ and $x_{3}$ to be able to
be asymptotically stable fixed points in any given range of $\lambda$. As the
middle panel of figure 5.1 shows, precisely $x_{0}$ and $x_{3}$ are the fixed
points whose product distances cross the stability band $(-2,\,0)$ in the
range of $\lambda$ being plotted. As we recall, the product distance functions
are the “stability conditions” of the fixed points. As long as the product
distances remain within the stability interval, the fixed points are
attractors, as we can verify in the upper panel of figure 5.1; also in this
last panel, we can see the two attracting fixed points at the beginning of the
plotted range; then, first $x_{3}$ loses its stability and gives rise to the
period doubling bifurcations cascade which leads to chaotic behavior; later,
zero also loses its stability and also gives rise to period doubling and
chaos.

Figure 5.1: Bifurcation diagram (upper), Product Distance Functions (middle)
and fixed points plot of the quartic polynomial map example.

##  Chapter 6 An example application of the canonical map theory

We shall now consider an example application of the theory developed in
chapters 3 through 5 to a couple of well known examples of discrete dynamical
systems. This applications demonstrate a new method of solution for analyzing
the dynamics of real polynomial maps that simplifies upon the procedures found
in current literature.

###  6.1 The logistic map

The logistic map is the most immediate and obvious example application. We had
already encountered briefly the logistic map, $L_{\lambda}(x)=\lambda x(1-x)$,
in subsection 1.8 of chapter 1 and again in example 2.1 of chapter 2, where we
saw that the map undergoes a series of period doubling bifurcations starting
at the value of $\lambda=3$, ultimately achieving a chaotic nature at
$\lambda\approx 3.570$ [Elaydi(2008), p. 47]. We will now proceed to utilize
the theory developed in the previous chapters, particularly in chapter 3 and
generalized in chapter 5, to this particular map.

We already know (or can easily calculate) that the fixed points of the
logistic map are $y_{0}=0$ and $y_{1}=\frac{\lambda-1}{\lambda}$
[Elaydi(2008), p. 43]. With some minor algebra, we can then check that in the
corresponding Linear Factors Form of the logistic map is then

$h_{\lambda}(x)=x-\lambda x\left(x-\frac{\lambda-1}{\lambda}\right)$ (6.1)

where we can identify the functions of the parameter $M$, $y_{0}$ and $y_{1}$
from definition 3.2, on page 3.2, as

$\displaystyle s$ $\displaystyle=+1,$ (6.2) $\displaystyle\tilde{M}(\lambda)$
$\displaystyle=\lambda,$ (6.3) $\displaystyle y_{0}(\lambda)$
$\displaystyle=0,$ (6.4) $\displaystyle y_{1}(\lambda)$
$\displaystyle=\frac{\lambda-1}{\lambda}.$ (6.5)

The corresponding non-zero fixed point of the canonical logistic map,
$x_{1}(\lambda)=s\tilde{M}(y_{1}-y_{0})$ defined in (3.12), is then simply

$x_{1}(\lambda)=\lambda-1,$ (6.6)

from which we can state that the _canonical logistic map_ takes the explicit
form

$g_{\lambda}=x-x(x-\lambda+1).$ (6.7)

In order to determine the stability properties of these fixed points, both
zero and nonzero, in the canonical logistic map, it is then sufficient, as we
have proved in chapters 3 and 5, to observe the behavior of the Product
Distance Functions (PDFs) of these fixed points, viz.

$\displaystyle D_{g,0}(\lambda)$ $\displaystyle=x_{1}-0=x_{1}=\lambda-1,$
(6.8) $\displaystyle D_{g,1}(\lambda)$
$\displaystyle=0-x_{1}=-x_{1}=1-\lambda.$ (6.9)

By determining when these PDFs cross the stability bands whose boundaries are
shown in table 3.1 we can readily determine when these fixed points are stable
or unstable, when they bifurcate and when they reach any $2^{n}$ attracting
periodic orbit for any $n$, up to crossing the $b_{\infty}$ band. This whole
process is depicted in figure 6.1. In particular, we can see from table 3.2,
and again at figure 6.1, that when $-1<x_{1}<0$, the zero fixed point is
attracting since its PDF lies within the first stability band, then exchanges
stability at $x_{1}=0$, when this last FP becomes stable and proceeds to a
period doubling cascade upon its PDF, $-x_{1}$, crossing the bands defined by
the bifurcation values $-b_{1}$, $-b_{2}$, etc. until reaching
$x_{1}=b_{\infty}\approx 2.569941$. This last value, by the way, agrees quite
well and improves upon the approximation reported in [Elaydi(2008)] of 3.570
for the logistic map, since with the calculations of the present work
$\lambda_{\infty}=1+b_{\infty}\approx 3.569941\pm 5\times 10^{-7}$. Finally,
since these maps are topologically conjugate and it is known that the logistic
map is chaotic starting with $\lambda=4$ [Elaydi(2008)], we conclude that the
CQM must be so starting from $x_{1}=3$, which we may define as $b_{c}$, our
final “bifurcation” value.

Figure 6.1: Bifurcation diagram for the canonical logistic map.

###  6.2 Harvesting strategies

The connection of the logistic map with population models is old and well
known. In the book by [Sandefur(1990)], there are a few examples of uses of
second degree polynomials as recurrence functions which are used to model
“harvesting” –or hunting– strategies of a population of animals. The main idea
is that the population of animals grows whenever there is food and resources
in the environment which, by account of its finite resources has a certain
“carrying capacity”; this leads to a maximum population this environment can
hold, the population growing according to the logistic model. The population
may then be “harvested” –or hunted– at a certain rate yet to be specified and,
depending on this rate, it is not hard to imagine that the final fate of the
population of animals may be (i) extinction, if the rate is too high; (ii)
steady population below the carrying capacity, if the rate is “just right”; or
(iii) steady population at its maximum value dictated by the carrying capacity
of the environment. A final fourth possibility –perhaps more rare– is to (iv)
bring more individuals of the species from outside the system under analysis
and introduce them to it, therefore making it possible for the population to
surpass in number the carrying capacity of the system, but only to return
naturally to the maximum value after a finite number “time-steps”. To examine
this in detail consider the system defined by the recurrence relation

$\Delta y\equiv y_{n+1}-y_{n}=r(1-y_{n})y_{n}.$ (6.10)

Here the population growth in any given period $\Delta y$ is proportional to
both the initial population $y_{n}$ and the difference between this and the
maximum population, normalized to the value of 1. The proportion constant is
the growth rate $r$. After some simplification, we can rewrite system as

$y_{n+1}=(1+r)y_{n}-ry_{n}^{2}.$ (6.11)

We have yet to add the “harvesting term(s)”, which we might do so in several
ways.

If we consider a fixed rate, say each period we harvest a proportion $b$ of
the population (in terms of the maximum) then

$f(y)=(1+r)y-ry^{2}-b$ (6.12)

is the corresponding map of this discrete dynamical system. From here we see
that the Fixed Points Polynomial (FPP) of $f$ is111Remember
$f_{n}(y)=y-(-1)^{n}P_{f}(y)$ for the $n$-th degree polynomial.

$P_{f}(y)=ry^{2}-ry+b,$

whose FPs we can determine to be

$y_{\pm}=\frac{1\pm\sqrt{1-4b/r}}{2},$

from which we immediately calculate the Linear Factors Form (LFF) to be

$f(y)=y-ry(y-y_{+})(y-y_{-}).$ (6.13)

From this form it is also straightforward to determine that in the
corresponding canonical form we have

$x_{1}(b)=r(y_{+}-y_{-})=\sqrt{r(r-4b)}.$

Both systems are related then by the linear transformations
$y_{n}=y_{-}+x_{n}/r$ and $x_{n}=r(y_{n}-y_{-})$. With this choice, the zero
fixed point in the canonical map corresponds to $y_{-}$ and the nonzero fixed
point to $y_{+}$. To analyze the stability of this system, consider $r$ to be
fixed and given and $b$ to be the parameter of this family of systems. The one
immediate conclusion is that, for $x_{1}\in\mathbb{R}$, we necessarily have
that $x_{1}\geq 0$. Now, real $x_{1}$ implies $b\leq r/4$. Over the $r/4$
value $x_{1}$ becomes complex which does not give any fixed points (but would
mean “over-harvesting”). Since “harvesting” cannot be negative, it is clear
$b=0$ corresponds to the maximum value of $x_{1}=r$ and that $x_{1}=0$ when
$b=r/4$. Remember now that to analyze the stability of $x_{1}$ we consider its
PDF, $D_{1}=-x_{1}$. Analogously, $D_{0}=x_{1}$. Since the maximum value of
$x_{1}$ is $r$ and, being this the “unconstrained growth rate”, $0<r<1$, we
have that $-r<D_{1}<0$, therefore putting the $x_{1}$ in the stability range
between $b_{0}=0$ and $-b_{1}=-2$ determined by the first stability band. This
range is never left in any situation with physical meaning so we conclude that
the nonzero fixed point, i.e. $y_{+}$ in the original system, is always the
only asymptotically stable fixed point and the zero fixed point, i.e. $y_{-}$
in the original system, is always unstable. The only case to analyze with care
is $b=0$ since then the two fixed points collide, but proposition 3.9 (on page
3.9) guarantees that in this case $x_{1}=0$ is semistable from the right.

In conclusion:

  1. 1.

When $b=r/4$ the population faces extinction asymptotically. Over this value
extinction is achieved in a finite number of steps, there not being any more
fixed points.

  2. 2.

When $0<b<r/4$, $x_{n}\rightarrow r$ and the population tends to
$y_{+}=0.5(1+\sqrt{1-4b/r})$.

  3. 3.

When $b=0$, i.e. no harvesting, $x_{n}$ still tends to $x_{1}=0$ from the
right and, correspondingly, the population tends asymptotically to $y_{+}=1$.

###  6.3 Remarks on the application of the method

As somewhat instructive as the above examples may have been, the true
simplifying power of the Canonical Map Theory is not yet fully met. Therefore
we encourage the reader to apply this method to test the dynamics of any
discrete dynamical systems family with a polynomial map depending on a single
parameter, and evaluate the usefulness of the method by yourself. Families of
maps with particular parametric dependencies and of cubic order will give the
most useful applications for the time being, since the theory of higher degree
polynomials is so far restricted to analyzing the stability of fixed points
only.

##  Chapter 7 Conclusions and future work

Finally, lets sum up the main findings of this work and give some directions
on the future work that arises from it. Basically, we can summarize the
findings of this work as having successfully given conditions for the
stability of the fixed points of any real polynomial map with real fixed
points and that depends on a single parameter. In order to do this we have
defined “canonical polynomial maps” which are topologically conjugate to any
polynomial map of the same degree with real fixed points. Then, the stability
of the fixed points of the canonical polynomial maps has been found to depend
solely on a special function called “product distance” of a given fixed point.
The values of this product distance determine the stability of the fixed point
and when it bifurcates to give rise to attracting periodic orbits of period
$2^{n}$ for all $n$, and even ultimately when chaos arises through the period
doubling cascade, as it passes through different “stability bands”, although
the exact values and widths of these stability bands are yet to be calculated
for regions of type greater than one for higher order polynomials. The latter
must be done numerically.

As specific goals achieved we can remark that in this work we have

  * •

Generalized the existing results on parametric dependency of the coefficients
of real quadratic maps for a larger set of functions.

  * •

Studied in detail the dynamics of general real cubic maps when the parametric
dependency of its coefficients is given by continuous functions.

  * •

We have obtained transformations that take general quadratic and cubic maps
into one whose dynamics is known and easy to study, and that are also
topologically conjugate maps, so that the properties of stability and chaos
are preserved.

  * •

Explicitly given the set of fixed points for the the proposed ’canonical’
quadratic and cubic maps.

  * •

Created a method that allows to understand the dynamics of cubic maps, that
also includes quadratic maps.

  * •

Generalized the latter method to $n$-th degree real polynomial maps.

  * •

Proposed a method to generate a large class of regular-reversal cubic maps.

  * •

Reproduced the know stability characteristics of the logistic map through the
proposed method in a simpler way.

  * •

Proposed a methodology that allows to be used to create discrete dynamical
systems with some prescribed bifurcation diagram.

As future work we can point out the following:

  * •

The numerical values of the limits of the stability bands for still higher
degree canonical polynomial maps must be calculated with numerical methods.

  * •

Ultimately it is desired to obtain extensive tables of the bifurcation values
for higher order polynomials.

  * •

Algebraic multiplicity of the roots of the Fixed Points Polynomial, when
nonhyperbolic fixed points take place, must be analyzed for the general n-th
degree Canonical Polynomial Map.

  * •

The case of complex fixed points must be included in detail for completeness.

  * •

Once the preceding is accomplished, the possibility of extending the theory to
apply it to the Taylor Polynomial of any real analytical iteration function
can be analyzed.

  * •

The power and simplicity of the proposed methodology will best be appreciated
with 3rd or higher degree polynomials and when the implications on the Taylor
polynomial of any non-linear map are understood.

## Bibliography

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##  Appendix A Computational and numerical tools

For revision of the algebraic calculations, scripts in the wxMaxima scripting
language were used, running under Ubuntu GNU/Linux 11.10. Both wxMaxima and
Ubuntu are _free software_ 111“Free” as in “free speech”.. wxMaxima is a full-
featured Computer Algebra System (CAS) in which some of the plots for this
work were also made. Also, for most of the plots and bifurcation diagrams, a
Python script shown below was used to calculate and plot most of the
bifurcation diagrams presented in this work. The Python script was also ran
under Ubuntu GNU/Linux and it uses numpy and matplotlib as libraries for
numerical calculations and plotting, respectively.

    import numpy as np
    import matplotlib.pyplot as plt
    import time

    ###################### DEFINITION OF FUNCTIONS #################################
    def gn(n, xi, s=1, x0=0.1, niter=512, nplot=256): #iterates the canonical cubic map
    #niter:     total number of iterations
    #nplot:     number of points to return (for graphing)
    #x0:        initial condition
    #xi:        array containing the 3 fixed points
    #n:         order of the canonical map and number of f.p.’s
    #s:         sgn(M) from the original polynomial map

    x = np.zeros(nplot) #here we store the orbit to be returned for graphing
    xtemp = x0 #initialize orbit with initial condition
    ssn=np.power(-1,n-1)*np.power(s,n) #the sign parameter

    #part of the orbit that will not be graphed:
    for i in range(niter-nplot):
    prod=1
    for j in range(n-1):
    prod*=xtemp-xi[j+1]
    xtemp *= (1 + ssn*prod)

    x[0]=xtemp

    #part of the orbit that will be graphed:
    for i in range(nplot-1):
    prod=1
    for j in range(n-1):
    prod*=x[i]-xi[j+1]
    x[i+1] = x[i]*(1 + ssn*prod)

    return x

    def pdfn(n,xi): #Product Distance Function Values Array
    #xi: vector containing the n fixed points
    #k: number of fixed point whose stability we want to calculate
    #n: total number of fixed points
    pdf=np.zeros(n)     #array to store product distance values for each f.p.

    for k in range(n):
    prod=1 #initalize the product
    for j in range(n): #perform product
    if(j != k):
    prod *= xi[j] - xi[k]
    pdf[k]=prod

    return pdf

    def xl(i, p, lamb): #define the fixed points as a function of the parameter
    #i: number of fixed point to return value
    #p: NP 2-d array containing the extra parameters for the fuctional forms
    #   of the fixed points. 1st index is the number of fixed point, 2nd is
    #   the number of relevant parameter for that fixed point
    if(i>0):
    #return lamb*(3-lamb) #quadratic FP for QCM
    #return p[i][1]*np.sqrt(lamb) + p[i][2] #square root fixed points
    return p[i][0]*lamb + p[i][1] #linear fixed points
    #return (-1)**p[i,0]*p[i,1]+(-1)**(p[i,0]+1)*np.exp(-p[i,2]*lamb)
    else:
    return 0 #the zeroth fixed point is always zero in value

    ####################### DEFINITION OF PARAMETERS ###############################
    print time.clock(), "Defining parameters..."

    n=4         #number of fixed points and order of the canonical map
    npl=256    #number of points to graph for each value of lambda
    nit=512 + npl     #total number of iterations
    lmin=0     #starting value of lambda for the graph
    lmax=1.2      #ending value of lambda for the graph
    p=np.zeros([n,3])   #parameters for the definition of the fixed points
    p[1,0]=1
    p[1,1]=0
    #p[1,2]=-1
    p[2,0]=-1
    p[2,1]=0
    #p[2,2]=1
    p[3,0]=2
    p[3,1]=0
    #p=np.array([[0,0,0],[1,0,0],[-1,0,0]]) #for the linear FPs
    #p=np.array([[0,0,0],[1,0,0],[-1,0,0]]) #for the linear FPs
    #p=np.array([[0,0,0],[0,2,0],[6,1,0]])   #one linear FP another const. FP
    lval=np.linspace(lmin,lmax,600) #range of the parameter for the bif. diagram
    midl=(lmin+lmax)/2  #middle point of the lambda interval (bif.val. estimation)
    uncert=abs(lmax-lmin)/2 #uncertainty for the bif. value estimation
    xi=np.zeros(n) #the zero fixed point is always zero in value
    #x0 = 0.05 #initial condition
    pdf=np.zeros([n,lval.size])
    fps=np.zeros([n,lval.size])
    lines=[’-’,’--’,’-.’]*3
    linw=[2,2,2,3,3,3,4,4,4,5,5,5]

    ############# PRINT ESTIMATIONS TO SCREEN ######################################
    print time.clock(), "Print parameter estimates..."

    #we print the values found for the bifurcation
    #along with its corresponding propagated uncertainties
    print "l_i \in (",lmin,", ",lmax,")"
    print "l_i =",midl,"+-",uncert
    #print "b_i  \in (", -p[1][1]-np.exp(-lmin),",",-p[1][1]-np.exp(-lmax),")"
    #print "b_i =",-p[1][1]-np.exp(-midl),"+-",midl*np.exp(-midl)*uncert

    ############# BEGIN PLOT #######################################################
    print time.clock(), "Beginning Fixed Points plot..."

    ##draw the graph of the n fixed points varying with the parameter
    plt.figure(1)
    plt.subplot(313)
    plt.grid(True)
    plt.xlabel(’Parameter’)
    plt.ylabel(’FPs’)
    plt.xlim(lmin,lmax)
    #plt.title(’Fixed Points’)
    #plt.ylim(-2,2)
    for k in range(n):
    fps[k]=[xl(k,p,lam) for lam in lval]
    plt.plot(lval,fps[k,:],’k’,label=’FP’+str(k), ls=lines[k],linewidth=linw[k])

    plt.legend()

    print time.clock(), "Beginning PDFs plot..."

    #draw the product distance functions of each of the n FP’s
    plt.subplot(312)
    plt.grid(True)
    #plt.xlabel(’Parameter’)
    #plt.ylabel(’PDFs’)6
    #plt.ylabel(’Stability Conditions’)
    plt.ylabel(’PDFs’)
    #plt.title(’PDFs of the Fixed Points’)
    plt.xlim(lmin,lmax)
    plt.ylim(-4,1)
    #print bifurcation values boundaries
    plt.plot(lval,np.zeros_like(lval),’k--’)
    plt.plot(lval,-2*np.ones_like(lval),’k--’)
    if(n==2):
    plt.plot(lval,-2.449*np.ones_like(lval),’k--’)
    plt.plot(lval,-2.544*np.ones_like(lval),’k--’)
    plt.plot(lval,-2.5642*np.ones_like(lval),’k--’)
    #plt.plot(lval,-2.56871*np.ones_like(lval),’k--’)
    #plt.plot(lval,-2.56966*np.ones_like(lval),’k--’)
    #plt.plot(lval,-2.569881*np.ones_like(lval),’k--’)
    plt.plot(lval,-2.57*np.ones_like(lval),’k--’)

    if(n==3):
    plt.plot(lval,-3*np.ones_like(lval),’k--’)
    plt.plot(lval,-3.236*np.ones_like(lval),’k--’)
    plt.plot(lval,-3.288*np.ones_like(lval),’k--’)
    plt.plot(lval,-3.29925*np.ones_like(lval),’k--’)

    for j in range(lval.size):
    pdf[:,j]=pdfn(n,fps[:,j])

    for k in range(n):
    plt.plot(lval,pdf[k,:],’k’,label=’PDF’+str(k),ls=lines[k],linewidth=linw[k])
    #plt.plot(lval,pdf[k,:],’k’,label=’SC’+str(k),ls=lines[k],linewidth=linw[k])

    plt.legend()

    print time.clock(), "Begin bifurcation diagram plotting..."

    #draw the bifurcation diagram
    plt.subplot(311)
    plt.grid(True)
    #plt.xlabel(’Parameter’)
    plt.ylabel(’Asymptotic Value’)
    plt.title(’Bifurcation Diagram’)
    plt.xlim(lmin,lmax)
    #plt.ylim(-2, 2)
    m=0

    for lamb in lval: #draws the bifurcation diagram
    #    for j in range(n-1): #calculate the fixed points
    #        xi[j+1]= xl(j+1,p,lamb)
    #
    #    aux=pdfn(n,xi)  #calculate PDF of the FP’s
    #    for k in range(n):
    #        pdf[k][m]=aux[k] #store the values for graphing
    #plt.subplot(311)
    x = gn(n, fps[:,m], x0=0.9, niter=nit, nplot=npl)
    plt.plot(lamb*np.ones(npl),x,’k,’)
    x = gn(n, fps[:,m], x0=-0.01,niter=nit, nplot=npl)
    plt.plot(lamb*np.ones(npl),x,’k,’)
    m+=1

    print time.clock(), "Done."

    plt.show()

###### List of Tables

  1. 2.1 Table of regions of type $k$, $k=1,2,...,7$, for the logistic map. The logistic map is a regular map (see definition 2.5). Data reproduced from [Elaydi(2000), Table 1.1, p.41]
  2. 2.2 Regions of type $k$ for the quadratic map $x_{n+1}=x_{n}^{2}-b(\lambda)x_{n}$.
  3. 3.1 Bifurcation values for the Canonical Quadratic Map. $b_{0}=0$ is included only as a reference, although it is not a bifurcation value.
  4. 3.2 Periodic points and corresponding stability conditions for the canonical quadratic map.
  5. 4.1 Bifurcation values for the Canonical Cubic Map. $c_{0}=0$ is included only as a reference, although it is not a bifurcation value.

###### List of Figures

  1. 1.1 Graphs of $E_{\lambda}(x)=\lambda e^{x}$ where (a) $\lambda>1/e$, (b) $\lambda=1/e$, and (c) $0<\lambda<1/e$. Reproduced from [Devaney(1989)].
  2. 1.2 Phase portraits of $E_{\lambda}(x)=\lambda e^{x}$ for (a) $\lambda>1/e$, (b) $\lambda=1/e$, and (c) $0<\lambda<1/e$. Reproduced from [Devaney(1989)].
  3. 1.3 Bifurcation diagram of $E_{\lambda}(x)=\lambda e^{x}$. $x$ is plotted against $\lambda$. Reproduced from [Devaney(1989)].
  4. 2.1 The graphs of the one-parameter family of maps $E_{\lambda}(x)=\lambda e^{x}$ for (a) $-e<\lambda<0$, (b) $\lambda=-e$, and (c) $\lambda<-e$. Reproduced from [Devaney(1989)].
  5. 2.2 The graphs of $E^{2}_{\lambda}(x)$ for (a) $-e<\lambda<0$, (b) $\lambda=-e$, and (c) $\lambda<-e$. Reproduced from [Devaney(1989)].
  6. 2.3 The bifurcation diagram of $E_{\lambda}(x)=\lambda e^{x}$. Notice that the “vertical” curve is the curve of period 2 attracting points. Reproduced from [Devaney(1989)].
  7. 2.4 Feigenbaum’s sequence for $f(x)=1-2x^{2}$. $N$ denotes the cycle of order $2^{N}$. The parameter is denoted as $\lambda$. $\delta_{N}=(\lambda_{N+1}-\lambda_{N})/(\lambda_{N+2}-\lambda_{N+1})$ and $\delta_{N}^{\prime}=(\delta_{N+1}-\delta_{N})/(\delta_{N+2}-\delta_{N+1})$. Reproduced from [Feigenbaum(1978)]
  8. 2.5 $b(\lambda)$ functions of the quadratic regular reversal maps of example 2.3. The horizontal lines correspond to the values of $b_{k}$, for $k=1,2$ and $\infty$.
  9. 2.6 Bifurcation diagrams of the quadratic regular reversal maps of example 2.3.
  10. 2.7 Functions $b(\lambda)$ of the regular reversal maps of example 2.4 given by the equation 2.4. The horizontal lines correspond to the values of $b_{k}$, for $k=1,2$ and $\infty$.
  11. 2.8 Bifurcation the regular reversal maps of example 2.4.
  12. 2.9 Graphs of $b(\lambda)$, given by equation 2.8, for the regular reverse maps of example 2.8.
  13. 2.10 Bifurcation diagrams of the regular reversal maps of example 2.5.
  14. 2.11 Graphs of $b(\lambda)$, given by equation 2.9, for the quadratic regular reversal maps of example 2.6.
  15. 2.12 Bifurcation diagrams of the regular reversal maps of example 2.6.
  16. 3.1 Linear parametrization of fixed points (example 3.5.1).
  17. 3.2 Bifurcation diagram for the linear parametrization of the fixed points example.
  18. 3.3 Quadratic parametrization of fixed points with $p_{1}=1$ and $p_{2}=3$.
  19. 3.4 Bifurcation diagram for the quadratic parametrization of the fixed points with $p_{1}=1$ and $p_{2}=3$..
  20. 3.5 Quadratic parametrization of the fixed points with $p_{1}=1.1422$ and $p_{2}=3$.
  21. 3.6 Graph of corresponding $x_{1}$ for linear $y_{0}(\lambda)$ and quadratic $y_{1}(\lambda)$ with $p_{1}=1.2$ and $p_{2}=3$. The maximum value of $x_{1}(\lambda)$ is set to be slightly above $b_{\infty}$. Also, the bifurcation values $b_{1}$, $b_{2}$ and $b_{\infty}$ are shown.
  22. 3.7 Bifurcation diagram for the quadratic parametrization of the fixed points with $p_{1}=1.2$ and $p_{2}=3$. We have set here $x_{1}^{max}>b_{\infty}$.
  23. 3.8 Sinusoidal parametrization of fixed points.
  24. 3.9 Bifurcation diagram for the sinusoidal parametrization of the fixed points example.
  25. 3.10 Linear parametrization of fixed points crossing through all stability bands.
  26. 3.11 Bifurcation diagram for the linear parametrization of the fixed points example that crosses through all stability bands.
  27. 3.12 Bifurcation diagram (upper) and fixed points plot with stability bands for the exponential parametrization of the fixed point example (lower). In the lower panel, we have set $x_{1}\rightarrow-b_{\infty}$ (continuous thick line), the zero fixed point is marked by a dash-dotted line and the bifurcation values $-b_{1},\,-b_{2},...,\,-b_{5}$ are also shown as dashed lines.
  28. 4.1 Bifurcation diagram (upper), stability conditions plot (middle) and fixed points plot (lower) for the linear parametrization of the fixed points example of the CCM.
  29. 4.2 Bifurcation diagram (upper), stability conditions plot (middle) and fixed points plot (lower) for the square root parametrizations of the fixed points example of the CCM.
  30. 4.3 Bifurcation diagram (upper), stability conditions plot (middle) and fixed points plot (lower) for the constant and linear mixed parametrizations of the fixed points example of the CCM.
  31. 4.4 Bifurcation diagram (upper), stability conditions plot (middle) and fixed points plot (lower) for the exponential asymptotic parametrizations of the fixed points example 4.4 for the CCM.
  32. 5.1 Bifurcation diagram (upper), Product Distance Functions (middle) and fixed points plot of the quartic polynomial map example.
  33. 6.1 Bifurcation diagram for the canonical logistic map.