\documentclass[twoside]{book}
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\shtitle{Ant-resonance in complex systems}
\title{Anti-resonance\\
in complex systems}
\author{David B. Saakian\affil{Yerevan Physics Institute,\\
Institute of Physics,Academica Sinica\\
saakian@mail.yerphi.am}}
\date{30 June 2004}
\abstract{The notion of complex resonance is defined as a
situation, when global parameters of the system characterizing the
total state coincide for different hierarchic levels. Among those
global parameters are frequencies in classical mechanics systems,
temperatures (Nishimori like) in disordered systems and wave
function's phase. The more involved case could be connected with
the functions like replica symmetry breaking schemes. The
anti-resonance is defined as a situation, when high level
(hierarchic) parameter is chosen to suppress the motion in a
maximal away. We applied the last concept to the problems of the
modern history.}
\begin{document}
\maketitle
\section{Complex resonance}
\sectlabel{co1}
The concept of resonance is probably the
most noticable phenomenon in nature, culture and science. The
close notion of synchronization in complex system is becoming more
and more popular [1]. We are going to analyze the idea of
resonance in complex systems, to look for a possibility of, in
some sense, inverse situation with an exponential damping of
motion (anti-resonance). We suppose that this notion will
compliment our view to complex systems in previous section.
Originally, the simplest resonance situation has been investigated
in mechanics of classical deterministic system with some resonance
frequency, driven by external harmonic force. When two frequencies
coincide, the reaction of the system to external force increases
drastically. Even in this simple case we can observe two features
of phenomenon. Frequency is an essence of motion, and there is a
sharp peak in the ratio output-force.
The next step was parametric resonance in classical mechanics.
There is a hierarchy here. We observe a motion at given values of
parameters, and the resonance frequency depends on the value of
external parameters. If one changes the external parameter with
the same frequency, as the frequency of the pendulum, there
appears famous parametric resonance-the flow of energy from the
higher level of hierarchy to the lower level one. Let us
generalize this situation to other complex systems to define
complex resonance.
If there is a hierarchy in the system, and states at different
hierarchic levels have some essence (comparable logically with
each other), the generalized resonance happens, when
these essences coincide.
What about the essence of the state? In classical mechanics, there
is only one real number characterizing total state, i.e.
frequency. In general one should look for
other total parameters of the system. In modern physics
these are the following: temperature in statistical mechanics, replica system
breaking scheme in spin-glasses (edge of chaos parameter), and the wave function phase in
quantum mechanics.
The next famous example of such (generalized parametric resonance)
situation is related to the Nishimori line in disordered systems
[2,3]. A hierarchy (quenched disorder) is present here. Sometimes
it is possible to introduce some formal temperature to describe
this disorder. If two temperatures (real
one for the spins and the formal one for the quenched disorder)
coincide, the system reveals some interesting properties becoming
maximally analytic in some sense.
So we can define a hierarchy for the resonance. In the trivial
case, the system is not hierarchic, it is logically homogeneous.
The more involved case corresponds to the situation with
principally different kinds of motions or (and) hierarchy. It is
reasonable to define the second case as a complex resonance. In
several situations (i.e. stochastic resonance), when it is
impossible to define and compare clearly the essence of a state,
one considers a situation, when there is a sharp peak in the ratio
output-input at optimal value of external parameter.
An important moment should be mentioned regarding our concept. If
we consider some functional having different parameters,
functions, logical structures, and we optimize it over the entire
variables (besides some fixed group of parameters or functions)
it could be stated that the essence of the whole system is the
same, as that one of a fixed group.
\section{Anti-resonance}
Let me now analyze the resonance situation with the opposite goal:
to use the high levels of hierarchy to achieve a maximal negative
effect. This is a situation not too rare in living systems.
We define anti-resonance as a situation, when:\\
1) a resonance is possible for some value of external parameter;\\
2) it is possible to define the opposite phase transformation of the parameter;\\
3) at the opposite phase values of the parameter there is either\\
a) an exponential damping of a motion, or\\
b) a new feature (opposite in some sense to those at the
resonant parameter case) arose in a resonant way.
The phenomenon is very complex. Thus, we are investigating the
simplest models, trying to reveal those situations in complex
systems, when such phenomenon is possible. Let us consider the
pendulum with $x(0)=x_0,x'(0)=0$, when the frequency varies with
some small amplitude h [4]:
\begin{eqnarray}
\label{e49} \frac{d^2 x}{d t^2}=-w^2(t)(1+h\cos(2wt+\phi))x.
\end{eqnarray}
Taking an ansatz $x(t)=a(t)\cos(wt)+b(t)\sin(wt)$ we derive an
equation
$$
2a'+a\frac{hw}{2}\sin(\phi)+b\frac{hw}{2}\\
2b'-b\frac{hw}{2}\sin(\phi)+a\frac{hw}{2}. $$ For the unstable
solutions $a,b\sim \exp(st)$ one has an equation $ s=\pm
\frac{hw}{4}$.
For the amplified solution there is an expression
\begin{eqnarray}
\label{e50}
x(t)=\exp(\frac{hw}{4}t)[\cos(wt)+\tan(\phi/2)\sin(wt)],
\end{eqnarray}
and for the damped solution
\begin{eqnarray}
\label{e51}
x(t)=\exp(-\frac{hw}{4}t)[-\tan(\phi/2)\cos(wt)+\sin(wt)].
\end{eqnarray}
Thus it is possible to arrange a damping, choosing the phase
$\frac{x(0)}{wx'(0)}=-\tan(\phi/2)$ (for the amplification one
should choose the opposite phase). For the original amplitude A
the damping period $T$ is
\begin{eqnarray}
\label{e52} T\sim \frac{4\ln A}{hw}.
\end{eqnarray}
In this situation the picture is symmetric (both amplification and
damping are possible). The other situation is possible with only
resonant damping (like domino effect). For the case of parametric
resonance it is possible an exponential damping.
\section{ Nishimori line} One considers [2,3] $N$ spins $s_i$ with
interaction Hamiltonian
\begin{eqnarray}
\label{e53} H=-\sum_{I_1..i_p}j_{i_1..i_p}s_{i_1}s_{i_2}..s_{i_p}.
\end{eqnarray}
There is a p-spin interaction here, couplings $j_{i_1..i_p}$ are
random quenched variables $\pm 1$ with probability
$\frac{1+m_0}{2}$ for the values 1 and $\frac{1-m_0}{2}$ for the
values $-1$. It is possible to write the following probability
distribution:
\begin{eqnarray}
\label{e54} P(j_{i_1\dots i_p
})=\frac{\exp(\beta_0\tau_{i_1..i_p})}{2\cosh(\beta_0)}.
\end{eqnarray}
The parameter $\beta_0$ resembles an inverse temperature.
Using the invariance of the
Hamiltonian under transformation
\begin{eqnarray}
\label{e55} s_i\to s_iv_i,\qquad j_{i_1..i_p}\to
j_{i_1..i_p}v_{i_1}v_{i_2}..v_{i_p},
\end{eqnarray}
in [1,2] has been calculated exact energy of the model at
$\beta_0=\beta$. At $\beta=\beta_0$ our system has the best
ferromagnetic properties in a sense that the number of up spins
$\sum_i\frac{}{||}$ is maximal at Nishimori temperature
[3]. In opposite phase, we can take $\beta=-\beta_0$. While the
order parameter are different in ferromagnetic and
antiferromagnetic phases, free energy is the same in both models,
as $Z(j,\beta)=Z(j,-\beta)$ for the Hamiltonian (\ref{e53}).
For the odd values of $p$
one has an optimal properties for the configuration $s_i=-1$.
Thus, there is a trivial anti-resonance according to our
definition. For the even values of $p$ (i.e. $p=2$) and bonds on
the links of hypercube lattices in d -dimensional space, there is
an antiferromagnetic ordering (an anti-resonance situation).
\section{ Anti-resonance in complex systems} A search of
anti-resonance in stochastic resonance [5] is a very interesting
issue. The resonance is certainly a complex one, when the
deterministic harmonic motion has the same period as the
transition by noise. To construct the anti-resonance is
problematic, as stochastic resonance has not a phase to reverse
the resonance situation. In [6] the stochastic resonance
explanation for the crashes and bubbles in financial markets
(using the Ising spin model) has been considered. There is no
phase for the noise to be reversed in stochastic resonance, but
the information for the agents can certainly be positive or
negative, thus moving the market from the border of two phases to
one side.
During the last decade, the idea of evolution or development at
the edge of chaos [7-8], related to complex adaptive systems, was
very popular. What about anti-resonance aspect of the origin of
life? It is the case, at least, for the hyper cycle model by Eigen
and Schuster [9]. One tries to construct self-replicating system
overpassing simple problem,i.e. mutations, but, unavoidably,
parasite creatures appear. As a result, there is a chance to
consume all the information via those parasite creatures. We see,
in some sense, a resonance picture with a chance for
anti-resonance. The virus evolution is also often near the error
threshold (mutation catastrophe)[10]. At the top level of life
there is a phenomenon of apostasis, when the cell could be killed
by a simple command.
Our point of view is the following: even if complex systems are
walking at the edge of chaos and climbing the mountains of fitness
landscapes (in case of biological evolution), it is often a walk
near the precipice. For the evolution it is not so dangerous, as
only the survival of the species is crucial. One should be much
more careful with a rare or single systems, like humanity.
\section{Anti-resonance in history: the survival of humanity.}
Nowadays there are several mathematical methods to model politics,
see \cite{di03}. We will consider the problem of survival of
humanity using only heuristic arguments connected to the
phenomenon of anti-resonance.
As it is known from the experience of the history of mankind
in tense situations, when a
category connected with some symbol becomes urgent (paramount) the
sacrifice
of its
symbol unavoidably leads to destruction of the category.
A typical example is the 1938 Munich
agreement.
In 1938 England and France with the mute support of the US
sacrificed Czechoslovakia (the richest and democratic country).
This led to the fascist yoke almost all over Europe. Three years
later Pearl Harbor was attacked.
How one can understand this phenomena under the anti-resonance
point of view? From coding theory, especially as it is connected
with neuron-network and spin-glasses, it is known that any signal
can be encoded by a function having that signal as a extremum
(ground) state [11]. Therefore, Czechoslovakia can be considered
as an encoded democracy for that period of history. In the
considered case the protection and existence of democracy under
the fascist danger was most urgent during this period of history.
So there was one category, or one dimensionality in the
terminology of physics. Thus the action of three leaders of West
democracy to sacrifice Czechoslovakia was an accurate resonance
to their own interests,
but with opposite phase.
We have seen in the previous sections, that a resonant response
could be sometimes exponentially strong. Thus we can understand
why the reaction (consequences of Germany's attack ) was so
strong.
Let us consider another anti-resonance situation. Why do so many
disasters occur, placing into question the existence of mankind?
Because a symbol of eternity of humanity has been sacrificed.
In 1915 as with Armenian genocide (recognized by France, Italy,
Vatican)
Ottoman Turkey killed several hundred thousands Assyrians [12]. It was the end of
Assyrian civilization .
The Assyrian genocide (as well as Asian Greeks genocide) has not
even been debated.
I insist that this is extremely dangerous anti-resonance
situation, as the category connected with the Assyrians (eternity)
is the most crucial one nowadays: the survival of humanity.
The most ancient human civilizations come from the region of Mesopotamia. In addition,
these civilizations have had a strong influence on the Indo-European and Egyptian civilizations.
Through history, the Mesopotamian civilizations disappeared passing their culture to
successor civilizations. This continued for 6000 years without interruption. However, this
chain was broken in 1915, when by the massacre of Assyrians in the Ottoman Empire
has been destroyed the last Mesopotamian civilization.
Is there an exaggeration of the part of the Assyrians in today’s
disasters, when there are wealthier and numerous nations? It is
appropriate to note the pointed remark of Toynbee that the
appearance of a new God in a remote and poor province (Palestine)
of Rome would have been considered a bad joke by the ancient
Romans.
Humanity should respect the symbol of eternity. Assyrians genocide
should be recognized by UN (as well as the connected ones:
Armenians and Greeks of Asia), and should be done everything to
restore the Assyrians civilization. The last chance to rescue the
situation is, perhaps, the 100 year anniversary of 1915 year
Genocide.
This will give humanity a chance
to survive in this situation when one can hear the sinister
ticking of the clock, which measures the timeline of humanity’s
existence.
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\end{document}