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Life as a dynamical system: An eigenvalue analysis! Featured

Post 01 March 2014 by  Montasser Ghachem
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I have been reading lately about the stability of dynamic systems and I found the results at first very strange. It says that an equilibrium of a system is stable if and only if all of its corresponding eigenvalues have negative real parts. The word eigenvalue etymologically comes from the German 'Eigenwert' and means intrinsic value. I was wondering, how could a person be stable if all its intrinsic values are negative? I always thought that stability was related to positive appreciation about one's own worth i.e. own eigenvalue! After a while, I discovered that I was shortsighted and that the mathematical statement were very deep for my superficial understanding.

If someone reaches a state where he thinks that his intrinsic value is negative in all states of the world, then he enters in a state of rejection of life that it is impossible to escape. To live requires loving oneself as a living being i.e. a positive appreciation of oneself. Even the most depressed of us, occasionally thinks positively of himself/herself. If a person reaches a situation where he thinks he has a negative worth in all states of the world, then he loses hope of life and converges towards death. In fact, we can prove that death is asymptotically stable i.e. if we are close enough to death then we will certainly die. This a special case of the more general theorem that states that ‘in the long run we are all dead’! We proceed to prove that except death, all other states are unstable and therefore transitory.

Theorem: Death is asymptotically stable.

As said above, life requires hope i.e. a positive evaluation of oneself and of life and is therefore unstable. Even when we hate, we appreciate ourselves as haters. Life is never stable because being alive requires a certain feeling a self-worth i.e. a positive intrinsic value, a positive eigenvalue. This appreciation of ourselves as meaningful creatures in this world gives us energy to move forward in life and thus we leave this state. Given any positive eigenvalue in any state of the world, this state of the world will keep us moving. If I get rejected by twenty girls, I like and then I go to a bar and there a gorgeous girl smiles to me. There, I will continue going to that bar and wandering the streets of Cambridge because I know somewhere some time, I have a positive self-worth, I was appreciated. I will cling to life through this memory. The movement, the involvement in life following this positive intrinsic value proves that our state is not stable.

Theorem: As long as one thinks positively of oneself, his state is unstable.

What about states where the real part of all eigenvalues are zeroes? This is rather a tricky case, in the literature on dynamical systems, they are called centres. In real life, the centre are the states of coma or deep depressive state where a person becomes indifferent to life. Nothing matters anymore. One doesn’t reject life but doesn’t embrace it either. One could be a centre if one sees the absurdity of life as Mersault in the Stranger of Camus did when said “I had lived my life one way and I could just as well have lived it another. I had done this and I hadn’t done that… Nothing, nothing mattered”. These centres, following a certain perturbations, could lead to stable outcomes or unstable outcomes. Following a coma, one could die to go back to life. Following a deep depression, one could sink in the bottomless pit of depression or embrace life again despite its pains. The case of Mersault in the stranger is a mathematical puzzle. Mersault, after being a centre in all the novel, embraced life just to find death directly after it.

Theorem: Coma and deep state of depression are very sensitive to perturbations.

We have been very careful not to speak about the imaginary parts of the eigenvalues and we are very aware that one can keep on living based on purely imaginary sense of self-worth even if in reality the real part of it is zero or even negative. We leave this for future research.

The main takeout of our essay is that instability is inherent to life, one should not panic when things become unstable because as long as one is alive, the trajectory moves and no state except death is stable. Those of us who seek stability rely more on the imaginary part than the real part. Life is fundamentally change and no state in it is stable. I hope this note helps people to understand the experience of instability and never feel bad about it. Life never promised stability and it is sticking to its promise. Follow the trajectory of life, disregard stability.If we ever feel stable, this is due to the fact that our state is a rest point where we rest for a while. However, the winds of life make us move again proving that it was not stable. Go out now, enjoy life and try new things, talk to new people… Only death can trap you, otherwise, you are free as a bird. Completely unstable, completely free!

Mathematically Monty.

Montasser Ghachem

Montasser Ghachem

I am an Econ PhD student in Stockholm University. My interests are Poetry, Philosophy, Sociology, Economics and Game Theory. I am currently visiting Harvard University in USA. I share with you here my poems, my essays and my small discoveries in life. From Tunisia to USA, going through Germany and passing by Sweden; flows my life... A tale of metamorphosis!

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