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Applications of dynamical systems in climate sciences.

Sakshi Jain

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Climate Crisis is real!

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Why should we address climate crisis?

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Because

extreme climate is NOT FINE!!

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How do we address the crisis?

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Be climate conscious!

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as researchers, let’s also study the possible (optimal) methods to address it via

dynamical systems

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Dynamical systems

  • the study of the long-term behavior of the systems that change in time and the methods to control that behavior.

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Stochastic Differential Equations

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An SDE is a differential equation in which one or more terms are stochastic processes.

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Stochastic process

?????

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Stochastic Differential Equations

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An SDE is a differential equation in which one or more terms are noise/disturbance (brownian motion etc.).

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Why are SDEs relevant for studying Climate?

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Why to use SDE for Climate modelling?

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The climate system is highly sensitive to small changes in initial conditions, especially for long-term predictions. SDEs account for small, random variations like small temperature fluctuations, making the long-term predictions more realistic by reflecting the inherent uncertainty in such sensitive systems.

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SDE:dXt= b(Xt) dt + dWt

with X0 = x (initial conditions)

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SDE:dXt= b(Xt) dt + dWt

X0 = x (initial conditions)

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if the SDE satisfies certain nice properties (not difficult to achieve) then at each point there exists a nice continuous density which also satisfies some nice bounds:

kt(x,y)

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Goal: To perturb the SDE (change the initial system in some way) and to find the best perturbation giving optimal output after a certain time.

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Tool:

Transfer Operator

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1. Can you predict where individual ink particles will be after one minute?

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Tool:

Transfer Operator

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1. Can you predict where individual ink particles will be after one minute? NO: the motion of ink particles is chaotic.

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Tool:

Transfer Operator

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1. Can you predict where individual ink particles will be after one minute? NO: the motion of ink particles is chaotic.

2. Can you predict the density profile of ink after one minute?

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Tool:

Transfer Operator

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1. Can you predict where individual ink particles will be after one minute? NO: the motion of ink particles is chaotic.

2. Can you predict the density profile of ink after one minute? YES: it will be nearly uniform.

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Tool:

Transfer Operator

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1. Can you predict where individual ink particles will be after one minute? NO: the motion of ink particles is chaotic.

2. Can you predict the density profile of ink after one minute? YES: it will be nearly uniform.

The transfer operator: The action of a dynamical system on mass densities of initial conditions.

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SDE:

dXt = b(Xt) dt + dWt

X0 = x (initial conditions)

initial density kt(x,y)

Associated transfer operator:

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Tt: L1(Rd) L1(Rd)

Ttf(y)= ∫kt(x,y)f(x)dx.

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SDE:

dXt = b(Xt) dt + dWt

X0 = x (initial conditions)

initial density kt(x,y)

Associated transfer operator:

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Tt: L1(Rd) L1(Rd)

Ttf(y)= ∫kt(x,y)f(x)dx.

f:temperature (for example)

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SDE:

dXt = b(Xt) dt + dWt

X0 = x (initial conditions)

initial density kt(x,y)

Associated transfer operator:

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Tt: L1(Rd) L1(Rd)

Ttf(y)= ∫kt(x,y)f(x)dx=f(y).

Stationary measure: fixed point of Tt

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SDE:

dXt = b(Xt) dt + dWt

X0 = x (initial conditions)

perturbed density ktα (x,y)

Linear response: (study the

how does the stationary measure change with respect to the

change in the system (density).

Tt: L1(Rd) L1(Rd)

Ttfα(y)= ∫ktα (x,y)fα(x)dx = fα(y).

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Optimal Linear response

Study the optimality problem of how to modify the system to obtain desired future behaviour.

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Thank you!