# Is it possible to trace back a chaotic system, to its initial conditions from some given interval?

I am currently studying a maths module, as part of my second year on my physics degree, involving solutions to differential equations amongst other mathematical concepts. I have recently been (briefly) introduced to the Lorenz Equations and the idea of chaotic systems.

My question is not specifically about the Lorenz Equations, but more generally about chaotic systems.

I understand that a defining characteristic of a chaotic system is the fact that the system is extremely sensitive to its initial conditions, described simply by the notion of 'the butterfly effect'.

My Question

If I were to consider some system, such as a double pendulum, if I observed the trajectory of the pendulum over some time interval, would it in any way be possible to trace it back to its origin (i.e. its initial conditions)?

I'm not sure if I am asking a really stupid question, or if perhaps there is some theoretical/computational way to do this, or to at least estimate the initial conditions.

Thanks in advance :)

• First of all, in systems that have time-reversal symmetry, asking a backward-in time question like this is not different to asking the question with reversed (normal) time. Observing the pendulum will not help you, since you cannot determine all its "conditions" exactly, Being a chaotic system any prediction of the origin will be very sensitive to your "Observation" Commented Jan 17, 2018 at 2:21

In general: No.

One of the defining characteristics of chaotic systems is that nearby trajectories in phase space diverge from each other exponentially, i.e., the distance $\delta \mathbf{x}$ between them grows with time $t$ as

$$|\delta \mathbf{x}| = |\delta \mathbf{x}_0| e^{\lambda t},$$

where $\lambda$ is the system's largest Lyapunov exponent. This behavior implies that any improvement in the time length over which a meaningful prediction can be made requires an exponentially larger improvement in the determination of the system initial state.

For an $n$-dimensional system there are $n$ Lyapunov exponents and the Lorenz system is typical for a 3-D chaotic system in that it has a positive, a null, and a negative exponents ($\lambda_+$, $\lambda_0$, $\lambda_-$). The positive exponent, $\lambda_+$, characterizes the system's chaoticity, $\lambda_0=0$ is a consequence of smoothness, and $\lambda_-$ measures the contraction rate that prevents the system from diverging to infinity. Put in another way: the chaotic set that determines the evolution of the system has at almost every point both an unstable and a stable directions (related to the system's unstable and stable manifolds).

While the relationship between the dynamics of a system and that of its inverse is not trivial (arxiv), the reverse system (which you need for tracing it back to its initial state) will also have an unstable direction (a positive Lyapunov exponent), i.e., be chaotic and, therefore, subject to the same limitations to predictability as the original system.

I will refer specifically to continuous-time systems.

Chaos can occur in either dissipative systems, where the sum of all the Lyapunov exponents (LEs) is negative, or in conservative systems, where the sum of all the LEs is zero. In either of these cases, the system having a positive LE implies it also has a negative LE.

For continuous-time systems, reversing the flow of time (replacing $$t\to-t$$) will reverse the sign of all the Lyapunov exponents (see, e.g., here and here). Thus the negative exponents will become positive, and the time-reversed system will also have an unstable direction. Note that if the original system was dissipative and possessed an attractor, the time-reversed system will not be dissipative and thus not have an attractor.