### Dynamical systems: differential equations, maps, and chaotic behaviour

Consider stationary points of the dynamical systems 1,2,3. For our derivation, we assume that the system is chaotic or quasiperiodic if all these stationary points are unstable in at least one direction, i. We assume that for sufficiently high d , all Jacobian eigenvalues are statistically independent. This assumption of weakening correlations between dimensions as the number of dimensions increase is a rather strong approximation without which it seems impossible to derive analytical estimates, and which seems to result in reasonable results see below.

Denoting the probability that the real part of an eigenvalue is negative by P neg , the probability that at least one out of d eigenvalues of the Jacobian at a stationary point has a positive real part is.

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Hence the probability of chaos is. Specifically, consider the example of system 2 with cubic non-linearities. As above, we ignored low-order terms present in 5 , because such terms are irrelevant for large d. We assume that the distribution of is the same as for the coordinates x i themselves and is given by the universal invariant measure shown in Fig. We also consider the two terms and as statistically independent. Thus, the eigenvalues of are uniformly distributed on a disk with radius.

The probability distribution of the second, diagonal, term of the Jacobian, is defined by the invariant measure P y , given by 10 and shown in Fig.

Dynamical Systems And Chaos: Differential Equations Summary Part 1

It follows from scaling 8 that both and contribute terms of order d 3 to the eigenvalues of the Jacobian. The contribution from is always negative and has magnitude 5 y 4 with probability P y. It follows that the probability that the sum of the two contributions has negative real part is. Integration on dr produces. Using the numerical data for P y shown in Fig.

A similar analysis for Eqs 1 14 and 3 yields and , respectively. Substituting these values into Eq.

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Interestingly, a similar threshold was observed for the transition to ubiquity of chaos in a networks with tanh x non-linearity 11 , 12 , We note that the quadratic and cubic non-linearities considered here can be interpreted as the first few non-linear terms in the expansion of more complex non-linear dynamical systems, possibly extending the applicability of our results. We have also provided analytical explanations for the observed ubiquity of chaos and for the universality of the density distribution of chaotic trajectories.

The similarity of the three panels in Fig. Our observations may also provide important insights into chaotic behavior of continuous systems described by partial differential equations, which are often digitized as systems of many ordinary differential equations. Furthermore, our results are directly applicable and important not only to pure nonlinear physics, but also to other fields where non-linear interactions are common, such as hydrodynamics and plasma physics, optics, systems biology, evolution, and control theory.

One of the goals of this work was to illustrate the transition to chaos and ergodicity in high-dimensional phase space, a frequently used yet rarely precisely stated argument in the formal justification of statistical mechanics.

## Chaos and dynamical systems

Nevertheless, to explain our results we use scaling and probabilistic arguments borrowed from statistical physics. How to cite this article : Ispolatov, I. Chaos in high-dimensional dissipative dynamical systems. Author Contributions M. National Center for Biotechnology Information , U. Sci Rep. Published online Jul Author information Article notes Copyright and License information Disclaimer.

Received Feb 20; Accepted Jun This work is licensed under a Creative Commons Attribution 4. This article has been cited by other articles in PMC. Abstract For dissipative dynamical systems described by a system of ordinary differential equations, we address the question of how the probability of chaotic dynamics increases with the dimensionality of the phase space. Open in a separate window. Figure 1.

• Chaos in high-dimensional dissipative dynamical systems.
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Numerically measured probability of different types of dynamics as a function of dimension d of the phase space for Eq. Figure 2. Scaling of the size of chaotic trajectories. Figure 3. Additional Information How to cite this article : Ispolatov, I. Acknowledgments I. Footnotes Author Contributions M. References Kaneko K.