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I have come across a coupled nonlinear dynamical system given below $$ r\, \ddot{x} + \dot{x} = \sin y~,$$ $$ r\, \ddot{y} + \dot{y} = \sin x~,$$ where $r$ is some real number and $\dot{x}$ denotes $\frac{d\, x(t)}{d\,t}$. Despite the fact that the system is simple looking, numerical study (by finding the large time Lyapunov exponents) shows that the system exhibits periodic (limit cycles) and chaotic behaviors for the parameter $r = \mathcal{O}(1)$ at large time limit. Depending on the initial conditions $(x(0),y(0),\dot{x}(0),\dot{y}(0))$, for a particular value of the parameter $r \sim 1$, the system exhibits these interesting dynamical properties. However, theoretically, I am not able to explain why this happens. One resembling system I have seen is the Standard map. However, that too does not explain why my system shows these strange behaviours. Any insight will be helpful.

Qmechanic
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anu
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1 Answers1

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Chaos is very common — regular behavior is actually the special case.

Your system is nonlinear and has more than 3 dimensions, that is in principle enough for chaos to be possible or even likely — see, e.g., 1, 2 or the books suggested here, but, in a nutshell, 3 dimensions give you enough "room" for the continuous trajectories to be complicated, and nonlinearity is needed for the stretch and fold mechanism prototypical of chaos.

stafusa
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