# A coupled nonlinear dynamical system in four dimensional phase space

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.

• Title given must be specific and clear. Dec 23, 2021 at 6:55
• Hi anu. Welcome to Phys.SE. Where did you come across this system? Which reference? Which page? Have you found a Lagrangian or a constant of motion for the system? Dec 23, 2021 at 12:39
• @Qmechanic I have come across this in the context of particle motion in 2d viscous flows. r is the measure of particle inertia. The system is dissipative; I am not able to obtain a Hamiltonian or Lagrangian for the system.
– anu
Dec 27, 2021 at 14:40

## 1 Answer

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.

• Thank you for your valuable suggestions. However, I am looking for particularly some well-known dynamical systems that have similar properties. Especially some nonlinearity in terms of 'sine' function and inter-coupling between the parameters. In light of such systems (if it exists), probably I could better understand the dynamics of my system.
– anu
Dec 27, 2021 at 14:47