The Ikeda map originates in nonlinear optics. In 1979, Kensuke Ikeda showed that light circulating inside a ring cavity containing a nonlinear (Kerr) medium — a device closely related to a nonlinear Fabry-Pérot resonator — develops a phase shift that depends on the light's own intensity. Because the cavity feeds its output back in as input on the next round trip, this intensity-dependent phase turns a simple optical resonator into a genuine chaotic dynamical system. In 1980, Ikeda, Daido and Akimoto (with related work by Hammel, Jones and Moloney) reduced the continuous round-trip dynamics to the discrete two-dimensional real map used here: t_n = k − p/(1 + x_n² + y_n²), x_{n+1} = A + B(x_n cos t_n − y_n sin t_n), y_{n+1} = B(x_n sin t_n + y_n cos t_n). Physically, A is the amplitude of the external input field driving the cavity, B is the damping or dissipation per round trip (B < 1 means the cavity loses energy every pass), k is a fixed linear phase detuning, and p sets the strength of the nonlinear phase shift that grows with the local intensity x_n² + y_n².
For the classic parameters A = 1, B = 0.9, k = 0.4, p = 6, the map settles onto a beautiful spiral strange attractor: points near the origin get pushed outward and twisted by an intensity-dependent rotation, then pulled back in by the dissipation, tracing an infinitely detailed spiral that never closes on itself. This structure is directly relevant to nonlinear optics and laser physics, where it explains optical bistability, self-pulsing, and route-to-chaos phenomena observed in real ring-cavity and laser experiments. The simulation below seeds a scatter of parallel initial conditions near the origin and iterates the map forward for all of them simultaneously, accumulating every point on the canvas frame after frame so the spiral fills in progressively. The B (dissipation) and p (phase) sliders let you explore the map continuously between simple fixed-point behaviour and full chaos.
What is the Ikeda map and where does it come from?
The Ikeda map is a discrete-time model of light bouncing around inside a nonlinear optical ring cavity. Kensuke Ikeda introduced the underlying physics in 1979 to describe optical bistability in a cavity containing a Kerr-type nonlinear medium; Ikeda, Daido and Akimoto then derived the simplified two-dimensional real map used in most simulations, including this one, in 1980. It is one of the earliest and most influential examples of chaos discovered directly in an optics context rather than in fluid dynamics or population biology.
What do the parameters A, B, k, and p represent physically?
A is the amplitude of the external laser field injected into the cavity. B is the round-trip dissipation — the fraction of field amplitude that survives one pass through the cavity and its output mirror, so B < 1 always removes energy. k is a fixed linear phase shift accumulated per round trip. p sets the strength of the nonlinear (Kerr) phase shift, which depends on the local intensity x_n² + y_n²: brighter points twist through a different angle than dim ones, and this intensity-dependent twisting is what makes the map genuinely nonlinear and eventually chaotic.
Why does the attractor form a spiral?
Every iteration rotates a point by an angle t_n that depends on its distance from the origin, then rescales it by the dissipation factor B and re-centres it near A. Points far from the origin experience a larger nonlinear phase shift than points close to it, so the rotation angle varies continuously with radius. Repeating this radius-dependent rotation thousands of times winds nearby points into ever-tighter spiral arms, producing the characteristic whorled strange attractor instead of a simple closed loop.
Unlike area-preserving chaotic maps such as the standard map or the kicked rotor, the Ikeda map is dissipative whenever B < 1: each iteration shrinks areas in phase space by a factor of B² on average. This dissipation is essential — it is what allows the infinite spiral of iterations to collapse onto a finite-area strange attractor rather than filling the whole plane. At B = 1 the map would conserve area and behave very differently; as B is lowered further, the attractor becomes progressively more contracted and, eventually, dynamics simplify toward periodic or fixed-point behaviour.
Reducing p weakens the nonlinear phase term p/(1 + x² + y²), so the rotation angle t_n varies much less across the attractor and the twisting that builds the spiral is largely switched off. In this regime, most or all of the seeded initial conditions converge onto a single fixed point or a short periodic cycle rather than a diffuse spiral — you can see this directly with the "Low-p regime" preset, where the point cloud collapses to one or two tight clusters instead of spreading into an attractor.
As p approaches and exceeds the classic value of 6 (with B near 0.9), the nonlinear phase shift becomes strong enough that nearby trajectories are stretched and folded repeatedly — the two ingredients required for chaos. The result is the fully developed spiral strange attractor: an infinite, self-similar structure with a fractal cross-section, densely filled in by the simulation's parallel initial conditions after enough iterations.
Yes. The physics behind the map — intensity-dependent refractive index (the optical Kerr effect) combined with cavity feedback — is exactly the mechanism behind optical bistability, self-pulsing instabilities, and chaotic output intensity observed in real nonlinear ring-cavity and laser experiments since the late 1970s and 1980s. The Ikeda map became a standard theoretical benchmark precisely because its predictions could be, and were, tested against tabletop optics experiments.
In the chaotic regime, yes — two initial points that start extremely close together end up on wildly different parts of the spiral after enough iterations, the hallmark of deterministic chaos. This simulation makes that visible indirectly: seeding hundreds of nearby initial conditions and iterating them all forward causes the initially tight scatter to spread out and trace the full attractor rather than staying clustered together.
A single trajectory iterated for many steps eventually traces out the same attractor, since the map is ergodic on it, but that takes many iterations to fill in visually. Seeding a scatter of several hundred to a couple of thousand starting points and iterating them all forward in parallel for a few hundred steps each reveals the overall shape of the attractor — or the location of a fixed point — much more quickly, because every point contributes its own partial trace simultaneously.