Resonance Curves of Multi-dimensional Chaotic Systems

Alfred Hubler, Glen Foster, Vadas Gintautas, Karin DahmenPhysics, University of Illinois at Urbana-Campaign

- Resonant Forcing: Aperiodic function that induces the largest response o Conserved quantity = (Resonant forcing) ∙ (Displacement of nearby trajectories) = Power transfer

- Resonance Curve: Response of a dynamical system to the resonant forcing functions of a set of models

o Response is largest if the model system = dynamical system.o Lyapunov exponent large => large signal-to-noise ratio

Are mixed reality states resonances? How do SimCity, Second Life, and other even more realistic computer games impact the real world?

Initially funded by an ONR grant

Out-of-body experiences with video feedback

Blanke O et al.Linking OBEs and self processing to mental own body imagery at the temporo-parietal junction. J Neurosci 25:550-55 (2006).

- Subject sees video image of itself with 3D goggles

- Two sticks, one strokes person's chest for two

minutes, second stick moves just under the camera

lenses, as if it were touching the virtual body.

- Synchronous stroking => people reported the sense of being outside their own bodies, looking at themselves from a distance where the camera is located.

- While people were experiencing the illusion, the experimenter pretended to smash the virtual body by waving a hammer just below the cameras. Immediately, the subjects registered a threat response as measured by sensors on their skin. They sweated and their pulses raced.

Real system & similar virtual system & bi-directional instant. coupling = mixed reality

Experimental evidence for mixed reality states in physical systems

Objective: Understand synchronization between virtual and real systems. Approach: - Couple a real dynamical system to its virtual counterpart with an instantaneous bi-direction coupling.

- Measure the amplitudes of both systems and their phase difference, and then detect synchronization.

Experimental evidence for mixed reality states in physical inter-reality systemsResults:

- Experimental evidence for a phase transition from dual reality states to mixed reality states.

- Phase diagram of the inter-reality system is in good agreement with the phase diagram of the simulated inter-reality system.

Phase diagram of the inter-reality system: amplitude of the coupling versus the frequency ratio of the real and the virtual system. The phase boundary between mixed reality states (I) and dual reality states (II). The solid, dashed, and dotted lines indicate the critical points in the experiment, simulation, and analytic theory, respectively.

Resonance Curves of Inter-reality systems

Figure 1. Amplitude X of the real systemversus the frequency ratio for theexperimental system (squares) and forthe numerical system (triangles)

Perfect match between real and virtual system => largest amplitudes

Figure 2. The opposite of the amplitude of the real system versus the frequency ratio and versus the ratio of the third order terms

Mixed reality states in physical systems: Why are they important?

Publication: The paper "Experimental evidence for mixed reality states in an inter-reality system" by Vadas Gintautas and Alfred Hubler, in Phys. Rev. E 75, 057201 (2007),

was selected for the APS tip sheet: http://www.aps.org/about/tipsheets/tip68.cfm

- Virtual systems match their real counter parts with ever-increasing accuracy.

- New hardware for instantaneous bi-directional coupling

- In mixed reality states there is no clear boundary between the real and the virtual system. Mixed reality states can be used to analyze and control real systems with high precision. And then there is the possibility for time travel … by the virtual system.

Photo: A. Hubler and V. Gintautas at the inter-reality system

Resonant forcing of a multi-dimensional chaotic system

System: xn+1=f(xn)+Fn , n=0,1,..,N-1

Maximize final response: R2= (xN – yN)2

where yn+1=f(yn), y0=x0 (unperturbed system)and magnitude of force F2 = Σn Fn

2 = fixed

=> Optimal forcing: (Jn+1 )TFn+1=Fn where yN – xN=-μFN-1

Jn is the Jaccobian of f(xn)

The forcing function Fn which produces the largest final response complements the natural dynamics, i.e.

Fn dn+1 = P = constant

Magnitude of forcing: F2 = ΣFn2 = fixed

Map dynamics: xn+1= f(xn) + Fn

Separation of neighboring trajectories: dn+1=Jndn

Jaccobian of f(xn): Jn

Systems with an energy function: P = power transfer

The growth rate of the magnitude of the optimal forcing function is equal to the opposite of the largest Lyapunov exponent λ1: Fn = F0 exp(- λ1 n)

Resonant forcing of a chaotic logistic map, xn+1=3.61 xn (1-xn)

Fig.: Resonant forcing function and displacement of neighboring trajectories versus time for a chaotic logistic map

Large Lyapunov exponent => large response

R2 = F2 (1 − exp(2λ1N) )/(1 − exp(2λ1))The response R2/F2 to a resonant forcing function versus the largest Lyapunov exponent λ1 of a coupled shift map system, where N = 4, a2 = 0.5, k = .2, and F = 0.0001. (squares =numerical, cont. line= theory). The dashed line is the expectation value response to a random forcing function and the x-labels indicate numerical results.

This figure illustrates that the optimal forcing function is particularly

efficient if one Lyapunov exponent is significantly larger than the others.

Resonance CurvesSet of models:

zn+1=f(zn,am)+Fn(am)

am = model parameterSystem: xn+1=f(xn,a)+Fn(am)Response: R2= (xN – yN)2

Fn(am) = resonant forcing of a model where F2=ΣFn

2=fixed andyn+1=f(yn,a), y0=x0 (unperturbed dynamics)

Resonance curve: normalized response versus am

Deviation from expected response: versus am

Resonance curve of a chaotic Henon map.

Glenn Foster, Alfred W. Hübler, Karin Dahmen, Resonant forcing of multi-dimensional chaotic map dynamics, PRE, 2007

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Resonance Curve of a Chaotic Logistic Map, xn+1 = 3.61 xn (1-xn)+Fn

(circles): The resonance curve for a chaotic logistic map

(squares): deviation D between response and model response

number of time steps is N=4

the noise level is r=0.0005

magnitude of the forcing function is F=0.001.

The dashed line = theoretical result for the maximum of the resonance curve

The maximum of the resonance curve & one of the three roots of D are close to the parameter value of the dynamical system, a=3.61.

Resonance Curves of Chaotic Henon Maps

Figure: The resonance curve of a chaotic Henon map versus:

(a) model parameter am, where bm=b

(b) model parameter bm, where am=a

parameters: a=1.08, b=0.3, N=3, F=0.0001.

Henon map:

x1,n+1= 1 - a x1,n2 + x2,n + F1,n

x2,n+1= b x1,n + F2,n

State: xn = (x1,n,x2,n)

Forcing function: Fn = (F1,n, F2,n)

Parameters: a, b

Resonance Curve of a Chaotic Shift Map Dynamics

The resonance curve (a) and the difference between the model response and the system response versus the model parameter (b) for a chaotic shift map with parameter a=2 (bullets=numerical simulation, continuous lines = theoretical values)

model parameter = the system parameter =>

- resonance curve has a maximum

Coupled shift maps:

x1,n+1= mod(a1 x1,n + k x2,n + F1,n )

x2,n+1= mod(a2 x2,n + k x1,n + F2,n )

Signal-to-noise Ratio of the Resonance Curve

Signal-to-noise ratio R2/Rr2 versus the largest Lyapunov exponent of a

two coupled shift maps, where number of time steps N = 4, map parameter a2 = 0.5, coupling k = .2, forcing magnitude F = noise level r = 0.0001 (continuous line = theoretical value, bullets = expectation values determined from 1000 simulations, Rr=response to noise alone).

This figure illustrates that the signal to noise ratio is particularly large if one Lyapunov exponent is significantly larger than the others.

Application 1: Resonances of quantum mechanical wave functions in nonlinear potentials => Efficient energy transfer and energy storage in nano devices

F. Yamaguchi, K. Kawamura, A. Hubler, Sudden Drop of Dissipation in Field-Coupled Quantum Dot Resistors, Jpn. J. Appl. Phys. 34, L 105-108 (1995) H.Higuraskh, A. Toriumi, f. Yamaguchi, K. Kawamura, A. Hubler, Correlation Tunnel Device, U. S. Patent # 5,679,961 (1997) – the spacing of the energy levels has to be equal and opposite

Application 2: High-resolution system identification with aperiodic resonant waves, such as double diffraction systems

with two different aperiodic sets of scatterers

Schematics of a double diffraction system. Ri

0 = direction of the source scatterers

Rj1 = direction of the target scatterers

D = displacement from the source scatterers to the target scatterers.

Θ is the diffraction angle.

1) Single pulse plane wave is diffracted from a linear quasi-periodic array of scatterers: source Ri

0 .

2) The diffracted wave forms a quasi-periodic pulse train in the far field.

3) A screen prevents the quasi-periodic pulse train from reaching the detector.

4) The pulse train is then diffracted from a linear quasi-periodic array of scatterers: target Rj

1 .

5) After the quasi-periodic pulse train wave is diffracted from the target, it creates an interference pattern on the detector screen.

[J. Xu, A. Hubler, preprint 2007.]

Double diffraction from two similar quasi-periodic sets of scatterers: Time series

A typical time dependence of the intensity of the field I at the detector.

Double diffraction from two similar quasi-periodic sets of scatterers:

Diffraction pattern

The diffraction intensity versus the diffraction angle for a double diffraction system, where both the source scatterers and the target scatterers are a Fibonacci chain.

Number of source scatterers: N0=20

Number target scatterers: N1=10

Angles of projection-cut-method: α0=α1=1

The location of main diffraction peak is close to the theoretical value θ ≈ 1.12. The height of the main diffraction peak is close to the theoretical value max(Imax)=0.01

Double diffraction from two different quasi-periodic sets of scatterers:

Resonance curve

The intensity of the main peak Imax versus the structural parameter of the source α0

(angle in projection-cut method)

Model = System => large peak <-> high resolution system identification

Resonance Curves of Multidimensional Chaotic Systems

We use calculus of variations to determine the forcing function that induces the largest response:=>Conserved quantity: product of resonant forcing and the displacement of nearby trajectories (power transfer). Optimal forcing complements natural dynamics.

Compute the resonant forcing for a set of model systems and determine the response of the dynamical system to each forcing function:=>Response is largest if the model system matches the dynamical system.=>Signal to noise ratio is particularly large if one of the Lyapunov exponents is large. Applications: inter-reality systems, efficient energy storage and energy transfer, high-resolution system identification for aperiodic structures