# Signatures of Quantum Mechanics in Chaotic Systems

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## Abstract

**:**

## 1. Introduction

## 2. Background on Cupolets

#### 2.1. Generating Cupolets

#### 2.2. Cupolet Properties and Stability

## 3. Chaotic Entanglement

#### 3.1. Chaotic Entanglement through Periodic Orbits

#### 3.2. Chaotic Entanglement as an Analog of Quantum Entanglement

#### 3.3. Pure Chaotic Entanglement

## 4. Main Discussion: Parallels between Chaotic and Quantum Systems

#### 4.1. Hilbert Space Considerations

#### 4.2. Functional Representation of Cupolets

#### 4.3. Superposition of States

#### 4.4. Wave Function Collapse

#### 4.5. Natural Chaotic Entanglement

#### 4.6. Measurement Problem

“… quantum theory encounters questions that need to be answered, one of the most important of which is what it means to say, and how it can be ensured that the individual systems on which the repeated measurements are to be made are all in the ‘same’ state immediately before the measurement. This crucial problem of state preparation is closely related to the idea of a reduction of the state vector.”[51]

#### 4.7. Entropy

#### 4.8. Differences with Quantum Entanglement

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Cupolets of various periods belonging to the double scroll system. The control sequences that must be periodically applied in order to stabilize these periodic orbits are (

**a**) ‘00’, (

**b**) ‘11’, (

**c**) ‘00001’, and (

**d**) ‘001’.

**Figure 3.**Entangled cupolets (

**a**) $\mathbf{C}000011111$ (period 9) and (

**b**) $\mathbf{C}011101111$ (period 18). The visitation sequences of these cupolets are $\mathbf{V}000111111$ and $\mathbf{V}000000000111111111$, respectively.

**Figure 4.**(Color online) Schematic illustration of chaotic entanglement: in (

**a**) a control sequence is externally applied to Chaotic System I, via the indicated (yellow) control pump. System I subsequently stabilizes in (

**b**) onto cupolet $\mathbf{C}000011111$ according to the control method described in Section 2. This cupolet then evolves around the attractor to generate its visitation sequence, $\mathbf{V}000111111$. In (

**c**), an exchange function accepts this visitation sequence as an input and the outputted emitted sequence, $\mathbf{E}011101111$, is taken as a control sequence and used to control a second chaotic system, System II. System II subsequently stabilizes uniquely onto cupolet $\mathbf{C}011101111$ whose visitation sequence, $\mathbf{V}000000000111111111$, is then passed to the same exchange function in (

**d**). The resulting emitted sequence, $\mathbf{E}000011111$, is applied as control instructions to $\mathbf{C}000011111$ of System I. Note that each emitted sequence exactly matches each corresponding cupolet’s control sequence, and so the external control pumps seen in (

**a**–

**c**) are unnecessary and can be removed. Systems I and II are now dynamically engaged in a state of perpetual mutual-stabilization between their respective cupolets and are thus considered chaotically entangled. This entanglement is summarized in Table 1.

**Figure 5.**(Color online) Comparing the numerical solutions (solid black curves) of cupolets $\mathbf{C}00$ (period 1) and $\mathbf{C}00001$ (period 5) against their full and truncated functional representations (dashed red curves). In (

**a**,

**d**), the comparison is shown with the full version of the functional form (c.f., Equation (4)) for $\mathbf{C}00$ and $\mathbf{C}00001$, respectively. In (

**b**,

**e**), the comparison is shown with the truncated version using $Q=11$-many coefficients for $\mathbf{C}00$ and $Q=17$ for $\mathbf{C}00001$ (c.f., Equation (6)). The comparison between the ${v}_{{C}_{1}}$-component of the numerical solution and the corresponding truncated functional form is shown in (

**c**) for $\mathbf{C}00$ and in (

**f**) for $\mathbf{C}00001$. Cupolets are well-represented in both numerical and functional form, which is why the pairs of curves shown in each of these graphs are all effectively superimposed on each other. The simulated time periods of these cupolets are $T\approx 3.46558$ for $\mathbf{C}00$ and $T\approx 19.27239$ for $\mathbf{C}00001$.

**Table 1.**The following table summarizes the chaotic entanglement induced between two interacting cupolets, $\mathbf{C}000011111$ (of Chaotic System I) and $\mathbf{C}011101111$ (of Chaotic System II). The orbits of these cupolets are depicted in Figure 3, while the generation of the entanglement via a ‘preponderance’ exchange function is illustrated in Figure 4. Notice that the control sequence required to sustain the stability of cupolet $\mathbf{C}000011111$ is contributed by cupolet $\mathbf{C}011101111$ via this cupolet’s emitted sequence, $\mathbf{E}000011111$. Similarly, the stability of $\mathbf{C}011101111$ is maintained by the repeated application of emitted sequence $\mathbf{E}011101111$, which is generated by $\mathbf{C}000011111$ via the same exchange function. The font colors in this table are intended to accentuate the correspondence between the cupolets’ control sequences and their emitted sequences. Details of the generation of this entanglement are found in the text.

Cupolet | Visitation Sequence | Emitted Sequence | |
---|---|---|---|

Chaotic System I | $\mathbf{C}000011111$ | $\mathbf{V}000111111$ | $\mathbf{E}011101111$ |

Chaotic System II | $\mathbf{C}011101111$ | $\mathbf{V}000000000111111111$ | $\mathbf{E}000011111$ |

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Short, K.M.; Morena, M.A.
Signatures of Quantum Mechanics in Chaotic Systems. *Entropy* **2019**, *21*, 618.
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**AMA Style**

Short KM, Morena MA.
Signatures of Quantum Mechanics in Chaotic Systems. *Entropy*. 2019; 21(6):618.
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**Chicago/Turabian Style**

Short, Kevin M., and Matthew A. Morena.
2019. "Signatures of Quantum Mechanics in Chaotic Systems" *Entropy* 21, no. 6: 618.
https://doi.org/10.3390/e21060618