Quantum Convergence and Divergence with Bifucation (QCAD)

QCAD Region Estimator – User Guide

Overview

QCAD (Quantum Computational Analysis & Dynamics) is a visual tool that simulates dynamic regions by evaluating complex iterative expressions. It supports exploration of convergence, divergence, and bifurcation behaviors.

🔧 Inputs & Controls

Control	Description
P(x)	Parametric function in JavaScript syntax (e.g., x => 0.5 * x) that defines exponential weighting.
T(x)	Transform function (e.g., x => Math.sin(x)) applied at each point x.
Mode	Determines system behavior: • Convergence: damped system• Divergence: growth system• Bifurcation: chaotic/split behavior
X Min / X Max	Range of x values to analyze.
Depth (L)	Number of summation layers, controlling how deep the recurrence goes.
Steps	Resolution of the simulation — more steps give a smoother curve but take longer to compute.

📈 Computation Logic

The core expression computed for each x is:

Σ (k = 1 to L) exp(±k * P(x)) · T(x)

• In convergence mode, the exponent is negative: exp(-k * P(x))

• In divergence mode, the exponent is positive: exp(+k * P(x))

• In bifurcation mode, P(x) is modified: P(x) sin(k x)

▶️ Using the App

1. Adjust functions and settings as desired using the left panel.

2. Click “Run QCAD” to compute results.

3. View Σ result in the results box.

4. See the graph visualized dynamically on the canvas.

🧪 Example Presets

• Wave AnalysisP(x): x => 0.5 * xT(x): x => Math.sin(x)Mode: convergence

• Exponential GrowthP(x): x => 0.1 * xT(x): x => Math.exp(x / 10)Mode: divergence

• Chaos TheoryP(x): x => 0.3 xT(x): x => Math.sin(x) Math.cos(x / 2)Mode: bifurcation

💡 Tips

• Use Math. prefix for functions: Math.sin, Math.pow, etc.

• Large depth or steps values may take longer to compute.

• Try different P(x) and T(x) combinations to explore system behaviors.

 QCAD Region Estimator

Quantum Computational Analysis & Dynamics

Author: Travis Raymond-Charlie StoneModel Type: Quantum-inspired summation analysis toolCore Equation:

∑k=1Le±k⋅P(x)⋅T(x)\sum_{k=1}^{L} e^{\pm k \cdot P(x)} \cdot T(x)k=1∑L​e±k⋅P(x)⋅T(x)

📌 Executive Summary

QCAD is a computational modeling tool designed to explore mathematical systems using exponential summation behavior modulated by two key functions:

• P(x): a parameter mapping function

• T(x): a transformation function over the domain

The tool supports three analytical regimes:

1. Convergence (damped decay behavior)

2. Divergence (explosive growth behavior)

3. Bifurcation (chaotic transitions)

By evaluating the summation of exponential functions multiplied by a transformation, QCAD offers insights into dynamic system behavior across time, space, or abstract parameter domains.

🎯 Use Cases

1. Signal Dynamics and Frequency Tuning

• Objective: Analyze how an oscillating signal decays or amplifies under variable exponential weightings.

• Approach:

  • P(x) = x => 0.1 * x

  • T(x) = x => Math.sin(x)

  • Mode: convergence

• Outcome: See how frequency behavior dampens over iterations — useful in DSP simulations and waveform control.

2. Quantum Behavior Simulation (Bifurcation Points)

• Objective: Explore systems with multiple stable states or potential collapse into discrete paths.

• Approach:

  • P(x) = x => 0.3 * x

  • T(x) = x => Math.sin(x) * Math.cos(x/2)

  • Mode: bifurcation

• Outcome: Reveals phase transitions and chaotic oscillations akin to bifurcation diagrams in quantum mechanics and complex systems.

3. Growth & Threshold Modeling

• Objective: Simulate systems where recursive amplification results in rapid state change, e.g., viral growth or market bubbles.

• Approach:

  • P(x) = x => x / 10

  • T(x) = x => Math.exp(x / 10)

  • Mode: divergence

• Outcome: Produces curves that mimic exponential growth phases and saturation points.

4. Physics-Inspired Field Mapping

• Objective: Map and visualize energy potential fields or gradient convergence across a spatial domain.

• Approach:

  • P(x) = x => 0.05 * x^2

  • T(x) = x => Math.sin(x) + Math.cos(2 * x)

  • Mode: convergence

• Outcome: Field potentials emerge showing high/low energy regions or gravitational effects.

5. AI Behavior Feedback Modulation

• Objective: Design an AGI training feedback loop modulated by iterative reward transformations.

• Approach:

  • Use P(x) to represent model state weight.

  • Use T(x) to represent policy feedback function.

• Outcome: Dynamically adjusts learning behavior under reinforcement-type feedback.

🔬 Research & Education Applications

• Mathematical Research: Investigate series convergence and divergence behaviors under various non-linear mappings.

• Education: Demonstrates the effects of functional weight and recursion in a visual, interactive way.

• Model Verification: Use to test how new model functions behave under iterative pressure and transformation.

📐 Technical Parameters

Parameter	Description
xMin / xMax	Domain range of analysis
Depth (L)	Iterative summation depth
Steps	Resolution (discretization of x-axis)
P(x)	Determines exponential weighting
T(x)	Functional transformation applied to each x

🧰 Future Extensions

• Add symbolic interpretation or auto-simplification of functions.

• Export results as CSV, PNG, or LaTeX report.

• Introduce complex-valued or vector-valued functions.

• Allow user-defined recursion models (e.g., with memory or stochastic terms).

💡 Conclusion

QCAD is a lightweight yet powerful exploration platform for dynamic mathematical phenomena. With its ability to simulate convergence, divergence, and bifurcation in real time, it is ideal for theorists, developers, educators, and modelers investigating system response under recursive function pressure.