{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-discrete-dynamical-system",
  "semantic_family_id": "sf-discrete-dynamical-system",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "離散時間動力系統"
  },
  "summary": {
    "zh-Hant": "離散時間動力系統以狀態空間、自映射與自然數時間描述逐步演化；有限軌道核對不自動給出一般穩定性。"
  },
  "statement": {
    "zh-Hant": "離散時間動力系統由狀態空間 X 與自映射 T:X→X 組成。給定初始狀態 x_0∈X，以 x_(n+1)=T(x_n) 定義其前向軌道；本輪只在完整明示的有限 X 上精確迭代。"
  },
  "formula": {
    "tex": "x_{n+1}=T(x_n)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><msub><mi>x</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><mrow><mi>T</mi><mo>(</mo><msub><mi>x</mi><mi>n</mi></msub><mo>)</mo></mrow></mrow></math>",
    "semantic_ast": {
      "type": "equation",
      "lhs": {
        "type": "subscript",
        "base": {
          "type": "symbol",
          "name": "x",
          "glyph": "x"
        },
        "subscript": {
          "type": "addition",
          "operands": [
            {
              "type": "symbol",
              "name": "n",
              "glyph": "n"
            },
            {
              "type": "number",
              "value": 1
            }
          ]
        }
      },
      "rhs": {
        "type": "function_call",
        "callee": {
          "type": "symbol",
          "name": "T",
          "glyph": "T"
        },
        "arguments": [
          {
            "type": "subscript",
            "base": {
              "type": "symbol",
              "name": "x",
              "glyph": "x"
            },
            "subscript": {
              "type": "symbol",
              "name": "n",
              "glyph": "n"
            }
          }
        ]
      }
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "59384210b28cbd4e7b17f68884cd0b1adb0686fcfa5d20764dfcc1fa8c06871b"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "時間索引為包含 0 的自然數，T 是 X 上的總自映射。"
    },
    {
      "id": "a2",
      "zh-Hant": "題庫採有限剩餘類狀態與仿射更新；有限前綴不外推連續時間、漸近穩定或混沌。"
    }
  ],
  "symbols": [
    {
      "token": "x",
      "role": "state_sequence",
      "role_zh": "狀態軌道",
      "scope": "representative_formula"
    },
    {
      "token": "n",
      "role": "discrete_time",
      "role_zh": "離散時間索引",
      "scope": "representative_formula"
    },
    {
      "token": "T",
      "role": "state_update",
      "role_zh": "狀態更新自映射",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-set",
      "title_zh": "集合",
      "reason_zh": "狀態空間以集合表示。"
    },
    {
      "id": "mko-function-mapping",
      "title_zh": "函數映射",
      "reason_zh": "更新規則必須是狀態空間的總自映射。"
    },
    {
      "id": "mko-natural-number",
      "title_zh": "自然數",
      "reason_zh": "離散時間由自然數索引。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "初始狀態與更新映射共同決定軌道；只給狀態集合而沒有 T，不足以決定演化。",
      "固定點滿足 T(x)=x；正向不變子集 A 滿足 T(A)⊆A。有限狀態時可完整枚舉這些局部性質。",
      "有限軌道或有限狀態循環不等於連續時間模型，也不證一般參數族的穩定性、週期分類或混沌性。"
    ],
    "common_misconception_zh": "把幾個迭代值當成一般長期行為證明，或把離散更新直接解讀為連續微分方程。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "本輪只核對明示有限自映射的更新、軌道前綴、固定點與不變子集；不宣稱一般動力系統定理。",
    "evidence_refs": [],
    "producers": [
      {
        "id": "lean-mathlib",
        "adapter": "formal_system",
        "config_path": "formal/lean/lakefile.toml",
        "status": "configured"
      }
    ]
  },
  "formalization": {
    "status": "not_formalized",
    "target_systems": [
      "Lean 4",
      "Mathlib"
    ],
    "next_obligation_zh": "補充軌道、週期點、吸引性與穩定性的分層定義及形式化。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-discrete-dynamical-system.md 整理；2026-10-13 脈衝 R01（使用者手動接續 2026-10-20 工作序列；實際日期仍為 2026-10-13），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-13"
  }
}