{
  "analytic": "fusion-plasma-control",
  "kind": "declared_analytical_or_constructed_example",
  "experimental_model_result": false,
  "panels": [
    {
      "kind": "line",
      "x": [
        0,
        1,
        2,
        3,
        4,
        5,
        6,
        7,
        8,
        9,
        10,
        11,
        12,
        13,
        14,
        15,
        16,
        17,
        18,
        19,
        20,
        21,
        22,
        23,
        24,
        25,
        26,
        27,
        28,
        29,
        30,
        31,
        32,
        33,
        34,
        35,
        36,
        37,
        38,
        39,
        40,
        41,
        42,
        43,
        44,
        45,
        46,
        47,
        48,
        49,
        50
      ],
      "series": [
        {
          "label": "No delay, a=1.2 k=1.1",
          "y": [
            1.0,
            0.1,
            0.010000000000000002,
            0.0010000000000000002,
            0.00010000000000000002,
            1.0000000000000003e-05,
            1.0000000000000004e-06,
            1.0000000000000004e-07,
            1.0000000000000005e-08,
            1.0000000000000005e-09,
            1.0000000000000006e-10,
            1.0000000000000006e-11,
            1.0000000000000006e-12,
            1.0000000000000007e-13,
            1.0000000000000008e-14,
            1.0000000000000009e-15,
            1.0000000000000008e-16,
            1.000000000000001e-17,
            1.000000000000001e-18,
            1.000000000000001e-19,
            1.0000000000000011e-20,
            1.0000000000000012e-21,
            1.0000000000000012e-22,
            1.0000000000000013e-23,
            1.0000000000000014e-24,
            1.0000000000000014e-25,
            1.0000000000000015e-26,
            1.0000000000000015e-27,
            1.0000000000000015e-28,
            1.0000000000000016e-29,
            1.0000000000000017e-30,
            1.0000000000000016e-31,
            1.0000000000000018e-32,
            1.0000000000000018e-33,
            1.0000000000000019e-34,
            1.0000000000000019e-35,
            1.000000000000002e-36,
            1.0000000000000022e-37,
            1.000000000000002e-38,
            1.0000000000000022e-39,
            1.0000000000000022e-40,
            1.0000000000000023e-41,
            1.0000000000000023e-42,
            1.0000000000000025e-43,
            1.0000000000000024e-44,
            1.0000000000000025e-45,
            1.0000000000000026e-46,
            1.0000000000000026e-47,
            1.0000000000000027e-48,
            1.0000000000000028e-49,
            1.0000000000000027e-50
          ]
        },
        {
          "label": "One-step delay, same gain",
          "y": [
            1.0,
            1.2,
            0.33999999999999986,
            -0.9120000000000003,
            -1.4684000000000001,
            -0.7588799999999998,
            0.7045840000000005,
            1.6802688000000003,
            1.2412801599999996,
            -0.3587594880000011,
            -1.795919561600001,
            -1.7604680371199997,
            -0.13705012678399808,
            1.772054688691202,
            2.2772207658918404,
            0.783404761509886,
            -1.5648571286691615,
            -2.7395737920638688,
            -1.5661457089405644,
            1.1341563205415786,
            3.083747864484515,
            2.4529254847856814,
            -0.44861206919014895,
            -3.2365525162924285,
            -3.3903897434417503,
            -0.5082599242084282,
            3.119516808735812,
            4.302506087112246,
            1.731538814925301,
            -2.654910117913109,
            -5.090584837913562,
            -3.188300675791855,
            1.7736825107546936,
            5.6355497562766725,
            4.811608945701844,
            -0.425173997062128,
            -5.802978636746582,
            -6.4958829673275575,
            -1.4117830603718282,
            5.45133159161412,
            8.094559276345954,
            3.7170063808396128,
            -4.443607546973015,
            -9.421036075291193,
            -6.417274988679114,
            2.6624096964053763,
            10.253894123233477,
            9.376022281834258,
            -0.028056797355716867,
            -10.347292666844544,
            -12.385888723122164
          ]
        }
      ],
      "xlabel": "Discrete step",
      "ylabel": "Normalized state",
      "logx": false,
      "logy": false
    },
    {
      "kind": "line",
      "x": [
        0.05,
        0.06174496644295302,
        0.07348993288590605,
        0.08523489932885907,
        0.09697986577181208,
        0.1087248322147651,
        0.12046979865771813,
        0.13221476510067115,
        0.14395973154362418,
        0.15570469798657718,
        0.1674496644295302,
        0.17919463087248322,
        0.19093959731543625,
        0.20268456375838928,
        0.2144295302013423,
        0.2261744966442953,
        0.23791946308724832,
        0.24966442953020135,
        0.2614093959731544,
        0.2731543624161074,
        0.2848993288590604,
        0.2966442953020134,
        0.30838926174496645,
        0.3201342281879195,
        0.3318791946308725,
        0.3436241610738255,
        0.3553691275167785,
        0.36711409395973155,
        0.3788590604026846,
        0.3906040268456376,
        0.4023489932885906,
        0.4140939597315436,
        0.42583892617449665,
        0.4375838926174497,
        0.4493288590604027,
        0.4610738255033557,
        0.4728187919463087,
        0.48456375838926175,
        0.4963087248322148,
        0.5080536912751679,
        0.5197986577181208,
        0.5315436241610738,
        0.5432885906040269,
        0.5550335570469799,
        0.566778523489933,
        0.5785234899328859,
        0.590268456375839,
        0.602013422818792,
        0.6137583892617451,
        0.6255033557046981,
        0.637248322147651,
        0.6489932885906041,
        0.6607382550335571,
        0.6724832214765102,
        0.6842281879194632,
        0.6959731543624161,
        0.7077181208053692,
        0.7194630872483222,
        0.7312080536912753,
        0.7429530201342283,
        0.7546979865771812,
        0.7664429530201343,
        0.7781879194630873,
        0.7899328859060404,
        0.8016778523489934,
        0.8134228187919463,
        0.8251677852348994,
        0.8369127516778524,
        0.8486577181208055,
        0.8604026845637585,
        0.8721476510067114,
        0.8838926174496645,
        0.8956375838926175,
        0.9073825503355706,
        0.9191275167785236,
        0.9308724832214765,
        0.9426174496644296,
        0.9543624161073826,
        0.9661073825503357,
        0.9778523489932887,
        0.9895973154362416,
        1.0013422818791946,
        1.0130872483221476,
        1.0248322147651008,
        1.0365771812080538,
        1.0483221476510067,
        1.0600671140939597,
        1.071812080536913,
        1.0835570469798659,
        1.0953020134228189,
        1.1070469798657718,
        1.1187919463087248,
        1.130536912751678,
        1.142281879194631,
        1.154026845637584,
        1.165771812080537,
        1.1775167785234901,
        1.189261744966443,
        1.201006711409396,
        1.212751677852349,
        1.224496644295302,
        1.2362416107382552,
        1.2479865771812082,
        1.2597315436241612,
        1.2714765100671142,
        1.2832214765100671,
        1.2949664429530203,
        1.3067114093959733,
        1.3184563758389263,
        1.3302013422818793,
        1.3419463087248322,
        1.3536912751677854,
        1.3654362416107384,
        1.3771812080536914,
        1.3889261744966444,
        1.4006711409395973,
        1.4124161073825505,
        1.4241610738255035,
        1.4359060402684565,
        1.4476510067114094,
        1.4593959731543624,
        1.4711409395973156,
        1.4828859060402686,
        1.4946308724832216,
        1.5063758389261745,
        1.5181208053691275,
        1.5298657718120807,
        1.5416107382550337,
        1.5533557046979867,
        1.5651006711409396,
        1.5768456375838926,
        1.5885906040268458,
        1.6003355704697988,
        1.6120805369127518,
        1.6238255033557047,
        1.6355704697986577,
        1.647315436241611,
        1.659060402684564,
        1.6708053691275169,
        1.6825503355704698,
        1.6942953020134228,
        1.706040268456376,
        1.717785234899329,
        1.729530201342282,
        1.741275167785235,
        1.753020134228188,
        1.7647651006711411,
        1.776510067114094,
        1.788255033557047,
        1.8
      ],
      "series": [
        {
          "label": "No-delay spectral radius",
          "y": [
            1.15,
            1.138255033557047,
            1.126510067114094,
            1.114765100671141,
            1.1030201342281878,
            1.0912751677852348,
            1.0795302013422818,
            1.0677852348993289,
            1.0560402684563757,
            1.0442953020134227,
            1.0325503355704697,
            1.0208053691275167,
            1.0090604026845638,
            0.9973154362416107,
            0.9855704697986576,
            0.9738255033557046,
            0.9620805369127516,
            0.9503355704697987,
            0.9385906040268456,
            0.9268456375838925,
            0.9151006711409395,
            0.9033557046979865,
            0.8916107382550336,
            0.8798657718120805,
            0.8681208053691274,
            0.8563758389261744,
            0.8446308724832214,
            0.8328859060402685,
            0.8211409395973154,
            0.8093959731543623,
            0.7976510067114093,
            0.7859060402684563,
            0.7741610738255034,
            0.7624161073825503,
            0.7506711409395972,
            0.7389261744966442,
            0.7271812080536912,
            0.7154362416107383,
            0.7036912751677852,
            0.6919463087248321,
            0.6802013422818791,
            0.6684563758389261,
            0.6567114093959731,
            0.6449664429530201,
            0.633221476510067,
            0.621476510067114,
            0.6097315436241609,
            0.597986577181208,
            0.5862416107382549,
            0.5744966442953019,
            0.5627516778523489,
            0.5510067114093958,
            0.5392617449664429,
            0.5275167785234898,
            0.5157718120805368,
            0.5040268456375838,
            0.49228187919463073,
            0.48053691275167776,
            0.46879194630872467,
            0.4570469798657717,
            0.4453020134228187,
            0.43355704697986563,
            0.42181208053691266,
            0.4100671140939596,
            0.3983221476510066,
            0.3865771812080536,
            0.37483221476510054,
            0.36308724832214756,
            0.3513422818791945,
            0.3395973154362415,
            0.3278523489932885,
            0.31610738255033544,
            0.30436241610738246,
            0.2926174496644294,
            0.2808724832214764,
            0.2691275167785234,
            0.25738255033557034,
            0.24563758389261736,
            0.23389261744966428,
            0.2221476510067113,
            0.21040268456375832,
            0.19865771812080535,
            0.18691275167785237,
            0.17516778523489918,
            0.1634228187919462,
            0.15167785234899323,
            0.13993288590604025,
            0.12818791946308705,
            0.11644295302013408,
            0.1046979865771811,
            0.09295302013422813,
            0.08120805369127515,
            0.06946308724832195,
            0.05771812080536898,
            0.045973154362416,
            0.03422818791946303,
            0.02248322147650983,
            0.010738255033556854,
            0.0010067114093961216,
            0.012751677852349097,
            0.024496644295302072,
            0.03624161073825527,
            0.047986577181208245,
            0.05973154362416122,
            0.0714765100671142,
            0.08322147651006717,
            0.09496644295302037,
            0.10671140939597334,
            0.11845637583892632,
            0.1302013422818793,
            0.14194630872483227,
            0.15369127516778547,
            0.16543624161073844,
            0.17718120805369142,
            0.1889261744966444,
            0.20067114093959737,
            0.21241610738255057,
            0.22416107382550354,
            0.23590604026845652,
            0.2476510067114095,
            0.25939597315436247,
            0.27114093959731567,
            0.28288590604026864,
            0.2946308724832216,
            0.3063758389261746,
            0.31812080536912757,
            0.32986577181208077,
            0.34161073825503374,
            0.3533557046979867,
            0.3651006711409397,
            0.37684563758389267,
            0.38859060402684586,
            0.40033557046979884,
            0.4120805369127518,
            0.4238255033557048,
            0.43557046979865777,
            0.44731543624161096,
            0.45906040268456394,
            0.4708053691275169,
            0.4825503355704699,
            0.49429530201342287,
            0.5060402684563761,
            0.517785234899329,
            0.529530201342282,
            0.541275167785235,
            0.553020134228188,
            0.5647651006711412,
            0.5765100671140941,
            0.5882550335570471,
            0.6000000000000001
          ]
        },
        {
          "label": "One-delay spectral radius",
          "y": [
            1.1567764362830022,
            1.1461273052659489,
            1.1352663515616257,
            1.1241804085151799,
            1.1128548861307532,
            1.1012735458661616,
            1.0894182274315924,
            1.0772685144646867,
            1.064801321487338,
            1.0519903782310225,
            1.0388055783265178,
            1.025212146025389,
            1.0111695546663977,
            0.9966301000196666,
            0.9815369835267056,
            0.9658216824570472,
            0.9494002531664103,
            0.9321679853173671,
            0.9139914075684964,
            0.8946958391017636,
            0.8740450166321941,
            0.8517055913125224,
            0.8271799688683699,
            0.7996641475380106,
            0.7676925918731281,
            0.7279681168345243,
            0.6680505142024766,
            0.6058994091099047,
            0.6155152803973958,
            0.6249832212512889,
            0.6343098558974081,
            0.6435013284613665,
            0.6525633503151219,
            0.6615012415842086,
            0.6703199676724564,
            0.679024171516269,
            0.6876182021633143,
            0.6961061401749461,
            0.7044918202734612,
            0.7127788515908478,
            0.720970635822376,
            0.7290703835440538,
            0.7370811289159606,
            0.7450057429624151,
            0.7528469455938126,
            0.7606073165128546,
            0.7682893051291545,
            0.7758952395902374,
            0.7834273350233222,
            0.7908877010705745,
            0.7982783487904774,
            0.805601196989307,
            0.8128580780391846,
            0.8200507432327041,
            0.8271808677184592,
            0.8342500550568853,
            0.84125984143151,
            0.8482116995469481,
            0.8551070422416571,
            0.8619472258405548,
            0.8687335532700351,
            0.8754672769556461,
            0.8821496015206759,
            0.8887816863021201,
            0.8953646476989102,
            0.9018995613658685,
            0.9083874642656071,
            0.9148293565894419,
            0.9212262035574136,
            0.9275789371065724,
            0.9338884574758975,
            0.9401556346954818,
            0.9463813099869509,
            0.9525662970815051,
            0.9587113834614271,
            0.9648173315304179,
            0.9708848797176881,
            0.9769147435203251,
            0.9829076164881089,
            0.9888641711546075,
            0.9947850599180919,
            1.0006709158755414,
            1.0065223536127488,
            1.0123399699533258,
            1.018124344669183,
            1.0238760411548884,
            1.0295956070681145,
            1.035283574938245,
            1.0409404627450438,
            1.046566774469178,
            1.0521630006162408,
            1.0577296187158252,
            1.0632670937970752,
            1.068775878842066,
            1.0742564152182588,
            1.0797091330911937,
            1.0851344518185246,
            1.0905327803264067,
            1.095904517469198,
            1.1012500523733697,
            1.1065697647664616,
            1.1118640252918768,
            1.1171331958102437,
            1.122377629688048,
            1.127597672074182,
            1.1327936601650221,
            1.137965923458616,
            1.1431147839985156,
            1.148240556607772,
            1.1533435491135673,
            1.1584240625629427,
            1.1634823914300485,
            1.1685188238153197,
            1.1735336416369542,
            1.1785271208150638,
            1.1834995314488288,
            1.1884511379869813,
            1.1933821993919231,
            1.198292969297766,
            1.203183696162564,
            1.2080546234150018,
            1.2129059895957786,
            1.217738028493924,
            1.2225509692782637,
            1.2273450366242469,
            1.2321204508363328,
            1.2368774279661185,
            1.2416161799264032,
            1.2463369146013397,
            1.2510398359528523,
            1.2557251441234631,
            1.26039303553568,
            1.265043702988082,
            1.2696773357482414,
            1.2742941196425983,
            1.2788942371434229,
            1.2834778674529654,
            1.2880451865849132,
            1.2925963674432621,
            1.2971315798986893,
            1.301650990862536,
            1.3061547643584877,
            1.3106430615920297,
            1.3151160410177811,
            1.3195738584047634,
            1.3240166668996989,
            1.3284446170883981,
            1.332857857055318,
            1.3372565324413437,
            1.3416407864998738
          ]
        },
        {
          "label": "Unit-circle boundary",
          "y": [
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0
          ]
        }
      ],
      "xlabel": "Feedback gain k",
      "ylabel": "Spectral radius",
      "logx": false,
      "logy": false
    }
  ],
  "title": "A single delay changes the feedback stability boundary",
  "assumptions": "Scalar normalized system a=1.2; no reactor or actuator parameters. Delayed initial x[-1]=0,x[0]=1. No real device control.",
  "worked": "无延迟乘子0.1；一拍延迟特征方程z^2-1.2z+1.1的根模sqrt(1.1)≈1.04881，超过1。",
  "checks": {
    "delayed_root_radius": 1.0488088481701516
  },
  "generator_source_sha256": {
    "ops/datacenter/editorial_guides/science_figures.py": "01890ea429a67a880650c9b0602b1b66ed6ddeaa9704b7224897857a5f4803d7",
    "ops/datacenter/editorial_guides/deep_figures.py": "158969d9b1f5a4bd13906cf78dd71affc2692df9194593918059d861a6f59c85"
  },
  "source_ids": [
    "fusion-control",
    "fusion-tearing"
  ],
  "formula_source": "Article equations and this versioned script",
  "diagram_nodes": [
    "任务与操作包络",
    "仿真和模型误差",
    "冻结策略 / 时延测试",
    "实际执行与约束",
    "保护接管和放电日志",
    "工况留出与迁移"
  ]
}